diff --git a/.gitattributes b/.gitattributes index 97f7fdb8cf8725d3c2309666f74b6596605d9a07..0f09ff7034686be2ed80768e04ec92e2561e1e8f 100644 --- a/.gitattributes +++ b/.gitattributes @@ -243,3 +243,4 @@ VtFOT4oBgHgl3EQf6zSP/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex 9tAyT4oBgHgl3EQfqPjX/content/2301.00541v1.pdf filter=lfs diff=lfs merge=lfs -text x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf filter=lfs diff=lfs merge=lfs -text M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf filter=lfs diff=lfs merge=lfs -text +6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/1NAyT4oBgHgl3EQfofhG/content/tmp_files/2301.00507v1.pdf.txt b/1NAyT4oBgHgl3EQfofhG/content/tmp_files/2301.00507v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fda57111cf0850ac57408b3dc48075367d8a68b0 --- /dev/null +++ b/1NAyT4oBgHgl3EQfofhG/content/tmp_files/2301.00507v1.pdf.txt @@ -0,0 +1,1138 @@ +arXiv:2301.00507v1 [math.DG] 2 Jan 2023 +On Geodesics of Sprays and Projective Completeness +Guojun Yang +Abstract +Geodesics, which play an important role in spray-Finsler geometry, are integral +curves of a spray vector field on a manifold. Some comparison theorems and rigidity +issues are established on the completeness of geodesics of a spray or a Finsler metric. In +this paper, projectively flat sprays with weak Ricci constant (eps. constant curvature) +are classified at the level of geodesics. Further, a geodesic method is introduced to +determine an n-dimensional spray based on a family of curves with 2(n−1) free constant +parameters as geodesics. Finally, it shows that a spray is projectively complete under +certain condition satisfied by the domain of geodesic parameter of all geodesics. +Keywords: Spray, Geodesic, Completeness, Path Space, Finsler Metric +MR(2000) subject classification: +53B40, 53C60 +1 +Introduction +Spray geometry studies the properties of sprays on a manifold, and it is closely related to +Finsler geometry. Every Finsler metric induces a natural spray but there are a lot of sprays +which are not Finsler-metrizable (not be induced by any Finsler metric) ([3, 5, 10]). So +a popular topic is to investigate whether a given spray is metrizable or not, and what’s +more important is to give necessary and sufficient conditions for certain class of sprays to +be metrizable ([2, 11, 12]). It is also important to investigate the properties of some special +classes of sprays, for example, (locally) projectively flat sprays, Berwald sprays, sprays of +scalar (resp. isotropic, constant) curvature, Hamel (resp. Funk) sprays ([2, 6, 11, 12]). +A spray G on a manifold M defines a special vector filed on a conical region C of +T M \ {0}, and it naturally defines its integral curves and the projections of the integral +curves onto the manifold M are called geodesics. Geodesics play an important role in the +studies of comparison theorems and rigidity issues on spray or Finsler manifolds. In [8], Z. +Shen studies two pointwise projectively related Einstein Finsler metrics and determine the +metrics along geodesics. In [10], the present author obtains a comparison theorem on the +Ricci curvatures of a spay and a Finsler metric which are pointwise projectively related and +the corresponding projective factor is estimated. In [1], R. Bryant proves that a geodesically +reversible Finlser metric on S2 with positive constant flag curvature is a Riemann metric. +In [7], C. Robles classifies geodesics of Randers metrics of constant flag curvature. In [4], +L. Huang and X. Mo obtain the relation between the geodesics of two Finsler metrics F +and ˜F, where ˜F is defined by the navigation data (F, V ) with V being a homothetic vector +field of F. In this paper, we study projectively flat sprays with weak Ricci constant, the +construction of sprays from a geodesic method and the projective completeness of sprays. +In [11], it introduces sprays of constant curvature and a spray G of constant curvature +is weakly Ricci constant (the Ricci curvature is constant along any geodesic of G). For +two pointwise projectively related sprays, they have same geodesics as point sets and their +geodesic parameters are closely related by the projective factor. Starting from this fact, we +can determine a projectively flat spray with weak Ricci constant at geodesic level. +1 + +We consider a projectively flat spray manifold (G, M), that is, +Gi = �Gi + Pyi, +(1) +where �G is a locally Minkowski spray on M. We have the following theorem. +Theorem 1.1 If the spray G in (1) is weakly Ricci constant Ric;0 = 0 or of constant +curvature, then along any geodesic x = x(s) of G, P(s) := P(x(s), x′(s)) is given by one of +the following cases: +P(s) = +1 +s + κ, +P(s) = −c · tan(cs + κ), +P(s) = −c(1 − κe2cs) +1 + κe2cs , +(2) +where c, κ are constant. Further, if G is complete, then P(s) is given by +P(s) = −c(1 − κe2cs) +1 + κe2cs . +(3) +In Theorem 1.1, we can further give the relation between the geodesic parameters of G +and �G by (2) (see Proposition 3.1, Corollary 3.3). +The family of geodesics of an n-dimensional spray considered as point sets or paths is +dependent on 2(n − 1) free constant parameters. A path space is a family of curves satis- +fying certain conditions (Definition 4.1). We can freely give many interesting path spaces, +especially in dimension two. Starting from a path space, we can construct its corresponding +spray. +Theorem 1.2 In an n-dimensional path space G, all paths in a local coordinate system (xi) +can be parameterized under a variable t with 2(n−1) free constant parameters u, v as follows: +x = x(t) = σ(t; u, v), +(u, v ∈ Rn−1). +(4) +Further, the parametric equation (4) induces a spray G whose geodesics are given by (4) +with t as its geodesic parameter, and if a new variable s = s(t) = s(t; u, v) is given with +s′(t) > 0, then it gives a spray ¯G ∈ Proj(G) with s as its geodesic parameter. +If a family of curves can be parameterized in the form (4), then with an auxiliary pa- +rameter c > 0 multiplied by t in (4), we can obtain the corresponding spray by eliminating +the parameters u, v, c, t. We give some examples to show how to solve the sprays from given +path spaces (see Examples 4.5-4.8). +In the study of rigidity issues on a Finsler or spray manifold, it is important to assume +that the (Finsler) spray in consideration be (positively/negatively) complete. A given spray +is not necessarily (positively/negatively) complete. So a natural problem is whether a spray +can be projectively (positively/negatively) complete or not. We solve this problem under +certain conditions in the following result (Theorem 1.3). +Theorem 1.3 Let G be a spray on a manifold M with its each geodesic x = x(t) being +defined on the maximal interval I given by one case of the following +I = (a, b), +or (a, +∞), +or (−∞, b), +(5) +where a = a(u, v) < 0, b = b(u, v) > 0 with u = x(0), v = x′(0) are C∞ functions on a +conical region C of T M \ {0}. Then G is projectively (positively/negatively) complete on C. +In Theorem 1.3, usually we can also put u, v as that in (4) (see Example 5.5). If (5) +is not satisfied, it is uncertain that G is projectively complete (cf. +Example 5.5)). +We +give Examples 5.2-5.5 as an application of Theorem 1.3. A Finsler metric is not necessarily +projectively (positively/negatively) complete, namely, if G in Theorem 1.3 is a Finsler spray, +the corresponding spray projective to G may not be a Finsler spray. +2 + +2 +Geodesic parameters in projective relations +A spray on M, in our consideration, is a smooth vector field G on a conical region C of +T M \ {0} (an important case is C = T M \ {0}) expressed in a local coordinate system +(xi, yi) in T M as follows +G = yi ∂ +∂xi − 2Gi ∂ +∂yi , +where Gi are local homogeneous functions satisfying Gi(x, λy) = λ2Gi(x, y) for λ > 0. If +C = T M \ {0}, G is called regular; otherwise, it is called singular. +The integral curves of G projected onto M are the geodesics of G. Let x = x(s) be a +geodesic of G. Then it satisfies the following ODE: +d2xi +ds2 + 2Gi(x, dx +ds ) = 0, +where s is called a geodesic parameter of the geodesic x = x(s). Reparameterizing a geodesic +x = x(s) by a general parameter t with ds/dt > 0, we have +d2xi +dt2 + 2Gi(x, dx +dt ) = γ(t)dxi +dt , +(6) +where γ(t) is given by +γ(t) = d2s +dt2 +�ds +dt = − d2t +ds2 +� +( dt +ds)2. +(7) +Let G, ¯G be two sprays pointwise projectively related by ¯Gi = Gi + Pyi. Let x = x(t) +be a geodesic of G or ¯G as a point set for a general parameter t. Then along the geodesic +x = x(t), it follows from (6) and (7) that +2P(t) = ¯s′′(t) +¯s′(t) − s′′(t) +s′(t) , +� +P(t) := P(x(t), x′(t)) +� +, +(8) +where s, ¯s are the geodesic parameters of the curve x = x(t) in G, ¯G respectively. +In +particular, along a geodesic x = x(s) of G, it follows from (8) that +2P(s) = ¯s′′(s) +¯s′(s) , +� +P(s) := P(x(s), x′(s)) +� +, +(9) +If we express the geodesic x = x(s) of G as the geodesic x = x(¯s) of ¯G, by (9), we have +2P(¯s) = 2P(x(s), x′(s))ds +d¯s = +¯s′′(s) +� +¯s′(s) +�2 , +� +P(¯s) := P(x(¯s), x′(¯s)) +� +. +(10) +So if P(s) or P(¯s) is known, the relation ¯s = ¯s(s) can be obtained from (9) or (10). +Example 2.1 Let F be the Funk metric on a strongly convex domain Ω ⊂ Rn. Define a +projectively flat spray G by +Gi = Pyi, +P := cF, +where c is a constant. Any geodesic x = x(t) (as a point set) of G is given by +x = x(t) = vt + u, +� +− +1 +F(u, −v) < t < +1 +F(u, v) +� +, +3 + +where u, v ∈ Rn are constant vectors. We have +F(vt + u, v) = +F(u, v) +1 − tF(u, v). +(11) +Let s be a geodesic parameter of G. Then by (9) and (11) we have +s′′(t) +s′(t) = 2cF(vt + u, v) = +2cF(u, v) +1 − tF(u, v), +(12) +integration of which with s(0) = 0 gives +s = s(t) = +� +κ ln +� +1 − tF(u, v) +� +, +(c = 1 +2), +κ +� +1 − +� +1 − tF(u, v) +�1−2c� +, +(c ̸= 1 +2), +(13) +where κ is a constant with κ < 0 for c ≥ 1/2, and κ > 0 for c < 1/2. Thus the spray +is positively complete for c ≥ 1/2, and any geodesic is defined on a finite open interval for +c < 1/2. Besides, the spray G is (locally) metrizable if and only if c = 0, 1, 1/2 (see [10]). +Example 2.2 In Example 2.1, if the spray G is given by +Gi(y) := Pyi, +P := c +� +F(y) − F(−y) +� +, +then by (11) and +F(vt + u, −v) = +F(u, −v) +1 + tF(u, −v). +(14) +it follows from (9) that +s′′(t) +s′(t) = +2cF(u, v) +1 − tF(u, v) − +2cF(u, −v) +1 + tF(u, −v), +integration of which with s(0) = 0 gives +s = s(t) = κ +� t +0 +�� +1 − tF(u, v) +�� +1 + tF(u, −v) +��−2c +dt, +(15) +where κ > 0 is constant. From (15), it is clear to conclude that G is complete if c ≥ 1/2; s +is bounded in a finite open interval if c < 1/2. +3 +Projective flat sprays with weak Ricci constant +For a spray G, the Riemann curvature tensor Ri +k is defined by +Ri +k := 2∂kGi − yj(∂j ˙∂kGi) + 2Gj( ˙∂j ˙∂kGi) − ( ˙∂jGi)( ˙∂kGj), +where we define ∂k := ∂/∂xk, ˙∂k := ∂/∂yk. The trace of Ri +k is called the Ricci curvature, +Ric := Ri +i. +For a spray tensor T = Tidxi as an example, the horizontal and vertical +derivatives of T with respect to Berwald connection are given by +Ti;j = δjTi − TrGr +ij, +Ti.j = ˙∂jTi, +(δi := ∂i − Gr +i ˙∂r, Gk +ir := ˙∂r ˙∂iGk)). +4 + +A spay is called weakly Ricci constant if Ric;0 := Ric;ryr = 0. A spray G is said to be of +constant curvature if Ri +k is given by Ri +k = Rδi +k − τkyi with ([11]) +τi;k = 0 ( ⇔ R = τk = 0, or R;i = 0(R ̸= 0)). +By definition, it is clear that a spray of constant curvature is weakly Ricci constant. For +two pointwise projectively related sprays G, ¯G with ¯Gi = Gi + Pyi, their Ricci curvatures +Ric, ¯Ric are related by +¯Ric = Ric − (n − 1)(P;0 − P 2). +(16) +We consider a projectively flat spray manifold (G, M) given by (1), that is, +Gi = �Gi + Pyi, +where �G is a locally Minkowski spray on M ( �G has local straight lines as geodesics). If +G is weakly Ricci constant, then we can determine the projective factor P along geodesics, +which is shown in Theorem 1.1. +Proof of Theorem 1.1 : By (16) and �Gi = Gi − Pyi, the Ricci curvature Ric of G is +given by +Ric = −(n − 1)(P 2 + P;0). +Therefore, Ric;0 = 0 is equivalent to P;0;0 + 2PP;0 = 0. Then along a geodesic x = x(s) of +G, we have +P ′′(s) + 2P(s)P ′(s) = 0. +whose solution is given by one of the three cases in (2). Further, if G is complete, it is clear +that (3) follows from (2). +Q.E.D. +If the spray G in (1) is weakly Ricci constant Ric;0 = 0, then applying (2) and (10), we +obtain the following proposition. +Proposition 3.1 Let the spray G in (1) be weakly Ricci constant (esp. of constant curva- +ture). For any geodesic σ, let s and t be the geodesic parameters of σ with respect to G and +�G respectively. Then s = s(t) is given by one of the following cases: +s = at, (a > 0); +s = b ln(1 + at), (ab > 0); +(17) +s = +bt +1 + at, (a ̸= 0, b > 0); +s = c +� +arctan(at + b) − arctan b +� +, (ac > 0); +(18) +s = c ln 1 + bt +1 + at, +� +(b − a)c > 0, ab ̸= 0 +� +, +(19) +where a, b, c are constant, and in (19), it further requires s′(t) > 0 (see Remark 3.2). +Proof : By (10) we need to solve the following ODE with initial conditions: +s′′(t) +s′(t) = 2P(s)s′(t), +(s(0) = 0, s′(t) > 0), +integration of which gives +s′(t) = ae2 � P (s)ds, +� +e−2 � P (s)dsds = at + b, +(20) +5 + +where a, b are two constants. Now P(s) is given by (2) from Theorem 1.1, and thus we can +obtain s = s(t) by plugging P(s) into (20). +If P(s) = 0, then (20) gives s = at+b. Since s(0) = 0, s′(t) > 0, we obtain s = at (a > 0), +which gives the first formula in (17). +If P(s) = c ̸= 0 is constant, then (20) gives the second formula in (17) with ab > 0. +If P(s) is given by the first formula in (2), then (20) gives +s = −κ + +1 +at + b, +which can be rewritten as the form of the first formula in (18) by s(0) = 0, s′(t) > 0. +If P(s) is given by the second formula in (2) (c ̸= 0), then (20) gives +s = −κ − arctan(at + b) +c +, +which can be rewritten as the second formula in (18) by s(0) = 0, s′(t) > 0. +If P(s) is given by the third formula in (2) (cκ ̸= 0), then (20) gives +s = 1 +2c ln +� +1 +at + b − 1 +κ +� +which can be rewritten as the formula in (19) by s(0) = 0, s′(t) > 0. +Q.E.D. +In (19), by s′(t) > 0, we have further restriction on the constant parameters a, b, c, which +is shown in the following remark. +Remark 3.2 In (19), let t be defined on the maximal interval (κ1, κ2) with κ1 < 0 < κ2. It +is easy to conclude the following cases from s′(t) > 0: +a > 0, b > 0 : +� +t ∈ (κ1, κ2) ⊂ (− 1 +a, +∞), +(b < a) +t ∈ (κ1, κ2) ⊂ (− 1 +b, +∞), (b > a), +a < 0, b < 0 : +� +t ∈ (κ1, κ2) ⊂ (−∞, − 1 +a), +(b > a) +t ∈ (κ1, κ2) ⊂ (−∞, − 1 +b), (b < a), +a > 0, b < 0 : t ∈ (κ1, κ2) ⊂ (−1 +a, −1 +b), +a < 0, b > 0 : t ∈ (κ1, κ2) ⊂ (−1 +b , −1 +a). +By Proposition 3.1 and Remark 3.2, we directly obtain the following corollary. +Corollary 3.3 If the spray G in Proposition 3.1 (P ̸= 0) is complete, then s = s(t) is given +by one of the following two cases: +s = b ln(1 + at), (ab > 0), +(21) +s = c ln 1 + bt +1 + at, +� +(b − a)c > 0, ab < 0 +� +, +(22) +where in (21) and (22), we respectively have +t ∈ (−∞, −1 +a) if a < 0, and t ∈ (−1 +a, +∞) if a > 0; +t ∈ (−1 +a, −1 +b) if a > 0, b < 0, and t ∈ (−1 +b , −1 +a) if a < 0, b > 0. +6 + +Now in the following, we give some projectively flat sprays to verify the above results on +the projective factors and the geodesic parameters. +Example 3.4 Consider the spray G in Example 2.1. A direct computation shows that G +is weakly Ricci constant or of constant curvature if and only if c = 0, 1, 1/2. Let x = x(t) = +vt + u be a geodesic (as a point set) of G, and the geodesic parameter s in G is given by +(13). Then it follows from (10) and (13) that P = cF is given by +P(s) = +� +− 1 +2κ, +(c = 1 +2) +c +2c−1 · +1 +s−κ, +(c ̸= 1 +2). +(23) +It is clear from (23) that P(s) is in one of the forms in (2) if and only if c = 0, 1, 1/2. +Meanwhile, s = s(t) is given in Proposition 3.1 if and only if c = 1, 1/2, 1, and in this case, +s = s(t) is in the respective forms shown in (17) and the first formula in (18). +Example 3.5 Let G be the spray in Example 2.2 with c = 1/2. G is complete and it is of +constant curvature. Then it follows from (15) that +s = κ ln 1 + tF(u, −v) +1 − tF(u, v) , +(24) +where κ > 0 is a constant. In this case, s = s(t) is in the form (22), and it is easy to verify +that P(s) is in the form of the third formula in (2) by plugging (24) into (10). +Example 3.6 Let G be a spray on Rn defined by +Gi := Pyi, +P := − ⟨x, y⟩ +1 + |x|2 . +G is metrizable and it is of constant curvature. Let x = x(t) = vt + u be a geodesic (as a +point set) of G, and by (9), the geodesic parameter s of G satisfies +s′′(t) +s′(t) = − +2(|v|2t + ⟨u, v⟩) +1 + |u|2 + 2⟨u, v⟩t + |v|2t2 . +Solving the ODE, we obtain +s = s(t) = κ1 + κ2 arctan +|v|2t + ⟨u, v⟩ +� +(1 + |u|2)|v|2 − ⟨u, v⟩2 , +where κ1, κ2 are constant. It is clear that s = s(t) is in the form of the second formula in +(18) if s(0) = 0, and P(s) is in the form of the second formula in (2) by plugging the above +s = s(t) into (10). +4 +Construction of sprays from geodesics +Given a family of curves G on a manifold, if G can constitute the geodesics of a spray G +on M, how can we solve G (at least locally)? A spray induces a (local) semispray and two +pointwise projectively related sprays induce a same semispray (cf. [9]). A semispray can +also be considered as a special parameterized family of curves, which forms a path space. +In this section, we will start from a path space and introduce some ways to construct +sprays based on a path space and its parameterization. We call it the geodesic method of +construction of sprays. +Similarly to a spray, a path space G on a manifold M is usually defined on a conical +region C of T M \ {0} (see Definition 4.1), and G is called singular if C ̸= T M \ {0}. +7 + +Definition 4.1 Let G be a family of C∞ parameterized curves (called paths) on an n- +dimensional manifold M. G or (M, G) is called an n-dimensional path space if on a conical +region C of T M it satisfies +(i) for y ∈ Cx, there is a curve σ : (−ǫ, ǫ) → M in G with σ′(0) = y; +(ii) for any σ, τ in G with σ′(0) = τ ′(0), σ and τ coincide in a small intervals of 0; +(iii) if a curve σ is in G, then for any constants λ > 0 and to, the curve η is also in G, +where η is defined by η(t) := σ(λt + to). +An equivalent version of Definition 4.1 in regular case is refereed to [9] (P52). +Example 4.2 Consider a set G of a family of curves x = x(s) on R2 in the form +x(s) = σ(s; xo, yo), +� +x(0) = xo = (a, b), +x′(0) = yo = (u, v) +� +, +σ(s; xo, yo) := (a, b) + (u, v)s − (0, 1) +�1 +3u3s3 + au2s2� +, +where a, b, u, v are arbitrary parameters. It can be directly verified that G is a path space on +R2, since Definition 4.1 (i) (ii) automatically hold, and Definition 4.1 (iii) follows from +σ(λs + so; xo, yo) = σ(s; ˆxo, ˆyo), +where we define +xo = (a, b), +yo = (u, v), +ˆxo = (ˆa,ˆb), +ˆyo = (ˆu, ˆv), +ˆa := a + uso, +ˆb := b + vso − 1 +3u2(3a + uso)(so)2, +ˆu := λu, +ˆv := λv − λu2so(2a + uso). +For a path space G, we have different ways to parameterize the paths in G under a para- +metric variable and some constant parameters (see Theorem 1.2 and Lemma 4.3). Example +4.2 satisfies (25) and (26) in the following Lemma 4.3 with +f(s; xo, yo) := −(0, 1) +�1 +3u3s3 + au2s2� +, +xo = (a, b), yo = (u, v). +Lemma 4.3 An n-dimensional path space (M, G) is locally expressed as the following family +of curves x = x(s) with arbitrary constant parameters xo, yo ∈ Rn: +x(s) = σ(s; xo, yo) = xo + yos + f(s; xo, yo), +(25) +where f is a smooth function satisfying f(0; xo, yo) = f ′(0; xo, yo) = 0 and +f(s; ˆxo, ˆyo) = f(λs + so; xo, yo) − f(so; xo, yo) − λf ′(so; xo, yo)s, +(26) +� +ˆxo := xo + yoso + f(so; xo, yo), +ˆyo := λyo + λf ′(so; xo, yo) +� +. +It is clear that the collection of geodesics of a spray naturally forms a path space. Shen +proves the converse in the following lemma ([9]). We also give the proof for convenience. +Lemma 4.4 A path space G induces a spray G with the set of geodesics of G being G. +Proof : Let (G, M) be a path space on a conical region C. For a given y ∈ Cx, there is a +curve σ : (−ǫ, ǫ) → M in G with σ(0) = x, σ′(0) = y by Definition 4.1 (i). Define +Gi(y) := −1 +2 +d2σ +ds2 (0), +8 + +which is independent of the choice of σ by Definition 4.1 (ii). We are going to verify that G +is a spray. For any constant λ > 0, let η(s) := σ(λs) ∈ G (see Definition 4.1 (iii)). Then we +have +Gi(λy) = −1 +2 +d2η +ds2 (0) = −1 +2λ2 d2σ +ds2 (0) = λ2Gi(y), +which implies that Gi is positively homogeneous of degree two. Further, for any η : (a, b) → +M in G and any fixed t ∈ (a, b), define γ(s) := η(s + t). Then we have +η′(t) = γ′(0), +η′′(t) = γ′′(0). +So by the definition of Gi, we get +Gi� +η′(t) +� += Gi� +γ′(0) +� += −1 +2 +d2γi +ds2 (0) = −1 +2 +d2ηi +ds2 (t), +which implies that η satisfies the following ODE: +d2ηi +ds2 + 2Gi�dη +ds +� += 0. +Therefore, G is a spray, and the set of geodesics of G coincides with G. +Q.E.D. +In Lemma 4.4, by different choices of the parametric variables, it (locally) induces a +projective class Proj(G) of G, each of which is projective to G. +For a given path space G, it induces a spray G by Lemma 4.4. Then G defines a semispray +ˆG (see [9]: P37) and the geodesics of G and ˆG are closely related (see [9]: Lemma 3.1.1). +Therefore, in G, any path can be locally expressed as +xa = xa(x1; u, v), +(u, v ∈ Rn−1, 2 ≤ a ≤ n), +where u, v are free constant parameters. So all paths in an n-dimensional path space depend +only on 2(n−1) free constant parameters, where the Jaccobi determinant is not zero, namely, +det +�∂xa/∂u +∂xa/∂v +∂ya/∂u +∂ya/∂v +� +̸= 0, +� +ya := dxa +dx1 +� +. +Then we obtain Theorem 1.2 for the construction of sprays based on the parametric equations +of path spaces. +If we write (4) in the form +x(t) = σ(λt + µ, u, v), +(27) +where λ, µ are constant numbers, then under the 2n constant parameters λ, µ, u, v, this +family of curves satisfies Definition 4.1 (i)(ii)(iii). +For instance, the 2-dimensional path +space in Example 4.2 can be written as the following family of curves +x(s) = τ(λs + µ; bo, vo) = (0, bo) + (1, vo)(λs + µ) − 1 +3(0, 1)(λs + µ)3, +� +λ := u, µ := a, vo := v +u + a2, bo = b − avo + 1 +3a3� +. +By Theorem 1.2, if the set A of a family of curves on an n-dimensional manifold defines +a path space, then A just depends on 2(n − 1) free constant parameters. For example, in +Rn, all circles with fixed radius cannot define a path space when n ≥ 3, because in this case, +the circles depend on more than 2(n − 1) free constant parameters. +9 + +Now we introduce a method of constructing a spray G determined by a path space +considered as the geodesics of G, which is similar to Okubo’s method for the construction of +a Finlser metric from a hypersurface as its indicatrix. We can start from a family of curves +given by (25) or (4) to determine a corresponding spray. +Method (I): For a family of curves given by (25) satisfying (26), actually we can reduce +one constant parameter since (25) can be written as (if y1 +o ̸= 0) +x(t) = σ(t; xo, ¯yo) = xo + ¯yot + f(t; xo, ¯yo), +� +t := y1 +os, ¯ya +o := ya +o/y1 +o, ¯yo := (¯ya +o) +� +. +Let a path space be determined by (25) and we put +x = xo + yos + f(s; xo, yo), +y(= dx +ds ) = yo + f ′(s; xo, yo)), +(28) +Gi := −1 +2 +d2x +ds2 = −1 +2f ′′(s; xo, yo)). +(29) +Then we obtain a spray G from (29) by eliminating xo, yo, s in (29) from (28), where s is a +geodesic parameter of the spray G. +Method (II): Suppose that a family of curves are given by the parametric equation (4) with +2(n − 1) free constant parameters u, v. This case is more convenient to construct sprays. +With an auxiliary parameter c > 0, we put +x = σ(cs; u, v), +y = dx +ds = cdσ +dˆs (cs; u, v), +ˆs := cs. +(30) +Theoretically, we can express c, s, u, v as functions of x, y from (30). Then plugging them +into the following +Gi := −1 +2 +d2xi +ds2 = c2 d2σi +dˆs2 (cs; u, v), +(31) +we obtain a spray G given by (31), where s is a geodesic parameter of the spray G. +Now in the following Examples 4.5-4.8, we use Method (I) or Method (II) to show how +we construct sprays from given path spaces by eliminating the corresponding parameters. +Example 4.5 Consider a set G of a family of curves on R3: +x(s) = (a, b, c) + (u, v, w)s − (0, 1, 0)h(s), +� +h(s) := −1 +3(u3 + w3)s3 − (au2 + cw2)s2� +, +where a, b, c, u, v, w are constant parameters. G is a path space. By (28) we get +x1 = a + us, +x3 = c + ws, +y1 = u, +y3 = w. +(32) +By (29), the induced spray G is given by +G1 = −1 +2 +d2x1 +ds2 = 0, +G3 = −1 +2 +d2x3 +ds2 = 0, +G2 = −1 +2 +d2x2 +ds2 = (u3 + w3)s + (au2 + cw2) += (u3 + w3)s + +� +(x1 − us)u2 + (x3 − ws)w2� � +by (32) +� += x1u2 + x3w2 = x1(y1)2 + x3(y3)2 � +by (32) +� +. +10 + +G has zero Riemann curvature and so it is metrizable (a Finsler spray) ([11]). +Example 4.6 Let G be the set of all circles with fixed radius r on R2. We parameterize G +by +x1(s) = a + r cos s, +x2(s) = b + r sin s, +where a, b are arbitrary constant parameters. G depends on just two free constant parameters. +By Theorem 1.2, G defines a spray G on R2 with s as a geodesic parameter of G. We show +the spray as follows. With an auxiliary parameter c > 0, it follows from (30) that +x1 = a + r cos cs, +x2 = b + r sin cs, +y1 = −cr sin cs, +y2 = cr cos cs. +(33) +Then plugging the latter two formulas of (33) into (31) yields a spray G given by +G1 = −c2r cos cs = −1 +r y2� +(y1)2 + (y2)2, +G2 = −c2r sin cs = 1 +r y1� +(y1)2 + (y2)2. +This circle spray first appears in [9] (P49), and even locally it is not metrizable ([11]). +Example 4.7 Consider a family of semicircles G on the positive semi-plane R2 ++ with center +on x1-axis and arbitrary radius. Note that G is singular at the direction parallel to x2-axis. +We can parameterize G by +x1 = a + b coss, +x2 = b sin s, +(x2 > 0, b ≥ 0), +where a, b are arbitrary constant parameters. G depends on just two free constant parameters. +By Theorem 1.2, G defines a spray G on R2 ++ with s as a geodesic parameter of G. With an +auxiliary parameter c > 0, by (30) we get +x1 = a + b cos cs, +x2 = b sin cs, +y1 = −bc sincs, +y2 = bc coscs. +(34) +Then similarly, by the elimination of the parameters a, b, c, s in (31) from (34), the spray G +with s being a geodesic parameter is given by +G1 = −y1y2 +2x2 , +G2 = (y1)2 +2x2 . +(35) +The spray G is regular on R2 ++ (any straight lines parallel to x2-axis are geodesics of G). G +is of isotropic curvature, and locally it is not metrizable by the method in [2, 11]. +Example 4.8 Let Bn be the unit ball in Rn and G be all circle arcs in Bn which are +perpendicular to the boundary Sn−1 = ∂Bn. Let s be the arc-length parameter of a circle arc +induced by the Euclidean metric. What is the spray G induced by G with s being a geodesic +parameter of G (see Example 4.1.4 in [9])? We will show that G is given by +Gi = ⟨x, y⟩yi − |y|2xi +1 − |x|2 +, +(36) +which is not metrizable by [11]. Now for arbitrarily given p, q ∈ Sn−1, there is a circle arc +γ in G, in which γ is perpendicular to Sn−1 at p, q. Let C be the circle with γ ⊂ C. The +center and radius of C are respectively given by +τ(p + q), +|p − τ(p + q)|, +(τ := (1 + pq)−1), +11 + +where pq is the Euclidean inner product of p, q. Then γ is parameterized by the equation +x(s) = x(s; p, q) = [p − τ(p + q)] cos s + |p − τ(p + q)|p sin s + τ(p + q). +(37) +Since there are just 2(n − 1) free constant parameters in (37), the family of curves in the +form (37) define a path space by Theorem 1.2. Now based on (30) and (31), we can give +the spray G from (37) with s being a geodesic parameter of G. With an auxiliary parameter +c > 0, by (30) we put +x = [p − τ(p + q)] cos cs + |p − τ(p + q)|p sin cs + τ(p + q), +(38) +y = −c[p − τ(p + q)] sin cs + c|p − τ(p + q)|p cos cs. +(39) +By (31) we have +2Gi := c2� +[p − τ(p + q)]i cos cs + |p − τ(p + q)|pi sin cs +� +. +(40) +By a direct lengthy computation, we can eliminate the parameters p, q, c, s in (40) from (38) +and (39) (the details are omitted). Finally, the spray G is given by (36). +5 +Projectively complete sprays +For a given spray G, if we know the general solutions of all geodesics of G, then under another +parameter as a geodesic parameter, we can determine a corresponding spray projectively +related to G. Now suppose that the general solutions of geodesics of G are locally given by +x = σ(t) = σ(t; u, v), +� +u, v ∈ Rn−1� +, +(41) +where t is a geodesic parameter of G and u, v are free constant parameters. Sometimes, it is +also convenient to put u = σ(0), v = σ′(0) for the elimination of parameters. Make a change +of the variables from t to s with +t = t(s) = t(s; u, v), +dt +ds > 0. +(42) +With an auxiliary parameter c > 0, we put +x = σ(t(cs); u, v), +y = dx +ds = cdσ +dt +dt +ds, +(43) +where dt/ds, as a function of s, takes the value at cs. Further, we have +d2xi +ds2 = c2 d2σi +dt2 +� dt +ds +�2 + c2 dσi +dt +d2t +ds2 += −2Gi(x, dσ +dt )c2� dt +ds +�2 + c2 dσi +dt +d +dt +� dt +ds +� dt +ds += −2Gi(x, y) + c d +dt +� dt +ds +� +yi. +(44) +Expressing c, t in terms of x, y from (43), and then plugging c, t into (44), we obtain a spray +¯G given by +¯Gi = Gi − 1 +2 +d +dt +� dt +ds +� +cyi = Gi + Pyi, +(45) +� +P = P(x, y) := −1 +2 +d +dt +� dt +ds +� +c +� +, +with s being a geodesic parameter of ¯G. +12 + +Lemma 5.1 Suppose that the general solutions of geodesics of a spray G are given by (41). +Let s be another parameter related to t by (42). Then a spray ¯G projective to G with s being +its geodesic parameter is given by (45), where c, t are determined by (43). +Under certain condition, a spray can be projectively (positively/negatively) complete, +which is shown in Theorem 1.3. Now we give the proof of Theorem 1.3. +Proof of Theorem 1.3 : Let G be a spray on a manifold M. For an arbitrary geodesic +x = x(t), suppose that t belongs to the maximal interval I given by (5). +If I = (a, +∞) or I = (−∞, b), we respectively make a change of the variables from t to +s by +s = ln(1 − t +a), +or +s = − ln(1 − t +b), +(46) +either of which gives s(0) = 0, s′(t) > 0 and the maximal interval of s with s ∈ (−∞, +∞). +If I = (a, b), make a change by (46) and then we respectively have +s ∈ +� +− ∞, ln(1 − b +a) +� +, +or +s ∈ +� +− ln(1 − a +b ), +∞). +If I = (a, b), make a change of the variables from t to s by +s = ln 1 − t/a +1 − t/b , +or +s = tan +� +π +b − a(t − a + b +2 +) +� ++ tan +�b + a +b − a +π +2 +� +, +(47) +either of which gives s(0) = 0, s′(t) > 0 and the maximal interval of s with s ∈ (−∞, +∞). +Therefore, by the change (46) or (47), we obtain a (positively/negatively) complete spray +which is projective to G. This completes the proof. +Q.E.D. +As an application of Theorem 1.3, we give the following Examples 5.2-5.4 to show the +construction of the (positively/negatively) complete sprays projective to given sprays. +Example 5.2 Let F be the Funk metric on a strongly convex domain Ω ⊂ Rn. +The +Minkowski spray G = 0 on Ω has its geodesics given by +x(t) = vt + u, +� +− +1 +F(u, −v) < t < +1 +F(u, v) +� +, +where u, v ∈ Rn are arbitrary constant vectors. By (46), put t = t(s) as +s = − ln +� +1 − tF(u, v) +� +. +(48) +With s being a geodesic parameter, we obtain a projectively flat and positively complete spray +¯G, which will be shown to be the Finsler spray induced by F, namely, +¯Gi = 1 +2Fyi. +(49) +Actually, it follows from (2) and (11) that +dt +ds = +1 +F(u, v) − t = +1 +F(vt + u, v). +(50) +Then (43) gives +x = vt + u, +y = cv dt +ds. +(51) +13 + +It is clear from (51) and (50) that +F(x, y) = cF(vt + u, v) dt +ds = c. +(52) +Therefore, by (45), (50) and (52), the spray G is given by (49). +Example 5.3 In Example 5.2, by (47), put t = t(s) as +s = ln 1 + tF(u, −v) +1 − tF(u, v) . +(53) +With s being a geodesic parameter, we obtain a projectively flat and complete spray ¯G, which +will be shown to be the Finsler spray induced by the Klein metric ¯F(x, y) := +� +F(x, y) + +F(x, −y) +� +/2, namely, +¯Gi(x, y) = 1 +2 +� +F(x, y) − F(x, −y) +� +yi. +(54) +Firstly, by (53), we get +dt +ds = +� +1 − tF(u, v) +�� +1 + tF(u, −v) +� +F(u, v) + F(u, −v) +, +(55) +d +dt +� dt +ds +� += F(u, −v) − F(u, v) − 2tF(u, v)F(u, −v) +F(u, v) + F(u, −v) +. +(56) +Secondly, (43) gives +x = vt + u, +y = cv dt +ds, +from which we have +F(x, y) = F(vt + u, v)c dt +ds, +F(x, −y) = F(vt + u, −v)c dt +ds. +(57) +Plugging (11), (14) and (55) into (57), we obtain +c = F(x, y) + F(x, −y), +t = F(x, y)F(u, −v) − F(x, −y)F(u, v) +F(u, v)F(u, −v)[F(x, y) + F(x, −y)]. +(58) +Finally, by (58) and (56), it follows from (45) that the spray ¯G is given by (54). +Example 5.4 For the spray G in Example 5.2, we will introduce a different way from that +in Example 5.3 to make G be complete, which is actually to use (46) to make complete the +Finsler spray induced by the Funk metric F. For a geodesic x(t) = vt + u of G, put +s = ln +� +1 − ln(1 − tF(u, v)) +a +� +, +a := ln +� +1 + F(u, v) +F(u, −v) +� +. +In a similar way to that for the computation in Example 5.3, we obtain a projectively flat +and complete spray ¯G given as follows: +¯Gi(y) = Gi +F (y) + 1 +2 +F(y) +ln +F (−y) +F (y)+F (−y) +yi, +� +Gi +F (y) := 1 +2F(y)yi� +. +¯G is of scalar curvature and actually we can verify that ¯G is not metrizable by using the +method in [2]. +14 + +Example 5.5 For the family of semicircles G on R2 ++ as shown in Example 4.7, we can +parameterize them in the following form +x1 = u − v sin t, +x2 = v cos t, +(x2 > 0, v ≥ 0), +(59) +where u, v are arbitrary constant parameters. We have shown in Example 4.7 that the spray +G determined by G is given by (35), that is, +G1 = −y1y2 +2x2 , +G2 = (y1)2 +2x2 . +(60) +We can make G be projectively complete on the conical region C with the direction (0, 1) +being deleted from T R2 ++ \ {0}. Since −π/2 < t < π/2 for any u, v in (59), by (47), we let +s = tan t. +Then by (45), we get a complete spray ¯G projective to G with the projective factor P being +given by +P = c +� +− 1 +2 +d +dt +� dt +ds +�� +t=cs = c +� +cos t sin t +� +t=cs = +c2s +1 + c2s2 . +(61) +Now it follows from (43) that +x1 = u − +vcs +√ +1 + c2s2 , +x2 = +v +√ +1 + c2s2 , +y1 = +−vc +(1 + c2s2)3/2 , +y2 = +−bc2s +(1 + c2s2)3/2 , +from which we get +s = − +x2y2 +(y1)2 + (y2)2 , +c = −(y1)2 + (y2)2 +x2y1 +. +Plugging s, c in the above into (61) yields P = −y2/x2. Thus the spray ¯G is given by +¯G1 = G1 + Py1 = −3y1y2 +2x2 , +¯G2 = G2 + Py2 = (y1)2 − 2(y2)2 +2x2 +. +¯G is complete on the conical region C but not complete in the direction (0, 1). We don’t know +whether the spray G in (60) can be projectively complete or not on T R2 \ {0}. Besides, ¯G +is of isotropic curvature, and locally it is not metrizable by the method in [2, 11]. +References +[1] R. Bryant, Geodesically reversible Finlser 2-spheres of constant curvature, Inspired by +S. S. Chern, 95-111, Nankai Tracts Math., 11, World Sci. Publ., Hackensack, NJ, 2006. +[2] I. Bucataru and Z. Muzsnay, Finsler metrizable isotropic sprays and Hilbert’s Fourth +Problem, J. Aust. Math. Soc. 97 (2014), 27-47. +[3] S. G. Elgendi and Z. Muzsnay, Metrizability of holonomy invariant projective defor- +mation of sprays, Cana. Math. Bulletin, 2020. +15 + +[4] L.Huang and X. Mo, On geodesics of Finsler metrics via navigation problem, P. Am. +Math. Soc., 139 (8) (2011), 3015-3024. +[5] Y. Li, X. Mo and Y. Yu, Inverse problem of sprays with scalar curvature, Intern. J. +Math. 30(6), 2019. +[6] B. Li and Z. Shen, Sprays of isotropic curvature, Intern. J. math. 2019. +[7] C. Robles, Geodesics in Randers spaces of constant curvature, Trans. Amer. Math. +Soc. 359 (2007), 1633-1651. +[8] Z. Shen: On projectively related Einstein metrics in Riemann-Finsler geometry, Math. +Ann., 320(2001), 625-647. +[9] Z. Shen: Differential geometry of spray and Finsler spaces, Kluwer Academic Publish- +ers, Dordrecht, 2001. +[10] G. Yang, Some classes of sprays in projective spray geometry, Diff. Geom. Appl., 29 +(2011), 606-614. +[11] G. Yang, On sprays of scalar curvature and metrizability, J. Geom. Anal., 2022 (in +press). +[12] G. Yang, Sprays on Hamel-Funk functions model, preprint. +Guojun Yang +Department of Mathematics +Sichuan University +Chengdu 610064, P. R. China +yangguojun@scu.edu.cn +16 + diff --git a/1NAyT4oBgHgl3EQfofhG/content/tmp_files/load_file.txt b/1NAyT4oBgHgl3EQfofhG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..112362487c13a19857b6f8a5491d44f9a3dbd890 --- /dev/null +++ b/1NAyT4oBgHgl3EQfofhG/content/tmp_files/load_file.txt @@ -0,0 +1,482 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf,len=481 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='00507v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='DG] 2 Jan 2023 On Geodesics of Sprays and Projective Completeness Guojun Yang Abstract Geodesics, which play an important role in spray-Finsler geometry, are integral curves of a spray vector field on a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Some comparison theorems and rigidity issues are established on the completeness of geodesics of a spray or a Finsler metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In this paper, projectively flat sprays with weak Ricci constant (eps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' constant curvature) are classified at the level of geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Further, a geodesic method is introduced to determine an n-dimensional spray based on a family of curves with 2(n−1) free constant parameters as geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Finally, it shows that a spray is projectively complete under certain condition satisfied by the domain of geodesic parameter of all geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Keywords: Spray, Geodesic, Completeness, Path Space, Finsler Metric MR(2000) subject classification: 53B40, 53C60 1 Introduction Spray geometry studies the properties of sprays on a manifold, and it is closely related to Finsler geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Every Finsler metric induces a natural spray but there are a lot of sprays which are not Finsler-metrizable (not be induced by any Finsler metric) ([3, 5, 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' So a popular topic is to investigate whether a given spray is metrizable or not, and what’s more important is to give necessary and sufficient conditions for certain class of sprays to be metrizable ([2, 11, 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' It is also important to investigate the properties of some special classes of sprays, for example, (locally) projectively flat sprays, Berwald sprays, sprays of scalar (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' isotropic, constant) curvature, Hamel (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Funk) sprays ([2, 6, 11, 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' A spray G on a manifold M defines a special vector filed on a conical region C of T M \\ {0}, and it naturally defines its integral curves and the projections of the integral curves onto the manifold M are called geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Geodesics play an important role in the studies of comparison theorems and rigidity issues on spray or Finsler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In [8], Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Shen studies two pointwise projectively related Einstein Finsler metrics and determine the metrics along geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In [10], the present author obtains a comparison theorem on the Ricci curvatures of a spay and a Finsler metric which are pointwise projectively related and the corresponding projective factor is estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In [1], R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Bryant proves that a geodesically reversible Finlser metric on S2 with positive constant flag curvature is a Riemann metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In [7], C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Robles classifies geodesics of Randers metrics of constant flag curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In [4], L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Huang and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Mo obtain the relation between the geodesics of two Finsler metrics F and ˜F, where ˜F is defined by the navigation data (F, V ) with V being a homothetic vector field of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In this paper, we study projectively flat sprays with weak Ricci constant, the construction of sprays from a geodesic method and the projective completeness of sprays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In [11], it introduces sprays of constant curvature and a spray G of constant curvature is weakly Ricci constant (the Ricci curvature is constant along any geodesic of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' For two pointwise projectively related sprays, they have same geodesics as point sets and their geodesic parameters are closely related by the projective factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Starting from this fact, we can determine a projectively flat spray with weak Ricci constant at geodesic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' 1 We consider a projectively flat spray manifold (G, M), that is, Gi = �Gi + Pyi, (1) where �G is a locally Minkowski spray on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 If the spray G in (1) is weakly Ricci constant Ric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='0 = 0 or of constant curvature, then along any geodesic x = x(s) of G, P(s) := P(x(s), x′(s)) is given by one of the following cases: P(s) = 1 s + κ, P(s) = −c · tan(cs + κ), P(s) = −c(1 − κe2cs) 1 + κe2cs , (2) where c, κ are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Further, if G is complete, then P(s) is given by P(s) = −c(1 − κe2cs) 1 + κe2cs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (3) In Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1, we can further give the relation between the geodesic parameters of G and �G by (2) (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' The family of geodesics of an n-dimensional spray considered as point sets or paths is dependent on 2(n − 1) free constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' A path space is a family of curves satis- fying certain conditions (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We can freely give many interesting path spaces, especially in dimension two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Starting from a path space, we can construct its corresponding spray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2 In an n-dimensional path space G, all paths in a local coordinate system (xi) can be parameterized under a variable t with 2(n−1) free constant parameters u, v as follows: x = x(t) = σ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' u, v), (u, v ∈ Rn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (4) Further, the parametric equation (4) induces a spray G whose geodesics are given by (4) with t as its geodesic parameter, and if a new variable s = s(t) = s(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' u, v) is given with s′(t) > 0, then it gives a spray ¯G ∈ Proj(G) with s as its geodesic parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' If a family of curves can be parameterized in the form (4), then with an auxiliary pa- rameter c > 0 multiplied by t in (4), we can obtain the corresponding spray by eliminating the parameters u, v, c, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We give some examples to show how to solve the sprays from given path spaces (see Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='5-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In the study of rigidity issues on a Finsler or spray manifold, it is important to assume that the (Finsler) spray in consideration be (positively/negatively) complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' A given spray is not necessarily (positively/negatively) complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' So a natural problem is whether a spray can be projectively (positively/negatively) complete or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We solve this problem under certain conditions in the following result (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3 Let G be a spray on a manifold M with its each geodesic x = x(t) being defined on the maximal interval I given by one case of the following I = (a, b), or (a, +∞), or (−∞, b), (5) where a = a(u, v) < 0, b = b(u, v) > 0 with u = x(0), v = x′(0) are C∞ functions on a conical region C of T M \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then G is projectively (positively/negatively) complete on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3, usually we can also put u, v as that in (4) (see Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' If (5) is not satisfied, it is uncertain that G is projectively complete (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We give Examples 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='5 as an application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' A Finsler metric is not necessarily projectively (positively/negatively) complete, namely, if G in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3 is a Finsler spray, the corresponding spray projective to G may not be a Finsler spray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' 2 2 Geodesic parameters in projective relations A spray on M, in our consideration, is a smooth vector field G on a conical region C of T M \\ {0} (an important case is C = T M \\ {0}) expressed in a local coordinate system (xi, yi) in T M as follows G = yi ∂ ∂xi − 2Gi ∂ ∂yi , where Gi are local homogeneous functions satisfying Gi(x, λy) = λ2Gi(x, y) for λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' If C = T M \\ {0}, G is called regular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' otherwise, it is called singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' The integral curves of G projected onto M are the geodesics of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Let x = x(s) be a geodesic of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then it satisfies the following ODE: d2xi ds2 + 2Gi(x, dx ds ) = 0, where s is called a geodesic parameter of the geodesic x = x(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Reparameterizing a geodesic x = x(s) by a general parameter t with ds/dt > 0, we have d2xi dt2 + 2Gi(x, dx dt ) = γ(t)dxi dt , (6) where γ(t) is given by γ(t) = d2s dt2 �ds dt = − d2t ds2 � ( dt ds)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (7) Let G, ¯G be two sprays pointwise projectively related by ¯Gi = Gi + Pyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Let x = x(t) be a geodesic of G or ¯G as a point set for a general parameter t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then along the geodesic x = x(t), it follows from (6) and (7) that 2P(t) = ¯s′′(t) ¯s′(t) − s′′(t) s′(t) , � P(t) := P(x(t), x′(t)) � , (8) where s, ¯s are the geodesic parameters of the curve x = x(t) in G, ¯G respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In particular, along a geodesic x = x(s) of G, it follows from (8) that 2P(s) = ¯s′′(s) ¯s′(s) , � P(s) := P(x(s), x′(s)) � , (9) If we express the geodesic x = x(s) of G as the geodesic x = x(¯s) of ¯G, by (9), we have 2P(¯s) = 2P(x(s), x′(s))ds d¯s = ¯s′′(s) � ¯s′(s) �2 , � P(¯s) := P(x(¯s), x′(¯s)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (10) So if P(s) or P(¯s) is known, the relation ¯s = ¯s(s) can be obtained from (9) or (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 Let F be the Funk metric on a strongly convex domain Ω ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Define a projectively flat spray G by Gi = Pyi, P := cF, where c is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Any geodesic x = x(t) (as a point set) of G is given by x = x(t) = vt + u, � − 1 F(u, −v) < t < 1 F(u, v) � , 3 where u, v ∈ Rn are constant vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We have F(vt + u, v) = F(u, v) 1 − tF(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (11) Let s be a geodesic parameter of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then by (9) and (11) we have s′′(t) s′(t) = 2cF(vt + u, v) = 2cF(u, v) 1 − tF(u, v), (12) integration of which with s(0) = 0 gives s = s(t) = � κ ln � 1 − tF(u, v) � , (c = 1 2), κ � 1 − � 1 − tF(u, v) �1−2c� , (c ̸= 1 2), (13) where κ is a constant with κ < 0 for c ≥ 1/2, and κ > 0 for c < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Thus the spray is positively complete for c ≥ 1/2, and any geodesic is defined on a finite open interval for c < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Besides, the spray G is (locally) metrizable if and only if c = 0, 1, 1/2 (see [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2 In Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1, if the spray G is given by Gi(y) := Pyi, P := c � F(y) − F(−y) � , then by (11) and F(vt + u, −v) = F(u, −v) 1 + tF(u, −v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (14) it follows from (9) that s′′(t) s′(t) = 2cF(u, v) 1 − tF(u, v) − 2cF(u, −v) 1 + tF(u, −v), integration of which with s(0) = 0 gives s = s(t) = κ � t 0 �� 1 − tF(u, v) �� 1 + tF(u, −v) ��−2c dt, (15) where κ > 0 is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' From (15), it is clear to conclude that G is complete if c ≥ 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' s is bounded in a finite open interval if c < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' 3 Projective flat sprays with weak Ricci constant For a spray G, the Riemann curvature tensor Ri k is defined by Ri k := 2∂kGi − yj(∂j ˙∂kGi) + 2Gj( ˙∂j ˙∂kGi) − ( ˙∂jGi)( ˙∂kGj), where we define ∂k := ∂/∂xk, ˙∂k := ∂/∂yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' The trace of Ri k is called the Ricci curvature, Ric := Ri i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' For a spray tensor T = Tidxi as an example, the horizontal and vertical derivatives of T with respect to Berwald connection are given by Ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='j = δjTi − TrGr ij, Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='j = ˙∂jTi, (δi := ∂i − Gr i ˙∂r, Gk ir := ˙∂r ˙∂iGk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' 4 A spay is called weakly Ricci constant if Ric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='0 := Ric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='ryr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' A spray G is said to be of constant curvature if Ri k is given by Ri k = Rδi k − τkyi with ([11]) τi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='k = 0 ( ⇔ R = τk = 0, or R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='i = 0(R ̸= 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' By definition, it is clear that a spray of constant curvature is weakly Ricci constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' For two pointwise projectively related sprays G, ¯G with ¯Gi = Gi + Pyi, their Ricci curvatures Ric, ¯Ric are related by ¯Ric = Ric − (n − 1)(P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='0 − P 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (16) We consider a projectively flat spray manifold (G, M) given by (1), that is, Gi = �Gi + Pyi, where �G is a locally Minkowski spray on M ( �G has local straight lines as geodesics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' If G is weakly Ricci constant, then we can determine the projective factor P along geodesics, which is shown in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 : By (16) and �Gi = Gi − Pyi, the Ricci curvature Ric of G is given by Ric = −(n − 1)(P 2 + P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Therefore, Ric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='0 = 0 is equivalent to P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='0 + 2PP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then along a geodesic x = x(s) of G, we have P ′′(s) + 2P(s)P ′(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' whose solution is given by one of the three cases in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Further, if G is complete, it is clear that (3) follows from (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' If the spray G in (1) is weakly Ricci constant Ric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='0 = 0, then applying (2) and (10), we obtain the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 Let the spray G in (1) be weakly Ricci constant (esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' of constant curva- ture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' For any geodesic σ, let s and t be the geodesic parameters of σ with respect to G and �G respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then s = s(t) is given by one of the following cases: s = at, (a > 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' s = b ln(1 + at), (ab > 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (17) s = bt 1 + at, (a ̸= 0, b > 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' s = c � arctan(at + b) − arctan b � , (ac > 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (18) s = c ln 1 + bt 1 + at, � (b − a)c > 0, ab ̸= 0 � , (19) where a, b, c are constant, and in (19), it further requires s′(t) > 0 (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Proof : By (10) we need to solve the following ODE with initial conditions: s′′(t) s′(t) = 2P(s)s′(t), (s(0) = 0, s′(t) > 0), integration of which gives s′(t) = ae2 � P (s)ds, � e−2 � P (s)dsds = at + b, (20) 5 where a, b are two constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Now P(s) is given by (2) from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1, and thus we can obtain s = s(t) by plugging P(s) into (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' If P(s) = 0, then (20) gives s = at+b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Since s(0) = 0, s′(t) > 0, we obtain s = at (a > 0), which gives the first formula in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' If P(s) = c ̸= 0 is constant, then (20) gives the second formula in (17) with ab > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' If P(s) is given by the first formula in (2), then (20) gives s = −κ + 1 at + b, which can be rewritten as the form of the first formula in (18) by s(0) = 0, s′(t) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' If P(s) is given by the second formula in (2) (c ̸= 0), then (20) gives s = −κ − arctan(at + b) c , which can be rewritten as the second formula in (18) by s(0) = 0, s′(t) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' If P(s) is given by the third formula in (2) (cκ ̸= 0), then (20) gives s = 1 2c ln � 1 at + b − 1 κ � which can be rewritten as the formula in (19) by s(0) = 0, s′(t) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In (19), by s′(t) > 0, we have further restriction on the constant parameters a, b, c, which is shown in the following remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2 In (19), let t be defined on the maximal interval (κ1, κ2) with κ1 < 0 < κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' It is easy to conclude the following cases from s′(t) > 0: a > 0, b > 0 : � t ∈ (κ1, κ2) ⊂ (− 1 a, +∞), (b < a) t ∈ (κ1, κ2) ⊂ (− 1 b, +∞), (b > a), a < 0, b < 0 : � t ∈ (κ1, κ2) ⊂ (−∞, − 1 a), (b > a) t ∈ (κ1, κ2) ⊂ (−∞, − 1 b), (b < a), a > 0, b < 0 : t ∈ (κ1, κ2) ⊂ (−1 a, −1 b), a < 0, b > 0 : t ∈ (κ1, κ2) ⊂ (−1 b , −1 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2, we directly obtain the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3 If the spray G in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 (P ̸= 0) is complete, then s = s(t) is given by one of the following two cases: s = b ln(1 + at), (ab > 0), (21) s = c ln 1 + bt 1 + at, � (b − a)c > 0, ab < 0 � , (22) where in (21) and (22), we respectively have t ∈ (−∞, −1 a) if a < 0, and t ∈ (−1 a, +∞) if a > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' t ∈ (−1 a, −1 b) if a > 0, b < 0, and t ∈ (−1 b , −1 a) if a < 0, b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' 6 Now in the following, we give some projectively flat sprays to verify the above results on the projective factors and the geodesic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='4 Consider the spray G in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' A direct computation shows that G is weakly Ricci constant or of constant curvature if and only if c = 0, 1, 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Let x = x(t) = vt + u be a geodesic (as a point set) of G, and the geodesic parameter s in G is given by (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then it follows from (10) and (13) that P = cF is given by P(s) = � − 1 2κ, (c = 1 2) c 2c−1 · 1 s−κ, (c ̸= 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (23) It is clear from (23) that P(s) is in one of the forms in (2) if and only if c = 0, 1, 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Meanwhile, s = s(t) is given in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 if and only if c = 1, 1/2, 1, and in this case, s = s(t) is in the respective forms shown in (17) and the first formula in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='5 Let G be the spray in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2 with c = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' G is complete and it is of constant curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then it follows from (15) that s = κ ln 1 + tF(u, −v) 1 − tF(u, v) , (24) where κ > 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In this case, s = s(t) is in the form (22), and it is easy to verify that P(s) is in the form of the third formula in (2) by plugging (24) into (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='6 Let G be a spray on Rn defined by Gi := Pyi, P := − ⟨x, y⟩ 1 + |x|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' G is metrizable and it is of constant curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Let x = x(t) = vt + u be a geodesic (as a point set) of G, and by (9), the geodesic parameter s of G satisfies s′′(t) s′(t) = − 2(|v|2t + ⟨u, v⟩) 1 + |u|2 + 2⟨u, v⟩t + |v|2t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Solving the ODE, we obtain s = s(t) = κ1 + κ2 arctan |v|2t + ⟨u, v⟩ � (1 + |u|2)|v|2 − ⟨u, v⟩2 , where κ1, κ2 are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' It is clear that s = s(t) is in the form of the second formula in (18) if s(0) = 0, and P(s) is in the form of the second formula in (2) by plugging the above s = s(t) into (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' 4 Construction of sprays from geodesics Given a family of curves G on a manifold, if G can constitute the geodesics of a spray G on M, how can we solve G (at least locally)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' A spray induces a (local) semispray and two pointwise projectively related sprays induce a same semispray (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' A semispray can also be considered as a special parameterized family of curves, which forms a path space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In this section, we will start from a path space and introduce some ways to construct sprays based on a path space and its parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We call it the geodesic method of construction of sprays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Similarly to a spray, a path space G on a manifold M is usually defined on a conical region C of T M \\ {0} (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1), and G is called singular if C ̸= T M \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' 7 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 Let G be a family of C∞ parameterized curves (called paths) on an n- dimensional manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' G or (M, G) is called an n-dimensional path space if on a conical region C of T M it satisfies (i) for y ∈ Cx, there is a curve σ : (−ǫ, ǫ) → M in G with σ′(0) = y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (ii) for any σ, τ in G with σ′(0) = τ ′(0), σ and τ coincide in a small intervals of 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (iii) if a curve σ is in G, then for any constants λ > 0 and to, the curve η is also in G, where η is defined by η(t) := σ(λt + to).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' An equivalent version of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 in regular case is refereed to [9] (P52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2 Consider a set G of a family of curves x = x(s) on R2 in the form x(s) = σ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo), � x(0) = xo = (a, b), x′(0) = yo = (u, v) � , σ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo) := (a, b) + (u, v)s − (0, 1) �1 3u3s3 + au2s2� , where a, b, u, v are arbitrary parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' It can be directly verified that G is a path space on R2, since Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 (i) (ii) automatically hold, and Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 (iii) follows from σ(λs + so;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo) = σ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' ˆxo, ˆyo), where we define xo = (a, b), yo = (u, v), ˆxo = (ˆa,ˆb), ˆyo = (ˆu, ˆv), ˆa := a + uso, ˆb := b + vso − 1 3u2(3a + uso)(so)2, ˆu := λu, ˆv := λv − λu2so(2a + uso).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' For a path space G, we have different ways to parameterize the paths in G under a para- metric variable and some constant parameters (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2 satisfies (25) and (26) in the following Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3 with f(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo) := −(0, 1) �1 3u3s3 + au2s2� , xo = (a, b), yo = (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3 An n-dimensional path space (M, G) is locally expressed as the following family of curves x = x(s) with arbitrary constant parameters xo, yo ∈ Rn: x(s) = σ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo) = xo + yos + f(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo), (25) where f is a smooth function satisfying f(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo) = f ′(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo) = 0 and f(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' ˆxo, ˆyo) = f(λs + so;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo) − f(so;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo) − λf ′(so;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo)s, (26) � ˆxo := xo + yoso + f(so;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo), ˆyo := λyo + λf ′(so;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' It is clear that the collection of geodesics of a spray naturally forms a path space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Shen proves the converse in the following lemma ([9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We also give the proof for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='4 A path space G induces a spray G with the set of geodesics of G being G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Proof : Let (G, M) be a path space on a conical region C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' For a given y ∈ Cx, there is a curve σ : (−ǫ, ǫ) → M in G with σ(0) = x, σ′(0) = y by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Define Gi(y) := −1 2 d2σ ds2 (0), 8 which is independent of the choice of σ by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We are going to verify that G is a spray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' For any constant λ > 0, let η(s) := σ(λs) ∈ G (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 (iii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then we have Gi(λy) = −1 2 d2η ds2 (0) = −1 2λ2 d2σ ds2 (0) = λ2Gi(y), which implies that Gi is positively homogeneous of degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Further, for any η : (a, b) → M in G and any fixed t ∈ (a, b), define γ(s) := η(s + t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then we have η′(t) = γ′(0), η′′(t) = γ′′(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' So by the definition of Gi, we get Gi� η′(t) � = Gi� γ′(0) � = −1 2 d2γi ds2 (0) = −1 2 d2ηi ds2 (t), which implies that η satisfies the following ODE: d2ηi ds2 + 2Gi�dη ds � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Therefore, G is a spray, and the set of geodesics of G coincides with G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='4, by different choices of the parametric variables, it (locally) induces a projective class Proj(G) of G, each of which is projective to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' For a given path space G, it induces a spray G by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then G defines a semispray ˆG (see [9]: P37) and the geodesics of G and ˆG are closely related (see [9]: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Therefore, in G, any path can be locally expressed as xa = xa(x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' u, v), (u, v ∈ Rn−1, 2 ≤ a ≤ n), where u, v are free constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' So all paths in an n-dimensional path space depend only on 2(n−1) free constant parameters, where the Jaccobi determinant is not zero, namely, det �∂xa/∂u ∂xa/∂v ∂ya/∂u ∂ya/∂v � ̸= 0, � ya := dxa dx1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then we obtain Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2 for the construction of sprays based on the parametric equations of path spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' If we write (4) in the form x(t) = σ(λt + µ, u, v), (27) where λ, µ are constant numbers, then under the 2n constant parameters λ, µ, u, v, this family of curves satisfies Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 (i)(ii)(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' For instance, the 2-dimensional path space in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2 can be written as the following family of curves x(s) = τ(λs + µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' bo, vo) = (0, bo) + (1, vo)(λs + µ) − 1 3(0, 1)(λs + µ)3, � λ := u, µ := a, vo := v u + a2, bo = b − avo + 1 3a3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2, if the set A of a family of curves on an n-dimensional manifold defines a path space, then A just depends on 2(n − 1) free constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' For example, in Rn, all circles with fixed radius cannot define a path space when n ≥ 3, because in this case, the circles depend on more than 2(n − 1) free constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' 9 Now we introduce a method of constructing a spray G determined by a path space considered as the geodesics of G, which is similar to Okubo’s method for the construction of a Finlser metric from a hypersurface as its indicatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We can start from a family of curves given by (25) or (4) to determine a corresponding spray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Method (I): For a family of curves given by (25) satisfying (26), actually we can reduce one constant parameter since (25) can be written as (if y1 o ̸= 0) x(t) = σ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, ¯yo) = xo + ¯yot + f(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, ¯yo), � t := y1 os, ¯ya o := ya o/y1 o, ¯yo := (¯ya o) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Let a path space be determined by (25) and we put x = xo + yos + f(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo), y(= dx ds ) = yo + f ′(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo)), (28) Gi := −1 2 d2x ds2 = −1 2f ′′(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' xo, yo)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (29) Then we obtain a spray G from (29) by eliminating xo, yo, s in (29) from (28), where s is a geodesic parameter of the spray G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Method (II): Suppose that a family of curves are given by the parametric equation (4) with 2(n − 1) free constant parameters u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' This case is more convenient to construct sprays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' With an auxiliary parameter c > 0, we put x = σ(cs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' u, v), y = dx ds = cdσ dˆs (cs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' u, v), ˆs := cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (30) Theoretically, we can express c, s, u, v as functions of x, y from (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then plugging them into the following Gi := −1 2 d2xi ds2 = c2 d2σi dˆs2 (cs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' u, v), (31) we obtain a spray G given by (31), where s is a geodesic parameter of the spray G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Now in the following Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='5-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='8, we use Method (I) or Method (II) to show how we construct sprays from given path spaces by eliminating the corresponding parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='5 Consider a set G of a family of curves on R3: x(s) = (a, b, c) + (u, v, w)s − (0, 1, 0)h(s), � h(s) := −1 3(u3 + w3)s3 − (au2 + cw2)s2� , where a, b, c, u, v, w are constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' G is a path space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' By (28) we get x1 = a + us, x3 = c + ws, y1 = u, y3 = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (32) By (29), the induced spray G is given by G1 = −1 2 d2x1 ds2 = 0, G3 = −1 2 d2x3 ds2 = 0, G2 = −1 2 d2x2 ds2 = (u3 + w3)s + (au2 + cw2) = (u3 + w3)s + � (x1 − us)u2 + (x3 − ws)w2� � by (32) � = x1u2 + x3w2 = x1(y1)2 + x3(y3)2 � by (32) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' 10 G has zero Riemann curvature and so it is metrizable (a Finsler spray) ([11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='6 Let G be the set of all circles with fixed radius r on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We parameterize G by x1(s) = a + r cos s, x2(s) = b + r sin s, where a, b are arbitrary constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' G depends on just two free constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2, G defines a spray G on R2 with s as a geodesic parameter of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We show the spray as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' With an auxiliary parameter c > 0, it follows from (30) that x1 = a + r cos cs, x2 = b + r sin cs, y1 = −cr sin cs, y2 = cr cos cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (33) Then plugging the latter two formulas of (33) into (31) yields a spray G given by G1 = −c2r cos cs = −1 r y2� (y1)2 + (y2)2, G2 = −c2r sin cs = 1 r y1� (y1)2 + (y2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' This circle spray first appears in [9] (P49), and even locally it is not metrizable ([11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='7 Consider a family of semicircles G on the positive semi-plane R2 + with center on x1-axis and arbitrary radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Note that G is singular at the direction parallel to x2-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We can parameterize G by x1 = a + b coss, x2 = b sin s, (x2 > 0, b ≥ 0), where a, b are arbitrary constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' G depends on just two free constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2, G defines a spray G on R2 + with s as a geodesic parameter of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' With an auxiliary parameter c > 0, by (30) we get x1 = a + b cos cs, x2 = b sin cs, y1 = −bc sincs, y2 = bc coscs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (34) Then similarly, by the elimination of the parameters a, b, c, s in (31) from (34), the spray G with s being a geodesic parameter is given by G1 = −y1y2 2x2 , G2 = (y1)2 2x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (35) The spray G is regular on R2 + (any straight lines parallel to x2-axis are geodesics of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' G is of isotropic curvature, and locally it is not metrizable by the method in [2, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='8 Let Bn be the unit ball in Rn and G be all circle arcs in Bn which are perpendicular to the boundary Sn−1 = ∂Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Let s be the arc-length parameter of a circle arc induced by the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' What is the spray G induced by G with s being a geodesic parameter of G (see Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='4 in [9])?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We will show that G is given by Gi = ⟨x, y⟩yi − |y|2xi 1 − |x|2 , (36) which is not metrizable by [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Now for arbitrarily given p, q ∈ Sn−1, there is a circle arc γ in G, in which γ is perpendicular to Sn−1 at p, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Let C be the circle with γ ⊂ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' The center and radius of C are respectively given by τ(p + q), |p − τ(p + q)|, (τ := (1 + pq)−1), 11 where pq is the Euclidean inner product of p, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then γ is parameterized by the equation x(s) = x(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' p, q) = [p − τ(p + q)] cos s + |p − τ(p + q)|p sin s + τ(p + q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (37) Since there are just 2(n − 1) free constant parameters in (37), the family of curves in the form (37) define a path space by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Now based on (30) and (31), we can give the spray G from (37) with s being a geodesic parameter of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' With an auxiliary parameter c > 0, by (30) we put x = [p − τ(p + q)] cos cs + |p − τ(p + q)|p sin cs + τ(p + q), (38) y = −c[p − τ(p + q)] sin cs + c|p − τ(p + q)|p cos cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (39) By (31) we have 2Gi := c2� [p − τ(p + q)]i cos cs + |p − τ(p + q)|pi sin cs � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (40) By a direct lengthy computation, we can eliminate the parameters p, q, c, s in (40) from (38) and (39) (the details are omitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Finally, the spray G is given by (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' 5 Projectively complete sprays For a given spray G, if we know the general solutions of all geodesics of G, then under another parameter as a geodesic parameter, we can determine a corresponding spray projectively related to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Now suppose that the general solutions of geodesics of G are locally given by x = σ(t) = σ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' u, v), � u, v ∈ Rn−1� , (41) where t is a geodesic parameter of G and u, v are free constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Sometimes, it is also convenient to put u = σ(0), v = σ′(0) for the elimination of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Make a change of the variables from t to s with t = t(s) = t(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' u, v), dt ds > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (42) With an auxiliary parameter c > 0, we put x = σ(t(cs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' u, v), y = dx ds = cdσ dt dt ds, (43) where dt/ds, as a function of s, takes the value at cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Further, we have d2xi ds2 = c2 d2σi dt2 � dt ds �2 + c2 dσi dt d2t ds2 = −2Gi(x, dσ dt )c2� dt ds �2 + c2 dσi dt d dt � dt ds � dt ds = −2Gi(x, y) + c d dt � dt ds � yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (44) Expressing c, t in terms of x, y from (43), and then plugging c, t into (44), we obtain a spray ¯G given by ¯Gi = Gi − 1 2 d dt � dt ds � cyi = Gi + Pyi, (45) � P = P(x, y) := −1 2 d dt � dt ds � c � , with s being a geodesic parameter of ¯G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' 12 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='1 Suppose that the general solutions of geodesics of a spray G are given by (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Let s be another parameter related to t by (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then a spray ¯G projective to G with s being its geodesic parameter is given by (45), where c, t are determined by (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Under certain condition, a spray can be projectively (positively/negatively) complete, which is shown in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Now we give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3 : Let G be a spray on a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' For an arbitrary geodesic x = x(t), suppose that t belongs to the maximal interval I given by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' If I = (a, +∞) or I = (−∞, b), we respectively make a change of the variables from t to s by s = ln(1 − t a), or s = − ln(1 − t b), (46) either of which gives s(0) = 0, s′(t) > 0 and the maximal interval of s with s ∈ (−∞, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' If I = (a, b), make a change by (46) and then we respectively have s ∈ � − ∞, ln(1 − b a) � , or s ∈ � − ln(1 − a b ), +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' If I = (a, b), make a change of the variables from t to s by s = ln 1 − t/a 1 − t/b , or s = tan � π b − a(t − a + b 2 ) � + tan �b + a b − a π 2 � , (47) either of which gives s(0) = 0, s′(t) > 0 and the maximal interval of s with s ∈ (−∞, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Therefore, by the change (46) or (47), we obtain a (positively/negatively) complete spray which is projective to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' As an application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3, we give the following Examples 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='4 to show the construction of the (positively/negatively) complete sprays projective to given sprays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2 Let F be the Funk metric on a strongly convex domain Ω ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' The Minkowski spray G = 0 on Ω has its geodesics given by x(t) = vt + u, � − 1 F(u, −v) < t < 1 F(u, v) � , where u, v ∈ Rn are arbitrary constant vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' By (46), put t = t(s) as s = − ln � 1 − tF(u, v) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (48) With s being a geodesic parameter, we obtain a projectively flat and positively complete spray ¯G, which will be shown to be the Finsler spray induced by F, namely, ¯Gi = 1 2Fyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (49) Actually, it follows from (2) and (11) that dt ds = 1 F(u, v) − t = 1 F(vt + u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (50) Then (43) gives x = vt + u, y = cv dt ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (51) 13 It is clear from (51) and (50) that F(x, y) = cF(vt + u, v) dt ds = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (52) Therefore, by (45), (50) and (52), the spray G is given by (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3 In Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2, by (47), put t = t(s) as s = ln 1 + tF(u, −v) 1 − tF(u, v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (53) With s being a geodesic parameter, we obtain a projectively flat and complete spray ¯G, which will be shown to be the Finsler spray induced by the Klein metric ¯F(x, y) := � F(x, y) + F(x, −y) � /2, namely, ¯Gi(x, y) = 1 2 � F(x, y) − F(x, −y) � yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (54) Firstly, by (53), we get dt ds = � 1 − tF(u, v) �� 1 + tF(u, −v) � F(u, v) + F(u, −v) , (55) d dt � dt ds � = F(u, −v) − F(u, v) − 2tF(u, v)F(u, −v) F(u, v) + F(u, −v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (56) Secondly, (43) gives x = vt + u, y = cv dt ds, from which we have F(x, y) = F(vt + u, v)c dt ds, F(x, −y) = F(vt + u, −v)c dt ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (57) Plugging (11), (14) and (55) into (57), we obtain c = F(x, y) + F(x, −y), t = F(x, y)F(u, −v) − F(x, −y)F(u, v) F(u, v)F(u, −v)[F(x, y) + F(x, −y)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (58) Finally, by (58) and (56), it follows from (45) that the spray ¯G is given by (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='4 For the spray G in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='2, we will introduce a different way from that in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3 to make G be complete, which is actually to use (46) to make complete the Finsler spray induced by the Funk metric F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' For a geodesic x(t) = vt + u of G, put s = ln � 1 − ln(1 − tF(u, v)) a � , a := ln � 1 + F(u, v) F(u, −v) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' In a similar way to that for the computation in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='3, we obtain a projectively flat and complete spray ¯G given as follows: ¯Gi(y) = Gi F (y) + 1 2 F(y) ln F (−y) F (y)+F (−y) yi, � Gi F (y) := 1 2F(y)yi� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' ¯G is of scalar curvature and actually we can verify that ¯G is not metrizable by using the method in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' 14 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='5 For the family of semicircles G on R2 + as shown in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='7, we can parameterize them in the following form x1 = u − v sin t, x2 = v cos t, (x2 > 0, v ≥ 0), (59) where u, v are arbitrary constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We have shown in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='7 that the spray G determined by G is given by (35), that is, G1 = −y1y2 2x2 , G2 = (y1)2 2x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (60) We can make G be projectively complete on the conical region C with the direction (0, 1) being deleted from T R2 + \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Since −π/2 < t < π/2 for any u, v in (59), by (47), we let s = tan t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Then by (45), we get a complete spray ¯G projective to G with the projective factor P being given by P = c � − 1 2 d dt � dt ds �� t=cs = c � cos t sin t � t=cs = c2s 1 + c2s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' (61) Now it follows from (43) that x1 = u − vcs √ 1 + c2s2 , x2 = v √ 1 + c2s2 , y1 = −vc (1 + c2s2)3/2 , y2 = −bc2s (1 + c2s2)3/2 , from which we get s = − x2y2 (y1)2 + (y2)2 , c = −(y1)2 + (y2)2 x2y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Plugging s, c in the above into (61) yields P = −y2/x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Thus the spray ¯G is given by ¯G1 = G1 + Py1 = −3y1y2 2x2 , ¯G2 = G2 + Py2 = (y1)2 − 2(y2)2 2x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' ¯G is complete on the conical region C but not complete in the direction (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' We don’t know whether the spray G in (60) can be projectively complete or not on T R2 \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content=' Besides, ¯G is of isotropic curvature, and locally it is not metrizable by the method in [2, 11].' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} +page_content='cn 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfofhG/content/2301.00507v1.pdf'} diff --git a/2tAzT4oBgHgl3EQfRvvX/content/tmp_files/2301.01222v1.pdf.txt b/2tAzT4oBgHgl3EQfRvvX/content/tmp_files/2301.01222v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..196994550883ea6599f266f672bfe85ea039f1e2 --- /dev/null +++ b/2tAzT4oBgHgl3EQfRvvX/content/tmp_files/2301.01222v1.pdf.txt @@ -0,0 +1,1230 @@ +A Multi-Source Information Learning Framework +for Airbnb Price Prediction +Lu Jiang1, Yuanhan Li1, Na Luo1, Jianan Wang2,∗, Qiao Ning3,∗ +1Information Science and Technology, Northeast Normal University, Changchun +2College of Physics, Northeast Normal University, Changchun +3Information Science and Technology, Dalian Maritime University, Dalian +{jiangl761, liyh447, luon110, wangjn}@nenu.edu.cn, ningq669@dlmu.edu.cn +Corresponding author* +Abstract—With the development of technology and sharing +economy, Airbnb as a famous short-term rental platform, has +become the first choice for many young people to select. The +issue of Airbnb’s pricing has always been a problem worth +studying. While the previous studies achieve promising results, +there are exists deficiencies to solve. Such as, (1) the feature +attributes of rental are not rich enough; (2) the research on +rental text information is not deep enough; (3) there are few +studies on predicting the rental price combined with the point of +interest(POI) around the house. To address the above challenges, +we proposes a multi-source information embedding(MSIE) model +to predict the rental price of Airbnb. Specifically, we first selects +the statistical feature to embed the original rental data. Secondly, +we generates the word feature vector and emotional score +combination of three different text information to form the text +feature embedding. Thirdly, we uses the points of interest(POI) +around the rental house information generates a variety of spatial +network graphs, and learns the embedding of the network to +obtain the spatial feature embedding. Finally, this paper combines +the three modules into multi source rental representations, and +uses the constructed fully connected neural network to predict +the price. The analysis of the experimental results shows the +effectiveness of our proposed model. +I. INTRODUCTION +Accommodation sharing systems are being introduced to +more and more cities recently, and therefore they have gener- +ated huge amounts of data. Airbnb is an online marketplace +for sharing home and experience which is suffering from the +chaotic pricing problem. Tenants need to know the reasonable +price of this rental house to prevent being deceived. The +homeowner needs to customize a reasonable price for their +short-term rental house to attract more customers. Therefore, +airbnb price prediction plays a key role in accommodation +sharing systems. However, rapid increase in the number +of tenants and homeowners makes traditional manual-based +methods [1] time-consuming and inefficient. Computational +methods have received more attention for accurate airbnb price +prediction [2]. +Computational methods for price prediction can be mainly +divided into two categories: (1) feature-based methods [3], and +(2) deep learning methods [4, 5]. In feature-based methods, +various types of features extraction strategies are utilized to +extract price correlated features for tenants and homeowners. +Feature-based methods transform price prediction into a ma- +chine learning methods, such as support vector machine(SVM) +and random forest. For instance, in order to distinguish from +the traditional method of formulating prices, Li et al. selects +rough set (RS) and SVM algorithms to establish a new math- +ematical model of pricing on the basis of hedonic price [6]. +PR Kalehbastiet al. proposed a price prediction model us- +ing machine learning, deep learning, and natural language +processing techniques to embed the features of the rentals, +owner characteristics, and the customer reviews [7]. Deep +learning methods, which use multi-layer neural network to +map the correlation between input features and output results. +For instance, Chen et al. applied auto regressive integrated +moving average model to generate the baseline while LSTM +networks to build prediction model [8]. +However, the research of airbnb price prediction based on +feature-based methods consider the single feature in most +cases. With the development of representation learning [9, 10], +the spatial embedding [11, 12] have received more attention. +There has been work to model the statistical features, text +feature and spatial features related to housing prices, but there +is no unified framework to integrate the above features. Based +on the above disadvantages, we proposes a prediction model +based on multi-source information embedding to study the +Airbnb price problem. The major contributions are summa- +rized below. +• Firstly, in order to obtain the best feature set, this paper +selects the features of the house itself to obtain statistical +information features. +• Secondly, the text information in this paper is divided into +three categories, and the house description and landlord +introduction are converted into feature matrix. The tenant +reviews are then converted into sentiment scores about +each house. +• Then, we uses different types of point-of-interest (POI) +data and houses to form various spatial network graphs +and learns their network embeddings to obtain spatial +information features. +• Finally, we combines these three types of feature embed- +dings are combined into multi-resource housing features +as input, and the neural network constructed in this paper +is used for price prediction. The effectiveness of our +model is demonstrated with two real data. +arXiv:2301.01222v1 [cs.LG] 1 Jan 2023 + +II. PRELIMINARY +We first introduce some key definitions and the problem +definition. Then, we present the overview of the proposed +method. +A. Definitions and Problem Statement +Definition 1. Statistic Feature The statistics feature con- +structed by our model is S =(s1, s2, ..., sn), where si +is the preprocessed listing features includes ’host since’, +‘host is superhost’, ’verification’, etc. +Definition 2. Text Feature There are three types of text +features: listing description, host introduction, and tenant +review. We convert listing description and host introduction +into feature vector L =(l1, l2, ..., ln) and H =(h1, h2, ..., hn), +and transform the tenant review to sentiment score R +=(r1, r2, ..., rn). Thus, we define the text features as T = +(L, H, R). +Definition 3. Spatial Feature We first combine each rental +house and the POI with in 1,000m around it into a spatial +network G = (V, E, W). Then, we learn network embedding +through SDNE [13], and get the spatial feature matrix P +=(p1, p2, ..., pn). +Definition 4. Problem Statement In this paper, we study the +problem of airbnb price prediction. We formulate the problem +as a multi-source feature embedding task. Formally, we aim +to find a mapping function f : (S, T, P) → V that takes +the statistic feature S, text feature T, spatial feature P as +input, and outputs a unified vectorized representations V , for +predicting the specific listing price. +B. Framework Overview +Figure 1 shows an overall framework for the multi-source +feature embedding. Specifically, we embed the original data +from three aspects. (1) For the statistical feature embedding, +we uses Lasso CV to select the feature set with the rental +house feature. (2) For the text feature embedding, we divides +the text feature into three categories, include house descrip- +tion, landlord introduction and tenant comments. Through the +negative sampling CBOW model, the house description and +landlord introduction are converted into word feature vectors, +and the Bayesian model based on naive Bayesian principle is +used to convert tenant comments into emotional scores, we +combine them as the text feature. (3) For the spatial feature +embedding, we collects different types of POIs, and combines +the POI of each house and the surrounding area within 1,000m +into a spatial network. Through the SDNE model to learn the +spatial feature. (4) Three different features are combined into +a multi-source feature and input into the neural network to +obtain the final rental price. +III. MULTI-SOURCE INFORMATION LEARNING +In this section, we introduce the core architecture of our +framework as follows: (1) statistic feature embedding; (2) text +feature embedding; (3) spatial feature embedding. +A. Statistics Feature Embedding +Each +house’s +statistic +feature +is +represented +by +a +245- +dimensional +vector +which +describes +the +listing +of +a +house, +including +listing id, +host id, +host since, +host response rate, host is superhost, host has profile pic, +host identity verified, +bathrooms, +bedrooms, +latitude, +longitude, accommodates, security deposit, guests included, +verification, etc. +We use the Lasso CV to do the feature set selection. The +loss function is defined as follows: +obj = 1 +2 +n +� +i=1 +� +yi − wT xi +�2 + α +m +� +j=1 +|wi| +(1) +where n is the number of houses, m is the number of +parameters, α is the regularization coefficient, α �m +j=1 |wi| is +the L1 regularization term, yi is the rental price, xi is the +statistical features of rental housing, w is the coefficient matrix +of rental housing features, xi = si. Statistical feature matrix +S =(s1, s2, ..., sn). Lasso CV can compress the coefficients of +unimportant features to 0, realizing the purpose of feature se- +lection, and ultimately leaving the important statistical feature +set that this paper wants. +B. Text Feature Embedding +We extract three types of text data from the original data, +there are listing description, host introduction and tenant re- +view. Listing description mainly about introducing the location +of the house, surrounding environment, indoor layout and +housing regulations, etc. Host introduction mainly introduces +the age, height, occupation, hobbies and personality of the +host, and tenant review expresses the tenants’ feelings about +housing rentals and the evaluation of the host’s attitudes. Since +tenant review contain emotional value, we use two different +methods to model text feature. We first use CBOW [14] +model to embed the text feature of listing description, host +introduction. We selects the Wikipedia Chinese thesaurus after +preprocessing as the training corpus W, the objective function +is defined as follows: +L = +� +c∈W +� +� +�log +� +σ +� +xT +c θc�� ++ +� +u∈NEG(c) +log +� +σ +� +−xT +c θu�� +� +� +� +(2) +Then the above objective function is optimized by using the +random gradient rise method to obtain: +L(c, u) = Lc(u) log +� +σ +� +xT +c θu�� ++[1 − Lc(u)] log +� +1 − σ +� +xT +c θu�� +(3) +Then calculate the gradient of L(c, u) to obtain: +v(˜c) := v(˜c) + η +� +u∈{c}∪NEG(c) +∂L(c, u) +∂xc +(4) + +Geospatial Information Embedding +Text Information Embedding +listing information +host introduction +tenant review +feature vector +feature vector +sentiment score +Original +Listing +Data +preprocess +Feature +Selection +Statistics Information Embedding +Real +Airbnb +Listing +Data +Multi +Source +Listing +Feature +listing +POI +Predict +Price +Fig. 1: Framework Overview. +In this paper, we set the dimension of the word vector as +100. l and h represent the embedding of listing description, +host introduction, respectively. +l = 1 +Z +Z +� +i=1 +v (˜cl) +(5) +h = 1 +z +z +� +i=1 +v (˜ch) +(6) +Therefore, we get the text information feature matrix of the +listing description and host introduction: L =(l1, l2, ..., ln) and +H =(h1, h2, ..., hn). +For the tenant review embedding, since it contains strong +emotional expression, in order to reflect whether the tenants’ +evaluation of the house is positive or negative, we uses the +naive Bayes method to generate the corresponding emotional +score r ∈ [0, 1] for each house, where 0 represents the negative +and 1 represents the positive. Specifically, the probability that +a tenant review text belongs to the positive class can be +expressed as: +P (pos | c1, . . . cd) = P (c1, . . . cd | pos) P(pos) +P (c1, . . . cd) +(7) +After simplifying the above formula, we can obtain: +P (pos | c1, . . . cd) = +1 +1 + γ +(8) +In this work, a text represents a tenant’s review, and a tenant +has many reviews, so the emotional score of a tenant can be +expressed as: +r = 1 +q +q +� +i=1 +P (pos | c1, . . . cd) +(9) +where q represents the number of reviews on a rental, +d represents the total number of words in a review, and +P (pos | c1, . . . cd) represents the probability that the review +belongs to the category of positive emotions. Therefore, the +emotional score vector of tenant reviews can be expressed as +R =(r1, r2, ..., rn). +C. Spatial Feature Embedding +We proposes a method to learn spatial embedding. First, POI +is divided into 8 different types, and the rented houses and the +surrounding different type POIs form a spatial network. Then +learn the network embedding of these spatial graphs through +SDNE model. This method can accurately capture the spatial +features related to important POIs such as scenic spots and +railway stations. +We uses Euclidean distance to calculate the weight W +between house and poi as follows: +W = R · arccos( dis ) · π/180 +(10) +where dis = sin(LatA) sin(LatB) cos(LonA − LonB) + +cos(LatA) cos(LatB), the two types of nodes, A and B, + +Net- +work1 +Learning Network Embedding +Net- +work2 +Net- +work3 +Net- +workkInput +layer +Hidden +layer +Output +layeirepresent the rental houses and POI respectively. LonA and +LonB are their longitudes, LatA and LatB are their latitudes, +and R is the average radius of the earth, taking the value of +6371.004km. +In SDNE model, the encoder is from xi to y(k) +i +, the decoder +is from y(k) +i +to �xi, y(k) +i +is the node embedding of vi, in this +paper,y(k) +i += pi. The formula of encoder is: +y(k) +i += σ +� +W (k)y(k−1) +i ++ b(k)� +(11) +Therefore, the spatial embedding can be expressed as P +=(p1, p2, ..., pn). After we get the statistic embedding, the text +embedding and the spatial embedding. We combine the above +features into a multi-source feature M = (S, T, P), and use +the fully connected neural network to predict the rental price. +We take the multi-source feature matrix M =(m1, m2, ..., mn) +as the input of the neural network, then obtain as follows: +y = wT m + b, +A = σ(y) +(12) +where y is the actual rental price, m is the multi-source +feature, w is the parameter matrix, and b is the offset term. +A is the activation function, we uses ReLU function as the +activation function. At last, the output layer uses a neuron to +output and get the predicted price �yi. +IV. EXPERIMENT +In this section, we first introduce two real dataset and +evaluation metrics. Then, we design experiments to answer +the following three questions: +• Q1. How is the performance of our proposed MSIE in +the airbnb price prediction task? +• Q2. How do the feature combination affect the price +prediction performance? +• Q3. What is the key influence on the airbnb price? +A. Dataset +We collect the dataset from an open online airbnb website. +Table I shows the statistics of our two real airbnb datasets +from two cities: Beijing and Shanghai after preprocess. +TABLE I: Statistics of the data +City +# Houses +# Reviews +Time Period +Beijing +10779 +191876 +01/2017-06/2019 +Shanghai +8638 +159069 +01/2020-07/2021 +Besides, we also collect the POI of the Beijing and Shanghai +in Table II. We divide it into 8 categories, include, Education, +Entertainment, Food, Beverage Shopping, Tourist, Transporta- +tion, Medical Service, and Public Service. +B. Evaluation Metrics +We evaluate the model performances in terms of the fol- +lowing metrics. +TABLE II: POI Categories +Number +POI Category Name +#Beijing +#Shanghai +1 +Education +8711 +2635 +2 +Entertainment +6501 +2607 +3 +Food +5744 +6301 +4 +Beverage Shopping +6601 +5632 +5 +Tourist +6713 +4176 +6 +Transportation +4322 +1753 +7 +Medical Service +3660 +2862 +8 +Public Service +5976 +3699 +(1) Mean Absolute Error(MAE) represents the average of +the absolute value of the error between the predicted value +and the true value. +MAE = 1 +n +n +� +i=1 +|ˆy − y| +(13) +(2) Mean Squared Error(MSE) is a measure of the close- +ness of the predicted value relative to the actual value. +MSE = 1 +n +n +� +i=1 +(ˆy − y)2 +(14) +(3) Root Mean Squared Error(RMSE) is defined as fol- +lows: +RMSE = +� +� +� +� 1 +n +n +� +i=1 +(ˆy − y)2 +(15) +where ˆy is the predicted price from the regression and y is the +actual price. The lower the RMSE, the better the method. +(4) The coefficient of determination(R2) convert the pre- +dicted results into accuracy, the results are between [0,1]. +R2 = 1 − +�n +i=1 (ˆyi − yi)2 +�n +i=1 (¯yl − yi)2 +(16) +The higher the value of R2 , the more accurate the estima- +tion method. +C. Baseline Algorithms +To prove the effectiveness of our model, we compare our +method with the following algorithms. +(1)Extreme Gradient Boosting XGBOOST [15] is an +improvement on the boosting algorithm based on Gradient +Boosting Decision Tree to make it faster and more efficient. +(2)Random Forest RF [16] is an algorithm that integrates +multiple trees through the idea of integrated learning. Its basic +unit is the decision tree. +(3)Support Vector Regression SVR [17] is an algorithm +that applies support vector machine to regression problems. +(4)TAPE TAPE [18] analyzed the relationship between +the description of each rental and the price, and added the +geographical factor component to recommend a reasonable +price for each new rental of the landlord. + +MAE +0.0 +0.1 +0.2 +0.3 +0.4 +XGB +RF +SVR +TAPE +GSNE +MSIE +(a) MAE +MSE +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +XGB +RF +SVR +TAPE +GSNE +MSIE +(b) MSE +RMSE +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +XGB +RF +SVR +TAPE +GSNE +MSIE +(c) RMSE +R2 +0.0 +0.2 +0.4 +0.6 +0.8 +XGB +RF +SVR +TAPE +GSNE +MSIE +(d) R2 +Fig. 2: Overall comparison on Beijing dataset. +MAE +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +XGB +RF +SVR +TAPE +GSNE +MSIE +(a) MAE +MSE +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +XGB +RF +SVR +TAPE +GSNE +MSIE +(b) MSE +RMSE +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +XGB +RF +SVR +TAPE +GSNE +MSIE +(c) RMSE +R2 +0.0 +0.2 +0.4 +0.6 +0.8 +XGB +RF +SVR +TAPE +GSNE +MSIE +(d) R2 +Fig. 3: Overall comparison on Shanghai dataset. +(5)GSNE GSNE [19] is a geospatial embedding framework, +which can accurately capture the geospatial neighborhood re- +lationship between houses and surrounding POIs. Essentially, +it is to learn the low dimensional Gaussian embedding on +the geospatial network node, and can be combined with the +regression method, which has a certain effect on house price +prediction. +Besides, our proposed model has three variants of the +feature set combination: (1) MSIE-S, where the model utilizes +the statistic feature; (2) MSIE-ST, where the model utilizes +the statistic feature and text feature; (3) MSIE-STP, where +the model utilizes the statistic feature, text feature and spatial +feature. We evaluate these three variants with our model. +In the experiment, we split the dataset into two nonover- +lapping sets: for all records, the earliest 80% of records +are the training set and the remaining 20% are testing set. +We implement the model by Pytorch and run the code on +Windows10, Inter(R) Core(TM) i7-7700HQ @2.80GHZ and +memory size 8G. +D. Overall Performances +We present the results for “MAE”, “MSE”, “RMSE” and +“R2”, compared with baseline algorithms. Figure 2 and Fig- +ure 3 show that our proposed method ”MSIE” outperform the +baselines over both the Beijing and Shanghai dataset. The +lower value of ”MAE”, ”MSE”, ”RMSE”, and the higher +value of ”R2”, means the performance better. In all cases, we +observe an improvement with respect to baseline algorithms, +especially on “MSE” and “R2”. One interesting observation is +that the traditional machine learning method(such as, ”XGB”, +”RF” and ”SVR”) performs better than ”TAPE”. We analysis +(a) Beijing +(b) Shanghai +Fig. 4: The loss curve on two datasets. +the reason is that our algorithm feature engineering proposed +in this paper is well done and has universality. +Besides, we uses the fully connected neural network con- +structed as the prediction model. In order to prevent over +fitting, we sets 128 neurons in the input layer, uses 2 hidden +layers and set Epoch=120, the size of Batch as 256. Figure 4 +show the loss curves of neural network training on the two +datasets respectively. +E. Robustness Check +We evaluate the feature embedding contribution on mod- +eling representations. To set the control group, we de- +velop a variant of the proposed ”MSIE”, namely ”MSIE-S”, +”MSIE-ST”, ”MSIE-STP”. ”MSIE-S”, ”MSIE-ST”, ”MSIE- +STP” takes the different combination of feature set as the +input, while other component of remains the same. Table III +and Table IV show the comparison results. We can observe + +LossCurve +1.0 +Training loss +0.8 +0.6 +loss +0.4 +0.2 +0.0 +Fo +20 +40 +60 +80 +100 +120 +epochLoss Curve +1.0 +Training loss +0.8 +0.6 +loss +0.4 - +0.2 +0.0 +0 +20 +40 +60 +80 +100 +120 +epochTABLE III: The feature combination on Beijing dataset. +Feature set +MAE +MSE +RMSE +R2 +MSIE-S +0.3652 +0.2341 +0.4839 +0.5545 +MSIE-ST +0.2941 +0.1688 +0.4109 +0.6786 +MSIE-STP +0.2905 +0.1669 +0.4086 +0.6824 +TABLE IV: The feature combination on Shanghai dataset. +Feature set +MAE +MSE +RMSE +R2 +MSIE-S +0.4003 +0.2852 +0.5340 +0.5824 +MSIE-ST +0.3512 +0.2371 +0.4869 +0.6527 +MSIE-STP +0.3310 +0.2065 +0.4544 +0.6977 +that the performance of ”MSIE-STP” outperforms ”MSIE- +S” and ”MSIE-ST” in terms of the four metrics over both +two datasets. The results validate that the integration of text +feature and spatial feature indeed enhances the modeling of +price prediction. +F. Analysis of Key Influence +In order to analysis the key feature-influence of price, ac- +cording to the previous studies, we use three feature selection +method, include manual selection, P-value [20] and Lasso +CV [21]. We use R2 as an indicator to analyze, and the results +are shown in Figure 5. The best result is to use Lasso CV to +select features from the original data. +(a) Beijing +(b) Shanghai +Fig. 5: The R2 with different feature selection method. +Then, we selects 20 features with the highest correlation for +rental price prediction according to P-value, and uses Lasso +CV to select features to obtain statistical information as input +for prediction. Figure 6 show the 10 features with the highest +correlation with rental prices in the two datasets. +V. RELATED WORK +In this work, we propose a model to predict the price of +listings on Airbnb. Some studies used sentiment analysis to +study the problem of Airbnb. Martinez R D et al. studied +the relationship between Airbnb host’s listing description and +occupancy rates by sentiment analysis [22]. Zhang et al. +proposed a text analytics framework to study the relationship +among self-description, trust perception and purchase behavior +(a) Beijing +(b) Shanghai +Fig. 6: The feature most relevant to price. +on Airbnb. They used text mining to extracted sentiment +intensity and use regression method to identify the impact +of linguistic and semantic features on trust perception [23]. +Kalehbast P R et al. used sentiment analysis and machine +learning to predict the price of listings on Airbnb [24]. +Most researches have proposed many works to study the +price by using the listing information. Wang et al. studied +the relationship between a price and its determinants by +various listing information (e.g., host identity verified, accom- +modates, wireless Internet, amenities and services, and free +parking) [25]. P.Choudhary et al. analyzed Airbnb listings in +San Francisco to better understand how different attributes +(e.g., bedrooms, location, and listing type) can be used to +accurately predict the price of a new listing, which is optimal +in terms of the host’s profitability yet affordable to their +guests [26]. Shen et al. analyzed the relationship between +the description of each listing and its price, and proposed a +text-based model to recommend a reasonable price for newly +added listings [27]. Tang et al. labeled text information for +nine handpicked classes, extracted image-related features, and +finally used all features to predict a listing’s neighborhood and +its price [28]. +VI. CONCLUSION +In this paper, we proposes a prediction model based on +multi-source information embedding to study the Airbnb price +problem. Specifically, in order to obtain the best feature set, +we first selects the features of the house itself to obtain +statistical information features. Secondly, the text information +in this paper is divided into three categories, and the house +description and landlord introduction are converted into feature +matrix. The tenant reviews are then converted into sentiment +scores about each house. Then, we uses different types of +point-of-interest (POI) data and houses to form various spatial +network graphs and learns their network embeddings to obtain +spatial information features. Finally, we combines these three +types of feature embeddings are combined into multi-resource +housing features as input, and the neural network constructed +in this paper is used for price prediction. The effectiveness +of our model is demonstrated with two real data. For future +work, we plan to combine some heuristic methods [29, 30] to +further improve performance. + +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 - +0.0 +Manual selection +P-value +Lasso CV0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 - +0.0 +Manual selection +p-value +Lasso CV1.0 +price +1.00 +0.63 +0.56 +0.56 +0.44 +0.41 +0.30 +0.24 +0.22 +0.21 +accommodates +0.63 +1.00 +0.82 +0.82 +0.63 +0.19 +0.34 +0.26 +0.22 +0.8 +bedrooms +0.56 +0.82 +1.00 +0.28 +0.72 +0.64 +0.14 +0.28 +0.23 +0.23 +Entirehome/apt +0.56 +0.38 +0.28 +1.00 +0.21 +0.09 +0.35 +0.18 +0.04 +-0.01 +0.6 +beds +0.44 +0.82 +0.72 +0.21 +1.00 +09:0 +0.10 +0.27 +0.23 +0.20 +bathrooms +0.41 +0.63 +0.64 +0.09 +0.60 +1.00 +0.06 +0.15 +0.25 +0.26 +0.4 +TV +0.30 +0.19 +0.14 +0.35 +0.10 +0.06 +1.00 +800 +0.08 +0.06 +guests_included +0.24 +0.28 +0.18 +0.27 +0.15 +0.08 +1.00 +0.04 +0.02 +0.2 +Suitable_for_events +0.22 +0.26 +0.23 +0.04 +0.23 +0.25 +0.08 +0.04 +1.00 +0.24 +latitude +0.21 +0.22 +0.23 +-0.01 +0.20 +0.26 +0.06 +0.02 +0.24 +1.00 +0.0 +price +commodates +bedrooms +ire home/apt +beds +bathrooms +2 +sts_included +_for_events +latitude1.00 +price +1.00 +0.72 +0.66 +0.61 +0.42 +0.34 +0.20 +0.20 +0.20 +0.19 +accommodates +0.72 +1.00 +0.87 +0.85 +0.49 +0.28 +0.18 +0.23 +0.22 +0.23 +0.75 +bedrooms +0.66 +0.87 +1.00 +0.89 +0.50 +0.23 +0.17 +0.22 +0.20 +0.21 +beds +0.61 +0.85 +0.89 +1.00 +0.47 +0.18 +0.18 +0.22 +0.21 +0.21 +0.50 +Entire villa +0.42 +0.49 +0.50 +0.47 +1.00 +0.18 +0.22 +0.22 +0.17 +0.22 +Entirehome/apt +0.34 +0.28 +0.23 +0.18 +0.18 +1.00 +-0.23 +-0.11 +-0.05 +-0.17 +0.25 +_Backyard +0.20 +0.18 +0.17 +0.18 +0.22 +-0.23 +1.00 +0.41 +0.27 +0.49 +_Barbecue_utensils +0.20 +0.23 +0.22 +0.22 +0.22 +-0.11 +0.41 +1.00 +0.48 +0.62 +0.00 +Board_games +0.20 +0.22 +0.20 +0.21 +0.17 +-0.05 +0.27 +0.48 +1.00 +0.33 +_BBQ_grill +0.19 +0.23 +0.21 +0.21 +0.22 +-0.17 +0.49 +0.62 +1.00 +price +commodates +bedrooms +speq +Entire villa +ire home/apt +cue_utensils +oardACKNOWLEDGMENTS +This work is supported by the Natural Science Research +Foundation of Jilin Province of China under Grant No. +YDZJ202201ZYTS415, the Fundamental Research Funds for +the Central Universities 2412019ZD013, NSFC (under Grant +Nos.61976050 and 61972384). +REFERENCES +[1] S. 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Hu, “An improved master-apprentice evolution- +ary algorithm for minimum independent dominating set +problem,” Frontiers of Computer Science, vol. 17, no. 4, +pp. 1–14, 2023. + diff --git a/2tAzT4oBgHgl3EQfRvvX/content/tmp_files/load_file.txt b/2tAzT4oBgHgl3EQfRvvX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a05c5a019d94470addbf1f09aabfcf7e87227794 --- /dev/null +++ b/2tAzT4oBgHgl3EQfRvvX/content/tmp_files/load_file.txt @@ -0,0 +1,802 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf,len=801 +page_content='A Multi-Source Information Learning Framework for Airbnb Price Prediction Lu Jiang1, Yuanhan Li1, Na Luo1, Jianan Wang2,∗, Qiao Ning3,∗ 1Information Science and Technology, Northeast Normal University, Changchun 2College of Physics, Northeast Normal University, Changchun 3Information Science and Technology, Dalian Maritime University, Dalian {jiangl761, liyh447, luon110, wangjn}@nenu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='cn, ningq669@dlmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='cn Corresponding author* Abstract—With the development of technology and sharing economy, Airbnb as a famous short-term rental platform, has become the first choice for many young people to select.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' The issue of Airbnb’s pricing has always been a problem worth studying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' While the previous studies achieve promising results, there are exists deficiencies to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Such as, (1) the feature attributes of rental are not rich enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (2) the research on rental text information is not deep enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (3) there are few studies on predicting the rental price combined with the point of interest(POI) around the house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' To address the above challenges, we proposes a multi-source information embedding(MSIE) model to predict the rental price of Airbnb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Specifically, we first selects the statistical feature to embed the original rental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Secondly, we generates the word feature vector and emotional score combination of three different text information to form the text feature embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Thirdly, we uses the points of interest(POI) around the rental house information generates a variety of spatial network graphs, and learns the embedding of the network to obtain the spatial feature embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Finally, this paper combines the three modules into multi source rental representations, and uses the constructed fully connected neural network to predict the price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' The analysis of the experimental results shows the effectiveness of our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' INTRODUCTION Accommodation sharing systems are being introduced to more and more cities recently, and therefore they have gener- ated huge amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Airbnb is an online marketplace for sharing home and experience which is suffering from the chaotic pricing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Tenants need to know the reasonable price of this rental house to prevent being deceived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' The homeowner needs to customize a reasonable price for their short-term rental house to attract more customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Therefore, airbnb price prediction plays a key role in accommodation sharing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' However, rapid increase in the number of tenants and homeowners makes traditional manual-based methods [1] time-consuming and inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Computational methods have received more attention for accurate airbnb price prediction [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Computational methods for price prediction can be mainly divided into two categories: (1) feature-based methods [3], and (2) deep learning methods [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' In feature-based methods, various types of features extraction strategies are utilized to extract price correlated features for tenants and homeowners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Feature-based methods transform price prediction into a ma- chine learning methods, such as support vector machine(SVM) and random forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' For instance, in order to distinguish from the traditional method of formulating prices, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' selects rough set (RS) and SVM algorithms to establish a new math- ematical model of pricing on the basis of hedonic price [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' PR Kalehbastiet al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' proposed a price prediction model us- ing machine learning, deep learning, and natural language processing techniques to embed the features of the rentals, owner characteristics, and the customer reviews [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Deep learning methods, which use multi-layer neural network to map the correlation between input features and output results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' For instance, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' applied auto regressive integrated moving average model to generate the baseline while LSTM networks to build prediction model [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' However, the research of airbnb price prediction based on feature-based methods consider the single feature in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' With the development of representation learning [9, 10], the spatial embedding [11, 12] have received more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' There has been work to model the statistical features, text feature and spatial features related to housing prices, but there is no unified framework to integrate the above features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Based on the above disadvantages, we proposes a prediction model based on multi-source information embedding to study the Airbnb price problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' The major contributions are summa- rized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Firstly, in order to obtain the best feature set, this paper selects the features of the house itself to obtain statistical information features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Secondly, the text information in this paper is divided into three categories, and the house description and landlord introduction are converted into feature matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' The tenant reviews are then converted into sentiment scores about each house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Then, we uses different types of point-of-interest (POI) data and houses to form various spatial network graphs and learns their network embeddings to obtain spatial information features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Finally, we combines these three types of feature embed- dings are combined into multi-resource housing features as input, and the neural network constructed in this paper is used for price prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' The effectiveness of our model is demonstrated with two real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='01222v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='LG] 1 Jan 2023 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' PRELIMINARY We first introduce some key definitions and the problem definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Then, we present the overview of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Definitions and Problem Statement Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Statistic Feature The statistics feature con- structed by our model is S =(s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=', sn), where si is the preprocessed listing features includes ’host since’, ‘host is superhost’, ’verification’, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Text Feature There are three types of text features: listing description, host introduction, and tenant review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' We convert listing description and host introduction into feature vector L =(l1, l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=', ln) and H =(h1, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=', hn), and transform the tenant review to sentiment score R =(r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=', rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Thus, we define the text features as T = (L, H, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Spatial Feature We first combine each rental house and the POI with in 1,000m around it into a spatial network G = (V, E, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Then, we learn network embedding through SDNE [13], and get the spatial feature matrix P =(p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=', pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Problem Statement In this paper, we study the problem of airbnb price prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' We formulate the problem as a multi-source feature embedding task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Formally, we aim to find a mapping function f : (S, T, P) → V that takes the statistic feature S, text feature T, spatial feature P as input, and outputs a unified vectorized representations V , for predicting the specific listing price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Framework Overview Figure 1 shows an overall framework for the multi-source feature embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Specifically, we embed the original data from three aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (1) For the statistical feature embedding, we uses Lasso CV to select the feature set with the rental house feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (2) For the text feature embedding, we divides the text feature into three categories, include house descrip- tion, landlord introduction and tenant comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Through the negative sampling CBOW model, the house description and landlord introduction are converted into word feature vectors, and the Bayesian model based on naive Bayesian principle is used to convert tenant comments into emotional scores, we combine them as the text feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (3) For the spatial feature embedding, we collects different types of POIs, and combines the POI of each house and the surrounding area within 1,000m into a spatial network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Through the SDNE model to learn the spatial feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (4) Three different features are combined into a multi-source feature and input into the neural network to obtain the final rental price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' MULTI-SOURCE INFORMATION LEARNING In this section, we introduce the core architecture of our framework as follows: (1) statistic feature embedding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (2) text feature embedding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (3) spatial feature embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Statistics Feature Embedding Each house’s statistic feature is represented by a 245- dimensional vector which describes the listing of a house, including listing id, host id, host since, host response rate, host is superhost, host has profile pic, host identity verified, bathrooms, bedrooms, latitude, longitude, accommodates, security deposit, guests included, verification, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' We use the Lasso CV to do the feature set selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' The loss function is defined as follows: obj = 1 2 n � i=1 � yi − wT xi �2 + α m � j=1 |wi| (1) where n is the number of houses, m is the number of parameters, α is the regularization coefficient, α �m j=1 |wi| is the L1 regularization term, yi is the rental price, xi is the statistical features of rental housing, w is the coefficient matrix of rental housing features, xi = si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Statistical feature matrix S =(s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=', sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Lasso CV can compress the coefficients of unimportant features to 0, realizing the purpose of feature se- lection, and ultimately leaving the important statistical feature set that this paper wants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Text Feature Embedding We extract three types of text data from the original data, there are listing description, host introduction and tenant re- view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Listing description mainly about introducing the location of the house, surrounding environment, indoor layout and housing regulations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Host introduction mainly introduces the age, height, occupation, hobbies and personality of the host, and tenant review expresses the tenants’ feelings about housing rentals and the evaluation of the host’s attitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Since tenant review contain emotional value, we use two different methods to model text feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' We first use CBOW [14] model to embed the text feature of listing description, host introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' We selects the Wikipedia Chinese thesaurus after preprocessing as the training corpus W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' the objective function is defined as follows: L = � c∈W � � �log � σ � xT c θc�� + � u∈NEG(c) log � σ � −xT c θu�� � � � (2) Then the above objective function is optimized by using the random gradient rise method to obtain: L(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' u) = Lc(u) log � σ � xT c θu�� +[1 − Lc(u)] log � 1 − σ � xT c θu�� (3) Then calculate the gradient of L(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' u) to obtain: v(˜c) := v(˜c) + η � u∈{c}∪NEG(c) ∂L(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' u) ∂xc (4) Geospatial Information Embedding Text Information Embedding listing information host introduction tenant review feature vector feature vector sentiment score Original Listing Data preprocess Feature Selection Statistics Information Embedding Real Airbnb Listing Data Multi Source Listing Feature listing POI Predict Price Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' 1: Framework Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' In this paper, we set the dimension of the word vector as 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' l and h represent the embedding of listing description, host introduction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' l = 1 Z Z � i=1 v (˜cl) (5) h = 1 z z � i=1 v (˜ch) (6) Therefore, we get the text information feature matrix of the listing description and host introduction: L =(l1, l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=', ln) and H =(h1, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=', hn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' For the tenant review embedding, since it contains strong emotional expression, in order to reflect whether the tenants’ evaluation of the house is positive or negative, we uses the naive Bayes method to generate the corresponding emotional score r ∈ [0, 1] for each house, where 0 represents the negative and 1 represents the positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Specifically, the probability that a tenant review text belongs to the positive class can be expressed as: P (pos | c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' cd) = P (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' cd | pos) P(pos) P (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' cd) (7) After simplifying the above formula, we can obtain: P (pos | c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' cd) = 1 1 + γ (8) In this work, a text represents a tenant’s review, and a tenant has many reviews, so the emotional score of a tenant can be expressed as: r = 1 q q � i=1 P (pos | c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' cd) (9) where q represents the number of reviews on a rental, d represents the total number of words in a review, and P (pos | c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' cd) represents the probability that the review belongs to the category of positive emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Therefore, the emotional score vector of tenant reviews can be expressed as R =(r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=', rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Spatial Feature Embedding We proposes a method to learn spatial embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' First, POI is divided into 8 different types, and the rented houses and the surrounding different type POIs form a spatial network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Then learn the network embedding of these spatial graphs through SDNE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' This method can accurately capture the spatial features related to important POIs such as scenic spots and railway stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' We uses Euclidean distance to calculate the weight W between house and poi as follows: W = R · arccos( dis ) · π/180 (10) where dis = sin(LatA) sin(LatB) cos(LonA − LonB) + cos(LatA) cos(LatB), the two types of nodes, A and B, Net- work1 Learning Network Embedding Net- work2 Net- work3 Net- workkInput layer Hidden layer Output layeirepresent the rental houses and POI respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' LonA and LonB are their longitudes, LatA and LatB are their latitudes, and R is the average radius of the earth, taking the value of 6371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='004km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' In SDNE model, the encoder is from xi to y(k) i , the decoder is from y(k) i to �xi, y(k) i is the node embedding of vi, in this paper,y(k) i = pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' The formula of encoder is: y(k) i = σ � W (k)y(k−1) i + b(k)� (11) Therefore, the spatial embedding can be expressed as P =(p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=', pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' After we get the statistic embedding, the text embedding and the spatial embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' We combine the above features into a multi-source feature M = (S, T, P), and use the fully connected neural network to predict the rental price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' We take the multi-source feature matrix M =(m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=', mn) as the input of the neural network, then obtain as follows: y = wT m + b, A = σ(y) (12) where y is the actual rental price, m is the multi-source feature, w is the parameter matrix, and b is the offset term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' A is the activation function, we uses ReLU function as the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' At last, the output layer uses a neuron to output and get the predicted price �yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' EXPERIMENT In this section, we first introduce two real dataset and evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Then, we design experiments to answer the following three questions: Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' How is the performance of our proposed MSIE in the airbnb price prediction task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' How do the feature combination affect the price prediction performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' What is the key influence on the airbnb price?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Dataset We collect the dataset from an open online airbnb website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Table I shows the statistics of our two real airbnb datasets from two cities: Beijing and Shanghai after preprocess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' TABLE I: Statistics of the data City # Houses # Reviews Time Period Beijing 10779 191876 01/2017-06/2019 Shanghai 8638 159069 01/2020-07/2021 Besides, we also collect the POI of the Beijing and Shanghai in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' We divide it into 8 categories, include, Education, Entertainment, Food, Beverage Shopping, Tourist, Transporta- tion, Medical Service, and Public Service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Evaluation Metrics We evaluate the model performances in terms of the fol- lowing metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' TABLE II: POI Categories Number POI Category Name #Beijing #Shanghai 1 Education 8711 2635 2 Entertainment 6501 2607 3 Food 5744 6301 4 Beverage Shopping 6601 5632 5 Tourist 6713 4176 6 Transportation 4322 1753 7 Medical Service 3660 2862 8 Public Service 5976 3699 (1) Mean Absolute Error(MAE) represents the average of the absolute value of the error between the predicted value and the true value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' MAE = 1 n n � i=1 |ˆy − y| (13) (2) Mean Squared Error(MSE) is a measure of the close- ness of the predicted value relative to the actual value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' MSE = 1 n n � i=1 (ˆy − y)2 (14) (3) Root Mean Squared Error(RMSE) is defined as fol- lows: RMSE = � � � � 1 n n � i=1 (ˆy − y)2 (15) where ˆy is the predicted price from the regression and y is the actual price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' The lower the RMSE, the better the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (4) The coefficient of determination(R2) convert the pre- dicted results into accuracy, the results are between [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' R2 = 1 − �n i=1 (ˆyi − yi)2 �n i=1 (¯yl − yi)2 (16) The higher the value of R2 , the more accurate the estima- tion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Baseline Algorithms To prove the effectiveness of our model, we compare our method with the following algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (1)Extreme Gradient Boosting XGBOOST [15] is an improvement on the boosting algorithm based on Gradient Boosting Decision Tree to make it faster and more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (2)Random Forest RF [16] is an algorithm that integrates multiple trees through the idea of integrated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Its basic unit is the decision tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (3)Support Vector Regression SVR [17] is an algorithm that applies support vector machine to regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (4)TAPE TAPE [18] analyzed the relationship between the description of each rental and the price, and added the geographical factor component to recommend a reasonable price for each new rental of the landlord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' MAE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='4 XGB RF SVR TAPE GSNE MSIE (a) MAE MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='30 XGB RF SVR TAPE GSNE MSIE (b) MSE RMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='6 XGB RF SVR TAPE GSNE MSIE (c) RMSE R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='8 XGB RF SVR TAPE GSNE MSIE (d) R2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' 2: Overall comparison on Beijing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' MAE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='5 XGB RF SVR TAPE GSNE MSIE (a) MAE MSE 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='35 XGB RF SVR TAPE GSNE MSIE (b) MSE RMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='6 XGB RF SVR TAPE GSNE MSIE (c) RMSE R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='8 XGB RF SVR TAPE GSNE MSIE (d) R2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' 3: Overall comparison on Shanghai dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (5)GSNE GSNE [19] is a geospatial embedding framework, which can accurately capture the geospatial neighborhood re- lationship between houses and surrounding POIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Essentially, it is to learn the low dimensional Gaussian embedding on the geospatial network node, and can be combined with the regression method, which has a certain effect on house price prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Besides, our proposed model has three variants of the feature set combination: (1) MSIE-S, where the model utilizes the statistic feature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (2) MSIE-ST, where the model utilizes the statistic feature and text feature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (3) MSIE-STP, where the model utilizes the statistic feature, text feature and spatial feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' We evaluate these three variants with our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' In the experiment, we split the dataset into two nonover- lapping sets: for all records, the earliest 80% of records are the training set and the remaining 20% are testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' We implement the model by Pytorch and run the code on Windows10, Inter(R) Core(TM) i7-7700HQ @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='80GHZ and memory size 8G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Overall Performances We present the results for “MAE”, “MSE”, “RMSE” and “R2”, compared with baseline algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Figure 2 and Fig- ure 3 show that our proposed method ”MSIE” outperform the baselines over both the Beijing and Shanghai dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' The lower value of ”MAE”, ”MSE”, ”RMSE”, and the higher value of ”R2”, means the performance better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' In all cases, we observe an improvement with respect to baseline algorithms, especially on “MSE” and “R2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' One interesting observation is that the traditional machine learning method(such as, ”XGB”, ”RF” and ”SVR”) performs better than ”TAPE”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' We analysis (a) Beijing (b) Shanghai Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' 4: The loss curve on two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' the reason is that our algorithm feature engineering proposed in this paper is well done and has universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Besides, we uses the fully connected neural network con- structed as the prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' In order to prevent over fitting, we sets 128 neurons in the input layer, uses 2 hidden layers and set Epoch=120, the size of Batch as 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Figure 4 show the loss curves of neural network training on the two datasets respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Robustness Check We evaluate the feature embedding contribution on mod- eling representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' To set the control group, we de- velop a variant of the proposed ”MSIE”, namely ”MSIE-S”, ”MSIE-ST”, ”MSIE-STP”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' ”MSIE-S”, ”MSIE-ST”, ”MSIE- STP” takes the different combination of feature set as the input, while other component of remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Table III and Table IV show the comparison results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' We can observe LossCurve 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='0 Training loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='6 loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='0 Fo 20 40 60 80 100 120 epochLoss Curve 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='0 Training loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='6 loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='0 0 20 40 60 80 100 120 epochTABLE III: The feature combination on Beijing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Feature set MAE MSE RMSE R2 MSIE-S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='3652 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='2341 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='4839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='5545 MSIE-ST 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='2941 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='1688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='4109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='6786 MSIE-STP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='2905 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='1669 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='4086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='6824 TABLE IV: The feature combination on Shanghai dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Feature set MAE MSE RMSE R2 MSIE-S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='4003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='2852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='5340 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='5824 MSIE-ST 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='3512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='2371 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='4869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='6527 MSIE-STP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='3310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='2065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='4544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='6977 that the performance of ”MSIE-STP” outperforms ”MSIE- S” and ”MSIE-ST” in terms of the four metrics over both two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' The results validate that the integration of text feature and spatial feature indeed enhances the modeling of price prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Analysis of Key Influence In order to analysis the key feature-influence of price, ac- cording to the previous studies, we use three feature selection method, include manual selection, P-value [20] and Lasso CV [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' We use R2 as an indicator to analyze, and the results are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' The best result is to use Lasso CV to select features from the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' (a) Beijing (b) Shanghai Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' 5: The R2 with different feature selection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Then, we selects 20 features with the highest correlation for rental price prediction according to P-value, and uses Lasso CV to select features to obtain statistical information as input for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Figure 6 show the 10 features with the highest correlation with rental prices in the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' RELATED WORK In this work, we propose a model to predict the price of listings on Airbnb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Some studies used sentiment analysis to study the problem of Airbnb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Martinez R D et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' studied the relationship between Airbnb host’s listing description and occupancy rates by sentiment analysis [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' proposed a text analytics framework to study the relationship among self-description, trust perception and purchase behavior (a) Beijing (b) Shanghai Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' 6: The feature most relevant to price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' on Airbnb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' They used text mining to extracted sentiment intensity and use regression method to identify the impact of linguistic and semantic features on trust perception [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Kalehbast P R et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' used sentiment analysis and machine learning to predict the price of listings on Airbnb [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Most researches have proposed many works to study the price by using the listing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' studied the relationship between a price and its determinants by various listing information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=', host identity verified, accom- modates, wireless Internet, amenities and services, and free parking) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='Choudhary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' analyzed Airbnb listings in San Francisco to better understand how different attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=', bedrooms, location, and listing type) can be used to accurately predict the price of a new listing, which is optimal in terms of the host’s profitability yet affordable to their guests [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' analyzed the relationship between the description of each listing and its price, and proposed a text-based model to recommend a reasonable price for newly added listings [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' labeled text information for nine handpicked classes, extracted image-related features, and finally used all features to predict a listing’s neighborhood and its price [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' CONCLUSION In this paper, we proposes a prediction model based on multi-source information embedding to study the Airbnb price problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Specifically, in order to obtain the best feature set, we first selects the features of the house itself to obtain statistical information features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Secondly, the text information in this paper is divided into three categories, and the house description and landlord introduction are converted into feature matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' The tenant reviews are then converted into sentiment scores about each house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Then, we uses different types of point-of-interest (POI) data and houses to form various spatial network graphs and learns their network embeddings to obtain spatial information features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' Finally, we combines these three types of feature embeddings are combined into multi-resource housing features as input, and the neural network constructed in this paper is used for price prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfRvvX/content/2301.01222v1.pdf'} +page_content=' The effectiveness 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Butchy, Cheryl A. Telmer, and Natasa Miskov-Zivanov +Abstract—There is significant interest in using existing repositories of biological entities, relationships, and models to automate +biological model assembly and extension. Current methods aggregate human-curated biological information into executable, +simulatable models, but these models do not resemble human curated models and do not recapitulate experimental results. Here, +we outline the process of automated model assembly and extension, while demonstrating it on both synthetic models and human- +curated models of biological signaling networks. We begin with an iterative, greedy, and combinatoric approach to automated +assembly and demonstrate the key difficulties inherent to contextless assembly. We publicly release the software used in this +paper to enable further exploration of this problem. +Index Terms— Automatic Model Creation; Biological Networks; Extending Biological Networks; Model Construction; Network +Reconstruction. +—————————— u —————————— +1 INTRODUCTION +omputational approaches to modeling large complex +systems standardize the representation of knowledge, +while simulation of computational models illuminates the +dynamics of systems, allowing for discoveries and theoret- +ical advances [1]. Due to the complexity and redundancy +of biological systems, computational models are difficult +and laborious to create and update. There are two main ap- +proaches to modeling these systems, bottom-up and top- +down [2]. In a bottom-up approach, known molecular in- +teractions are assembled into a model to help explain the +system’s behavior and predict how the system will re- +spond to new stimuli or inputs. This method has been used +extensively by biologists, biochemists, and molecular biol- +ogists to manually create models based on the interactions +within cells involved in signaling that are supported by sci- +entific literature. In a top-down approach, experimental +data—usually collected with high-throughput methods— +is used to infer correlations between element behavior and +determine causal relationships. Top-down approaches em- +ploy many different methods such as Bayesian Inference +[3], ANOVA calculations [4], and Fuzzy Logic [5]. In both +the mechanistic bottom-up approach and the data-driven +top-down approach, the model is used to predict the be- +havior of individual elements in the network [6, 7]. Re- +cently, there has been a push to integrate the two methods, +using experimental data to inform the bottom-up ap- +proach, and incorporating prior knowledge into the top- +down approach to reduce the number of potential models +[8-12]. Despite these hybrid approaches, this problem re- +mains a combinatoric one, with large, complex systems be- +ing prohibitively difficult to investigate and model manu- +ally. +It is a direct result of these factors that system and com- +putational biologists have endeavored to automate the +process of model creation and extension. To automatically +create models, information can be extracted from litera- +ture, queried from databases, or taken from existing path- +ways and models. Public databases such as Reactome [13], +MetaCyc [14], OmniPath [15], and STRING [16] offer easy +access to millions of interactions. Additionally, there exist +a number of model databases with published models that +are publicly available such as The Nature Pathway Interac- +tion Database [17], WikiPathways [18], BioModels [19], the +Cell Collective [20], and KEGG pathways [21]. These data- +bases contain highly targeted, curated published and un- +published models which are created for specific biological +context and may not be generalizable to explain other phe- +nomena. When new interactions are discovered, and de- +scribed in a scientific publication, state-of-the-art machine +reading engines such as REACH [22], TRIPS [23], and +EVEX [24] can extract them, together with other relevant +information. These automated readers are able to extract +tens of thousands of biological entity interactions from +hundreds of papers in a few hours, and produce a ma- +chine-readable, structured output [22]. Despite this abun- +dance of available interactions, there is still no efficient +way to assemble them into accurate models that correctly +reflect the system under investigation and the same biolog- +ical context and recapitulate the observed experimental be- +havior. +Recently, a few tools, such as Path2Models [25] and IN- +DRA [26, 27], have been created to help modelers collect +biological interactions, assemble a model, and perform +xxxx-xxxx/0x/$xx.00 © 200x IEEE Published by the IEEE Computer Society +———————————————— +• A.A. Butchy is with the Department of Bioengineering, University of Pitts- +burgh, Pittsburgh, PA 15213. E-mail: adam.butchy@pitt.edu. +• C.A. Telmer is with the Department of Biological Sciences, Carnegie +Mellon University, Pittsburgh, PA 15213. E-mail: ctelmer@cmu.edu. +• N. Miskov-Zivanov is with the Departments of Electrical and Computer +Engineering, Bioengineering, and Computational Biology, University of +Pittsburgh, Pittsburgh, PA 15213. E-mail: nmzivanov@pitt.edu. + +C + +2 +IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, MANUSCRIPT ID + +simulations. These tools assemble quantitative and quali- +tative models using available pathway information; how- +ever, the quality of the assembled models is dependent +upon the modeling approach, and the granularity of the +information they are given. These techniques rely on accu- +rate information, and their performance suffers when the +interaction information is incomplete, from a different bio- +logical context, or erroneous. Other methods have been +proposed to automatically expand, test, and select the best +model, with respect to a given performance metric. These +approaches integrate stochastic model simulations with +statistical model checking only [28], or also incorporating +Markov clustering [29], or genetic algorithm [30], and +therefore have different strengths and weaknesses. The +Markov clustering approach to model extension is well +suited for the combinatorial explosion in the number of +possible model extensions while the genetic algorithm ap- +proach is overwhelmed by large number of extensions. +Markov clustering prioritizes strongly connected compo- +nents at the expense of interactions involving nodes of low +degree. The genetic algorithm explores the effect of single +extensions distributed throughout the network. +In this work, we examine the complexities inherent to +automatic model assembly and extension. We use two +novel algorithms, Breadth First Addition (BFA) and Depth +First Addition (DFA), which utilize the same principles as +the breadth-first search and depth-first search algorithms +in network studies [31] to illustrate the key limitations of +iterative model assembly and extension. In contrast to pre- +vious work [28-30], these methods not only represent a +new approach to bottom-up model assembly but are also +used to demonstrate the existence of key biological prop- +erties which hinder automated modeling of biological sys- +tems. We demonstrate these properties using both syn- +thetic networks, Erdös-Rényi random networks (ER) [32] +and Barabási-Albert scale-free networks (BA) [33], as well +as two published expert curated and validated models, a T +cell large granular lymphocyte (TLGL) leukemia model +[34], and a model of naïve Tcell differentiation (Tcell) [35]. +By using different network structures, we are able to more +comprehensively explore automated model assembly and +identify the main difficulties with the BFA and DFA ap- +proaches. +2 METHODS +2.1 Discrete Models and Simulations +The underlying structure of models that we study here is a +network 𝐺(𝑉, 𝐸), where 𝑉 is a set of nodes (model ele- +ments), and 𝐸 is a set of directed edges (regulatory influ- +ences between elements). A few toy examples of such net- +works are shown in Figure 1 (A). Model elements usually +represent proteins, genes, chemicals, or biological pro- +cesses. For each model element 𝑣! ∈ 𝑉 (𝑖 = 1. . 𝑁, where +𝑁 = |𝑉|), we define an update rule 𝑣! = 𝑓"!(𝑣#, 𝑣$, … , 𝑣%), +which can either be a constant (input nodes in network 𝐺) +or it can depend on a subset of elements from 𝑉. In the lat- +ter case, for each element 𝑣! this subset is often referred to +as an influence set for 𝑣! and it consists of its positive (acti- +vating) and negative (inhibiting) regulators. Positive +regulators of 𝑣! comprise set 𝑉&'( +! and are represented with +regular arrowheads in Figure 1 (A). Negative regulators of +𝑣! comprise set 𝑉)*+ +! and are represented with blunt arrow- +heads in Figure 1 (A). +The high throughput retrieval of interaction infor- +mation from literature typically only includes knowledge +of the sign of influence (positive or negative) and rarely ad- +ditional information about relationships between regula- +tors. In such cases, logic functions and elements with two +levels, 0 (low) and 1 (high), have been found most suitable. +To broaden the application beyond just Boolean functions +to other cases where interactions were enriched either +through manual curation or more specific information re- +trieval, we will assume that each element 𝑣! can have 𝐿! +number of discrete levels. While the choice of function +does not affect the main algorithms described in Section +2.2, in order to simulate models, and closely approximate +different functions, including Boolean, we adopted the +common approach that computes a (weighted) sum of reg- +ulator values to determine element update values. The +general form of this function is: +𝑔"! = 𝑓"!(𝑣#, 𝑣$, … , 𝑣%) = ∑ +𝑤,𝑣, +""∈.#$% +! +− ∑ +𝑤/𝑣/ +"&∈.'() +! + +(1) +The weighting factors 𝑤, and 𝑤/ can be used to account for +different influence strengths for regulators. To remain +within boundaries of the allowed levels for element 𝑣! (0.. +𝐿! − 1), the function 𝑔"! is then used to determine a suitable +increment/decrement for 𝑣!, 𝛿"! = 𝑓(𝑔"!), such that: +𝑣!,)*12 = 8 +0 +𝑣! + 𝛿"! ≤ 0 +𝑣! + 𝛿"! +0 < 𝑣! + 𝛿"! < 𝐿! − 1 +𝐿! − 1 +𝑣! + 𝛿"! ≥ 𝐿! − 1 + +(2) +Together, the set of model elements 𝑉, element influences +forming the set 𝐸, and the set of element update rules 𝐹, +comprise an Executable Model, ℳ(𝑉, 𝐸, 𝐹), a model that in- +cludes all the necessary information for simulation and dy- +namic analysis. +We use the Discrete, Stochastic, Heterogeneous simula- +tor (DiSH) [36] which allows for simulations of discrete +models with various types of update functions, and has +several different simulation schemes, that can be either de- +terministic or stochastic. For the analysis we conducted +here, we used the USB-RSQ simulation scheme in DiSH +(uniform, step-based, random-order, sequential update +scheme, described in detail in [36]) . It has been shown pre- +viously [36, 37] that, by taking into account the random- +ness in timing of signaling events, the USB-RSQ simulation +scheme is able to recapitulate the network dynamics within +cells. DiSH simulates the models starting from an initial +state 𝒒ℳ,4 = A𝑠"*,4, 𝑠"+,4, … , 𝑠",,4C (assigned before simula- +tions), where 𝑠"!,4 denotes the state value of element 𝑣! at +time point 𝑡 = 0, and for a pre-defined number of time +steps, 𝑇 (e.g., when the steady state is reached). Each such +simulation run, 𝑟, yields for every model element 𝑣! ∈ 𝑉, a +trajectory of values, 𝒔"! +5 = A𝑠"!,# +5 , 𝑠"!,$ +5 , … 𝑠"!,6 +5 +C, where 𝑠"!,2 +5 is +the state value of element 𝑣! at time point 𝑡 (𝑡 = 1, . . , 𝑇) +within run 𝑟. Due to the randomness of the update scheme, +element trajectories may vary across multiple runs that + +BUTCHY ET AL.: TITLE +3 + +start with the same initial state. Therefore, for the same +time step 𝑡, following the approach from [36], we compute +the mean of values 𝑠"!,2 +5 across different runs, to obtain av- +erage trajectories for all elements. More formally, we com- +pute an average element trajectory of element 𝑣! as: +𝒔H"! = 1 +𝑅 J 𝒔"! +5 +7 +58# += 1 +𝑅 JA𝑠"!,# +5 , 𝑠"!,$ +5 , … 𝑠"!,6 +5 +C +7 +58# + += A𝑠̅"!,#, 𝑠̅"!,$, … 𝑠̅"!,6C + +(3) +where 𝑅 is the overall number of conducted simulation +runs. For example, in Figure 1 (B), we illustrate simulation +trajectories for elements of the toy models in Figure 1 (A). +We denote average model state for model ℳ(𝑉, 𝐸, 𝐹) at time +step 𝑡 as a vector of average element states at time step 𝑡: + +𝒒ℳ,2 +9"+ = A𝑠̅"*,2, 𝑠̅"+,2, … , 𝑠̅",,2C +(4) +We define model behavior resulting from a specific initial +model state 𝒒ℳ,4 = (S#, S$, … , S%) as: + +𝑸ℳ = A𝒒ℳ,4, 𝒒ℳ,# +9"+, … , 𝒒ℳ,6 +9"+C +(5) +2.2 Extension method inputs +We define here inputs used by extension methods and by +our evaluation methodology: Baseline Model, Golden +Model, and Candidate Knowledge. +Existing models of a system of interest are often lever- +aged and contextualized for a specific purpose. The Base- +line Model is the existing, high confidence model before +updating with extensions. As a special case, we can also +assume that the Baseline Model is an empty network with +no nodes or edges. The Golden Model is assumed to +contain all relevant knowledge about the system, including +accurate element relationships and update functions. The +Candidate Knowledge is a set of directed edges, including +their source and target nodes, which are candidates for ad- +dition to the Baseline Model. +Given the Golden Model knowledge, through simula- +tions, for different initial states representing different con- +ditions and scenarios, we can obtain Golden Model behav- +ior, 𝑸:;, as in Equation 5. 𝑸:; represents the true expected +behavior of the system being modeled. As part of 𝑸:;, we +also obtain the average Golden Model state at the final sim- +ulation time step 𝑇 (e.g., steady state), 𝒒:;,6 +9"+ . +The above definition of Golden Model is important for +the rest of our discussion since Golden Model is used as an +input to our evaluation methodology. However, in real sce- +narios, the Golden Model is usually not known in advance. +Instead, the goal of model assembly and extension algo- +rithms is to discover the Golden Model, while only the real +system behavior, i.e., measured state values for system +components, may be available. The system state data can +be used to form the target behavior 𝑸N. Ideally, the Golden +Model behavior is identical to the target behavior, 𝑸:; = +𝑸N. The target state at time 𝑇 is part of the target behavior +and is denoted as 𝒒O6. +As will be detailed in the following sub-sections, exten- +sion algorithms start with the Baseline Model for which +𝑸<; ≠ 𝑸N. Next, they add selected edges from the Candi- +date Knowledge to create new models, called Candidate +Models, which are then iteratively updated and simulated +to obtain 𝑸=; in each iteration, and to ultimately find a +model that most closely reproduces the target behavior 𝑸N. +Figure 1. A toy example illustrating directed cyclic network models explored in this work and the flow of the proposed meth- +odology for evaluating extension algorithms. (A) (top) An example Golden Model used in evaluation; (middle) Example input +graphs, Candidate Knowledge, and Baseline Model, used in extension methods ([28-30] and this work); (bottom) An example +Candidate Model recommended by extension methods. (B) Average element trajectories obtained from stochastic simulation +for the three example models (Golden, Baseline, and Candidate). (C) An example iterative procedure that uses the Total Model +Error (TME) metric to evaluate each intermediate Candidate Model. + + +A +Golden Model +B +c +Iterative Extension* +Golden Model Behavior +A +B +a +C +D +e +E +TME +F +Edge Removal +Candidate +Baseline Model +Baseline Model Behavior +Knowledge +F +0 +A +B +B +1 +2 +B +D +D +(Baseline) +E +E +Iteration +B +A +C +F +c +Where: +a = TME of Baseline Model +Model Extension +b = TME of Candidate Model 3 +c = TME of Candidate Model 1 +Candidate Model +Candidate Model Behavior +d = TME of Candidate Model 2 +A +e = TME of Candidate Model 4 +E +B +f = TME of Candidate Model 5 +D +→ = Path of minimizing TME +C +→ = Explored TME paths +F4 +IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, MANUSCRIPT ID + +2.3 Model evaluation metric +Given two models, ℳ# and ℳ$, if they have the same ele- +ment sets, 𝑉ℳ* ≡ 𝑉ℳ+ ≡ 𝑉 (𝑁 = |𝑉|), and if we simulate +them starting from the same initial state, 𝒒ℳ*,4 = 𝒒ℳ+,4 = +A𝑠"*,4, 𝑠"+,4, … , 𝑠",,4C, to obtain their behaviors, 𝑸ℳ* and +𝑸ℳ+, respectively, we can compute the difference between +the two model behaviors, Δ2A𝑸ℳ*, 𝑸ℳ+C at any simulation +time step 𝑡 as: + +Δ2A𝑸ℳ*, 𝑸ℳ+C = ∑ +S𝑠̅"!,2 +ℳ* − 𝑠̅"!,2 +ℳ+S +% +!8# + +(6) +In other words, Δ2 finds the absolute difference between an +element’s average state in time step 𝑡, in model ℳ# (𝑠̅"!,2 +ℳ*) +and in model ℳ$ (𝑠̅"!,2 +ℳ+) and sums these differences across +all model elements. +From (6), we derive the Total Model Error (TME) metric, +as Δ6, when 𝑡 = 𝑇, between a Candidate Model behavior +𝑸=; and known target behavior 𝑸N: + +TMEA𝑸=;, 𝑸NC = Δ6A𝑸=;, 𝑸NC = ∑ +S𝑠̅"!,6 +=; − 𝑠̂"!,6S +% +!8# + (7) +Or, in the case when a Golden Model is used: + TME(𝑸=;, 𝑸:;) = Δ6(𝑸=;, 𝑸:;) = ∑ +S𝑠̅>!,6 +=; − 𝑠̅>!,6 +:;S +% +!8# + (8) +Besides the above defined Δ2, other types of functions +could be used to compute the difference between two mod- +els, such as the squared error, or more statistic-based eval- +uation methods like the Chi-squared test to compare the +Figure 2. The Breadth and Depth First Addition (BFA and DFA, respectively) algorithms. Top: The pseudocode for the two +algorithms. Bottom: An example illustrating the Candidate Knowledge and Baseline Model inputs and steps for BFA and DFA +algorithms: (A, D) The inputs to the BFA and DFA algorithms. (B) In the BFA extension process, the Baseline Model is extended +with single interactions from Candidate Knowledge and the TME is calculated for each Candidate Model. The Candidate Model +with the lowest TME is selected and becomes the Baseline Model for the next iteration. (E) In the DFA extension process, the +Baseline Model is extended with a single interaction from Candidate Knowledge and the TME is calculated to determine if the +Candidate Model has a lower TME than the Baseline Model. As soon as the TME decreases, that edge of Candidate Knowledge +is incorporated into the Candidate Model, and it becomes the Baseline Model for the next iteration. (C, F) For both algorithms, +the process is repeated with the remaining Candidate Knowledge until all edges are added back, the TME reaches zero, or there +are no edges that reduce the TME below its current lowest value. + + + + +Algorithm: Breadth First Addition (BFA) +Algorithm: Depth First Addition (DFA) +Input: baseline model (MBM), list of edges (ENEw), TME of the baseline model +Input: baseline model, list of edges, expected performance of the golden +(TMEBM), expected performance of the golden model (QGM) +model, current TME +Output: extended baseline model that minimizes the TME +Output: extended baseline model that minimizes the TME +1: while (TMEBM!= 0) and (ENEw != [) +1: +EADDED = FALSE +2: +Initialize scores = [] +2: +for edge in ENEw: +3: +for edge in ENEw: +3: +4: +McM = a candidate model is created by adding the edge to the MBM +4: + simulate McM +5: +simulate McM +5: +TME(QGM,QcM ) use the TME function to compare +6: +TME(QGM,QcM ) use the TME function to compare the +6: +the candidate model to the expected performance +7: +7: +candidate model to the expected performance of the golden model +of the golden model +8: +8: +Append TMEcM to the scores list +if TMEcM < TMEBM: +9: +9: +end for +McM = a candidate model is created by +10: +10: +find index = min(scores) +adding the edge to the MBM +11: +11: +TMEcM = scores(index) +MBM = McM +12: +12: +if TMEcM < TMEBM: +ENEw.delete(edge) +13: +13: + McM = a candidate model is created by adding +14: +14: +the edge to the MBM +EADDED = TRUE +15: +15: +MBM = McM +exit for loop +16: +16: +17: +ENEw.delete(index) +17: +end if +18: +TMEBM = TMEcM +18: +end for +19: +else +19: +if (EADDED =- FALSE) +20: +return MBM +20: +return MBM +21: +end if +21: +end if +22: end while +22: +: end while +23: return MBM +23: return MBMCandidate +Baseline Model +Candidate +Baseline Model +A +Knowledge +TME = 5.0 +Knowledge +TME = 5.0 +BFA +DFA +B +D +Inputs +Inputs +2) +2) +3) +3) + B +E +First +First +Extension +Extension +Round +Round +Extension 1 +Extension 2 +Extension 3 +Extension 1 +TME = 3.0 +TME = 2.0 +TME = 6.0 +TME = 3.0 +c +: +Second +Second +Extension +Extension +Round +Round +Extension 2&1 +Extension 2&3 +Extension 1&2 +Extension 1&3 +TME = 0.0 +TME = 4.0 +TME = 0.0 +TME = 4.0BUTCHY ET AL.: TITLE +5 + +distribution of model states at time step 𝑡. We use the ab- +solute difference of the model’s end state (𝑡 = 𝑇) for a few +reasons: it would not exaggerate the effect of large differ- +ences (as would be observed in the squared error); it is less +computationally expensive than the Chi-squared test; and +it more accurately matches how computational biologists +compare computational model simulations against sparse +biological measurements, where the full time-course of the +model elements is often unknown. +2.4 Methodology for evaluating model extension +In this work, we are interested in evaluating automated +model extension, that is, the limitations of automatically +extending the Baseline Model with behavior 𝑸<; to +achieve the target or Golden Model behavior 𝑸:;. There- +fore, in our studies we assume that the Golden Model is +known, and to obtain Baseline Models we use the proce- +dure illustrated in Figure 1 (A) and described as follows. +For a given Golden Model, we create multiple Baseline +Models by removing edges from the Golden Model, in order +to disrupt its behavior and to determine whether the ex- +tension algorithms are able to recover the Golden Model +from a range of Baseline Models. The removed edges form +the Candidate Knowledge sets (Figure 1 (A)). The extension +algorithms are given the Baseline Model and the Candi- +date Knowledge and tasked with extending the Baseline +Model using edges from the Candidate Knowledge, to cre- +ate Candidate Models (Figure 1 (A)) and reproduce the +Golden Model behavior. +Using the DiSH simulator, we simulate the Golden, +Baseline, and Candidate Models to observe how elements +of each model behave over time, and to obtain model be- +haviors 𝑸:;, 𝑸<;, 𝑸=;, respectively (Figure 1 (B)). The goal +of this procedure is to find Candidate Model(s) with be- +havior similar to the Golden Model behavior. By tracking +TME (Equation 8) across consecutive extension iterations, +we can add Candidate Knowledge to the Baseline Model +to form new Candidate Models and determine whether +these new models perform more closely to the Golden +Model (Equation 8). If the TME decreases, the Candidate +Model is considered an improvement to the Baseline +Model. If the TME increases, the Candidate Model is +considered worse than the model from previous iteration, +and the Candidate Knowledge incorporated is removed +from the model. At each iteration, all Candidate +Knowledge is added one interaction at a time and the TME +is calculated. Candidate Knowledge with the largest de- +crease of TME is incorporated. +2.5 The Breadth First and Depth First Algorithms +In this analysis, we employ two algorithms to illustrate two +different philosophies in automated assembly and exten- +sion; namely (i) incorporating the least amount of infor- +mation necessary into the model that best improves the +model and (ii) incorporating the most amount of infor- +mation into the model as long as it relates to and improves +the model. These algorithms are called the: (i) Breadth First +Addition (BFA) algorithm that compares all potential ad- +ditions against each other to only add the best supported +information at any one time, and the (ii) Depth First Addi- +tion (DFA) algorithm that incorporates any new infor- +mation that improves the model. The pseudocode for the +two algorithms is shown in Figure 2 (top) and we depict +example demonstrations for both algorithms in Figure 2 +(bottom). +The Breadth First Addition (BFA) algorithm starts by +evaluating the contribution of each new edge to decreasing +TME, that is, it simulates the model that consists of the +original Baseline Model and a selected new edge, and then +computes TME of that extended model according to Eq 8. +Next, it permanently incorporates the new edge that leads +to the largest decrease in the original TME, and then it re- +peats the steps with this new extended model, i.e., similar +to what was done with the original model, it evaluates ad- +dition of the remaining edges to this new model by com- +puting their TME values. This process is repeated until at +least one of the following conditions is satisfied: (i) the ex- +tended model matches the expected end values of the +Golden Model; (ii) there are no more edges to evaluate; (iii) +no edge can be added to the Baseline Model without in- +creasing TME. The pseudocode and the toy example for the +BFA algorithm are shown in Error! Reference source not +found. (left). +The Depth First Addition (DFA) algorithm, similar to +Figure 3. Network structure illustration, standard graph attributes, and node degree distribution histograms for different net- +work types: Erdos-Renyi random networks, Barabasi-Albert scale-free networks, and two human-curated published biological +networks, TLGL and Tcell. + + +Erdos-Renyi +Barabasi-Albert +Published +Published +Network Type +Network +Network +TLGL Model +Tcell Model +Network Structure +Number of Nodes +48.4 ± 1.4 +50.0 ± 0.0 +87 +80 +Number of Edges +85.5 ± 8.9 +96.0 ± 0.0 +171 +122 +Model Density +0.04 ± 0.00 +0.04 ± 0.00 +0.024 +0.019 +Model Average Degree +3.53 ± 0.31 +3.84 ± 0.00 +4.07 +3.05 +Undirected Model Clustering +0.06 ± 0.03 +0.20 ± 0.06 +0.28 +0.0 +Undirected Model Diameter +6.9 ± 0.7 +5.0 ± 0.4 +4.0 +12.0 +Number of Models Used +50 +50 +1 +1 +Node Degree Distribution +0.0 +89 104 +0.0 +89 10+6 +IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, MANUSCRIPT ID + +the BFA algorithm, starts with evaluation of edges by com- +puting their contribution to decreasing TME of the Base- +line Model. Different from BFA, as soon as it finds an edge +which leads to a TME lower than the current TME, it adds +that edge to the Baseline Model. These steps are then re- +peated using the new extended model and the remaining +edges. Same as for the BFA algorithm, the DFA algorithm +stops when at least one of the three conditions above, (i)- +(iii) is satisfied. The pseudocode and the toy example for +the DFA algorithm are shown in Error! Reference source n +ot found. (right). +3 RESULTS +We describe here our experimental setup, including the set +of benchmarks that we created (Sections 3.1 and 3.2), and +we follow with a discussion of the outcomes of our study +(Sections 3.3-3.5). +3.1 Benchmarks: Synthetic and Curated Models +In this analysis, we explore how the BFA and DFA algo- +rithms affect automated assembly and extension of two +types of synthetic networks and two manually curated +published biological signaling pathway networks. +The Erdos-Renyi (ER) network type is considered a ran- +dom graph and does not share many similarities to biolog- +ical networks. The Barabasi-Albert (BA) network type is a +scale-free network that has many shared characteristics +with biological networks (most notably their node-degree +distribution). Since we generated the ER and BA networks +in a random manner, we created 50 models for each net- +work type. We employed the python package, NetworkX +[38] to create all synthetic networks. +The last two networks we used in our studies are the +human-curated biological model of T cell large granular +lymphocyte (TLGL) leukemia [34] and the biological +model of naïve T cell differentiation (Tcell) [35]. The TLGL +model has been used previously [39, 40] to perform struc- +tural and dynamic analysis in order to identify potential +therapeutic targets, while the Tcell model was created to +explore the control circuitry of naïve T cell differentiation +[41][42]. +In Figure 3, we show example networks illustrating dif- +ferent structure of these models. We also list several de- +scriptive statistics for networks to demonstrate the similar- +ities and differences between these network types. Model +Density is the fraction of edges present over all possible +edges between nodes. Model Average Degree is the sum of +each node’s degree across all model nodes (with degree be- +ing the number of edges that are incident to the node), di- +vided by the number of nodes in the graph. Undirected +Model Clustering [43] is a measure of the degree to which +nodes in a graph tend to cluster together in groups of local +triangles. Undirected Model Diameter is the maximum dis- +tance from any node in the network to any other node. In +the last row in Figure 3, we provide histograms of the Node +Degree Distribution metric. In the case of ER and BA net- +works, the histograms show average values for 50 gener- +ated models. +3.2 Experimental Setup +For the purposes of the evaluation discussed here, we +assume that each model element 𝑣! ∈ 𝑉 (𝑖 = 1. . 𝑁, where +𝑁 = |𝑉|), can be in one of the three states, OFF (value 0), +LOW activity (value 1), and HIGH activity (value 2). This +assumption makes the synthetic networks comparable to +the published biological models. We randomly initialized +the synthetic networks (as they are not based on human- +curated or biological knowledge) while we initialized the +Tcell [35] and TLGL [34] models based on the values listed +in their corresponding publications. As nodes and edges +are added back into the model, we assume that the initial +state value of each model element 𝑣!, is 𝑠"!,# +5 += 1. For each +created model, we conducted 𝑅 = 100 simulation runs. For +synthetic models, we simulated ER and BA models each +with T = 2,500 time steps, while we simulated human cu- +rated models—TLGL and Tcell— for T = 5,000 time steps. +The simulation length was governed by how long each net- +work type required to reach a steady state. +3.3 Network structure and baseline information +complicate model assembly +For each Golden Model, we used five different removal +probabilities 𝑝5*?'"9@ ∈ [0.10, 0.25, 0.50, 0.75, 1.00] to ran- +domly select edges for removal from the Golden Model. +Edges that were removed formed the Candidate +Knowledge and the remaining edges formed the Baseline +Model. When 𝑝5*?'"9@ = 1.00, the Baseline Model is empty +(no edges) and both the BFA and DFA algorithms will at- +tempt to reassemble the biological networks with only +Candidate Knowledge. In all conducted studies (𝑝5*?'"9@ ∈ +[0.10, 0.25, 0.50, 0.75, 1.00]), both the BFA and DFA algo- +rithms were given the exact same Baseline Models and +Candidate Knowledge and tasked to reconstruct the +Golden Model. The recall—or ratio of edges returned to the +Baseline Model out of all removed edges—is shown in Fig- +ure 4 for each network type (Erdos-Renyi - blue, Barabasi- +Figure 4. Recall distributions for all explored scenarios, for +each network type (Erdos-Renyi - blue, Barabasi-Albert - red, +TLGL - green, Tcell – purple) and at different edge removal +probability (𝑝5*?'"9@ ∈ [0.10, 0.25, 0.50, 0.75, 1.00]). (A) BFA +algorithm results and (B) DFA algorithm results. + + + +口 +ER +口 +BA +TLGL +Tcell +A +1.0- +0.8 +recall +0.6 +0.4 +0.2 +0.0 +10 +25 +50 +75 +100 +Premoval +B +1.0- +0.8 +recall +0.6 +0.4 +0.2 +美 +0.0 +10 +25 +50 +75 +100 +PremovalBUTCHY ET AL.: TITLE +7 + +Albert - red, TLGL - green, Tcell - purple) and each algo- +rithm (BFA – part A, DFA – part B). +In general, network type drastically affects recall rates, +and for the most part, each network’s recall trends down +with higher 𝑝5*?'"9@. This makes intuitive sense as the +more edges that are removed from each network, the more +information there is to add back, and therefore the recall +has a larger denominator (i.e., the size of the Candidate +Knowledge set). Even with many missing edges, both BFA +and DFA can still converge on local minima as long as each +edge reduces TME. Both ER and Tcell network types corre- +spond to higher rates of recall than in BA and TLGL. As +both BFA and DFA add edges back based on each edge’s +effect on TME, this points to ER and Tcell networks having +more edges which tangibly reduce TME. BA networks are +noted for their hub and spoke structure, with a small num- +ber of highly connected nodes, and a large number of +sparsely connected nodes. These networks are known for +their redundancy, with the removal of an edge often com- +pensated for by the rest of the network, the behavior that +is observed in our results (Figure 4). +3.4 Model performance is difficult to encapsulate +into one metric to optimize +We also examined the relationship between the selected +𝑝5*?'"9@ and TME. We expected the TME to be proportional +to the amount of the information removed from the model +(i.e., the number of edges in the Candidate Knowledge). To +explore the effect of network structure on automated as- +sembly and extension, we evaluated the starting TME of +each Baseline Model. For each Baseline Model of each net- +work type, we calculated the actual percentage of edges re- +moved based on the 𝑝5*?'"9@. This percentage was termed +the “Percent Removed”. For each network type, we plotted +the Percent Removed from the Golden Model and the TME +before extension started. Next, starting with a Golden +Model of each network type, we removed every combina- +tion of two edges and calculated the TME of the resultant +Baseline Models. The results of these two analyses are +shown in Figure 5. +We observed from our analysis that TME is not propor- +tional to missing information and that the contribution of +different edges to the model’s TME can vary. At higher lev- +els of Percent Removed, the relationship to TME is not lin- +ear. This points to the fact that even with only a few edges +missing, a model can have quite high TME. We found that +while TME does generally increase with more information +removed, this increase is not directly proportional or con- +sistent with information removed. TME functions as a sim- +plified error function that approximates the Baseline Mod- +els deviation from Golden Model behavior but does not +completely reflect how much information is missing from +the Baseline Model or indicate how much information the +algorithm must add back. +Additionally, network type has a large influence on the +TME response to missing information. Networks like the +Barabasi-Albert networks appear more robust to infor- +mation removal, with no single edge resulting in large +changes in TME. This same behavior is not observed in the +Erdos-Renyi or human curated models where only a few +edges can strongly affect TME. Indeed, returning to Figure +4, it appears that BA networks are some of the hardest to +assemble and extend with automated methods relying on +error evaluation, due to each edge only contributing a little +to TME. A more comprehensive error function would re- +quire more information about the Golden Model’s network +Figure 5. (A) Percent Removed plotted against TME for the ER, BA, TLGL, and Tcell network types. (B) The effect of the removal +of pairs of edges from the network, first edge index indicated by the x-axis value, second edge index indicated by the y-axis +value. The TME values are represented with shades of blue, from the minimum observed (i.e., no error, TME=0, shown in white) +to the maximum observed (TME=50, shown in blue). Solid blue lines show the importance of particular edges to model perfor- +mance and TME. + + +ER +BA +TLGL +Tcell +60 +607 +60 +601 +A +40 +40J +40 +40 +. +！ +. +: +20 +20} +20 +20 +: +0 +. +. +of +0+ +01 +0 +50 +100 +0 +50 +100 +0 +50 +100 +0 +50 +100 +Percent Removed +Percent Removed +Percent Removed +Percent Removed +82 +72- +50 +B +92 +162 +72- +82 +142- +62 +0 62 +Second Edge Removed + Second Edge Removed +72- +40 +122 +52 + 62- + 52- +42 +30 +82 +42- + 32- + 62 +32 +20 +22 +22 +42 +12 +A2 +22 +12- +10 +2 +2-4 +21 +2122232425262 +7282 +122232425262728292 +.、 +22 +42 +62 +82102122142162 +2 +12 +22 +62 +72 +Do +First Edge Removed +First Edge Removed +First Edge Removed +First Edge Removed8 +IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, MANUSCRIPT ID + +structure and dynamics; however, this proves elusive as +the more information about the Golden Model there is, the +easier this problem becomes. +3.5 Initialization values play a small but important +role in network assembly +Finally, when adding Candidate Knowledge back into +the model, if a new node is introduced into the Baseline +Model, there is no information surrounding how it should +be initialized. In Figure 6 we show the effects of different +initialization +assumptions +when +adding +Candidate +Knowledge back into the model. Each network type was +extended with BFA and DFA algorithms using one of five +different initialization schemes: initializing new model el- +ements with a fixed value (0, 1 or 2), initializing the model +with the correct initialization used in the Golden Model, +and randomly assigning an initial value. In general, initial- +ization does not play a large role in automated assembly or +extension. In Figure 6, there appears to be little difference +between initialization types for the ER and BA network +types, and the TLGL model. Although the human curated +models (TLGL and Tcell) do diverge slightly from this +trend, this is much more prominent for the Tcell model, +which is an outlier, with automated model assembly and +extension suffering due to the focused nature of the model. +This is not to say that initialization is a problem that can be +disregarded in model assembly, rather it is to be considered +after the correct structure of the model has been identified. +This is particularly true in the case of logical models where +initialization can impact downstream model elements de- +pending upon nature of the logic functions that are used. +For example, initializing to 0 a model element involved in +many “AND” operations will affect downstream model el- +ements. As described in Section 2.1, we used summation +functions in this analysis. This choice likely made the role +of initialization less important, as the inclusion of a new +edge (and thus, a new regulator for some element in the +model) would not impact the effect of other regulators in +such a substantial way as would be present with logic +update functions. +Taken together, the discussion in Sections 3.3-3.5 and +Figures 4-6 demonstrate the key difficulties to automated +model assembly and extension. Several methods exist +which create such automated pipelines but do not focus on +how they incorporate biological information into executa- +ble models [27, 44, 45]. To date, only a few methods have +been proposed to automatically assemble and extend mod- +els, while also evaluating the available information and its +impact on the created executable model [28-30]. Still, even +these methods do not fully assess the structural and dy- +namic impacts of adding new biological information to an +executable model, and therefore do not address the com- +plexities to this problem. +4 CONCLUSION +In this paper, we have presented an automated assembly +and extension pipeline to depict the types and magnitudes +of the problems facing computational and system biolo- +gists as they work to solve automated model assembly. +Through the largest assembly and extension analysis of +synthetic and human-curated models to date, we have +characterized the complexities of the automated model as- +sembly problem. Our findings demonstrate that iterative +model assembly, devoid of context, lacking starting struc- +tural information in the form of a baseline model, and +without robust dynamic information describing the golden +model’s behavior, is intractable. More often, model assem- +bly creates models which perform similarly in dynamics, +but do not represent the full information of a full “Golden” +network. +In this paper, we have demonstrated that particular fo- +cus must be paid to a model’s structure and baseline infor- +mation, as these can complicate model assembly. In pick- +ing a metric to optimize during model assembly, we have +illustrated that a single metric more often serves to sim- +plify the golden model, rather than recapitulate it. Lastly, +we have shown that initializing model elements only play +Figure 6. The ten networks for each network type were disassembled and then reassembled (either through BFA or DFA) under +different initialization schemes. In “0” new model elements are initialized with a starting value of 0. Similarly, “1” and “2” +follow similar schemes. “Golden” initializes the model element as it would be seen in the Golden Model, while “Random” +randomly initializes the model element. In each assembly, the number of edges added back were recorded and used to calculate +each assembly method’s recall. + + +ER +BA +TLGL +Tcell +1.0- +1.0- +1.0 +1.0- +0.8- +0.8- +0.8- +0.8- +50% +0.4- +0.4- +TT +IT +0.2- +0.2- +0.2- +0.2 +0.0 +0.0 +0.0 +0.0. +BFA +DFA +BFA +DFA +BFA +DFA +BFA +DFA +1.0- +1.0 +1.0- +1.0 +0.8- +0.8- +0.8- +0.8- + 0.6 + 0.6- +100% +& 0.4- +P 0.4 +0.2- +1 +0.2- +0.2- +0.2 +一 +0.0 +0.0 +0.0 +DFA +0.0 +BFA +DFA +BFA +BFA +DFA +BFA +DFA +0 +1 +□2 + Golden +RandomBUTCHY ET AL.: TITLE +9 + +a small role in network assembly. +In future work, we plan to further investigate the effect +of network type, additional parametrization of update +functions (e.g., timing effects), methods to determine initial +state for simulations, and other error functions on the qual- +ity of recommended Candidate Models. We will also ex- +plore the effect of erroneous Candidate Knowledge on ex- +tension methods. +ACKNOWLEDGMENT +NMZ is the corresponding author. 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Morel, and J. +Faeder, "Model Predicts Duration of T Cell Stimulation is a +Critical Determinant of Cell Fate and Plasticity, under +submission," 2013. +[43] +J. Saramäki, M. Kivelä, J.-P. Onnela, K. Kaski, and J. Kertesz, +"Generalizations of the clustering coefficient to weighted +complex networks," Physical Review E, vol. 75, no. 2, p. +027105, 2007. +[44] +F. Büchel et al., "Path2Models: large-scale generation of +computational models from biochemical pathway maps," +BMC Systems Biology, vol. 7, no. 1, pp. 116-116, 2013, doi: +10.1186/1752-0509-7-116. +[45] +B. M. Gyori, J. A. Bachman, K. Subramanian, J. L. Muhlich, +L. Galescu, and P. K. Sorger, "From word models to +executable models of signaling networks using automated +assembly," Molecular systems biology, vol. 13, no. 11, 2017. + +Adam A. Butchy. Adam is a PhD candi- +date in the Bioengineering Department +at the University of Pittsburgh. Adam +completed a B.S. in Chemical Engineer- +ing and a B.S. in Biochemistry at Villa- +nova University. He is working on Dis- +crete Modeling of Macrophage Activa- +tion and its role in the cancer microenvi- +ronment and lung. + +Cheryl A. Telmer Dr. Telmer is a Re- +search Biologist at Carnegie Mellon Uni- +versity. Cheryl and Natasa began work- +ing together as iGEM advisors in 2013 +and have expanded their collaborations +through the DARPA Big Mechanism and +World Modelers programs. Biologists +are constantly trying new tools that have +the potential to improve our understand- +ing of complex systems, and the standardized representation and +computational modeling approaches being developed by the Melody +Lab are a great contribution. + +Natasa Miskov-Zivanov Dr. Miskov-Zi- +vanov is an Assistant Professor of Elec- +trical and Computer Engineering, Bioen- +gineering, and Computational and Sys- +tems Biology at the University of Pitts- +burgh. She received a B.Sc. degree in +electrical engineering and computer sci- +ence from University of Novi Sad, Serbia +and M.Sc. and Ph.D. degrees in electri- +cal and computer engineering from Carnegie Mellon University. Be- +fore joining University of Pittsburgh as a faculty, she spent several +years as a postdoctoral researcher in Computational and Systems Bi- +ology at the University of Pittsburgh, and as research scientist and +instructor in Computer Science and in Electrical and Computer Engi- +neering at Carnegie Mellon University. Dr. Miskov-Zivanov’s research +interests include hybrid, knowledge-driven and data-driven, model +recommendation and reasoning for complex systems with applica- +tions in systems and synthetic biology. + + diff --git a/49FIT4oBgHgl3EQf7St_/content/tmp_files/load_file.txt b/49FIT4oBgHgl3EQf7St_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..35086d8395b7e18981507a28cff66df31b0abdea --- /dev/null +++ b/49FIT4oBgHgl3EQf7St_/content/tmp_files/load_file.txt @@ -0,0 +1,1001 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf,len=1000 +page_content='IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, MANUSCRIPT ID 1 Automating Knowledge-Driven Model Recommendation: Methodology, Evaluation, and Key Challenges Adam A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Butchy, Cheryl A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Telmer, and Natasa Miskov-Zivanov Abstract—There is significant interest in using existing repositories of biological entities, relationships, and models to automate biological model assembly and extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Current methods aggregate human-curated biological information into executable, simulatable models, but these models do not resemble human curated models and do not recapitulate experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Here, we outline the process of automated model assembly and extension, while demonstrating it on both synthetic models and human- curated models of biological signaling networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' We begin with an iterative, greedy, and combinatoric approach to automated assembly and demonstrate the key difficulties inherent to contextless assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' We publicly release the software used in this paper to enable further exploration of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Index Terms— Automatic Model Creation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Biological Networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Extending Biological Networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Model Construction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Network Reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' —————————— u —————————— 1 INTRODUCTION omputational approaches to modeling large complex systems standardize the representation of knowledge, while simulation of computational models illuminates the dynamics of systems, allowing for discoveries and theoret- ical advances [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Due to the complexity and redundancy of biological systems, computational models are difficult and laborious to create and update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' There are two main ap- proaches to modeling these systems, bottom-up and top- down [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In a bottom-up approach, known molecular in- teractions are assembled into a model to help explain the system’s behavior and predict how the system will re- spond to new stimuli or inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' This method has been used extensively by biologists, biochemists, and molecular biol- ogists to manually create models based on the interactions within cells involved in signaling that are supported by sci- entific literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In a top-down approach, experimental data—usually collected with high-throughput methods— is used to infer correlations between element behavior and determine causal relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Top-down approaches em- ploy many different methods such as Bayesian Inference [3], ANOVA calculations [4], and Fuzzy Logic [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In both the mechanistic bottom-up approach and the data-driven top-down approach, the model is used to predict the be- havior of individual elements in the network [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Re- cently, there has been a push to integrate the two methods, using experimental data to inform the bottom-up ap- proach, and incorporating prior knowledge into the top- down approach to reduce the number of potential models [8-12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Despite these hybrid approaches, this problem re- mains a combinatoric one, with large, complex systems be- ing prohibitively difficult to investigate and model manu- ally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' It is a direct result of these factors that system and com- putational biologists have endeavored to automate the process of model creation and extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' To automatically create models, information can be extracted from litera- ture, queried from databases, or taken from existing path- ways and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Public databases such as Reactome [13], MetaCyc [14], OmniPath [15], and STRING [16] offer easy access to millions of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Additionally, there exist a number of model databases with published models that are publicly available such as The Nature Pathway Interac- tion Database [17], WikiPathways [18], BioModels [19], the Cell Collective [20], and KEGG pathways [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' These data- bases contain highly targeted, curated published and un- published models which are created for specific biological context and may not be generalizable to explain other phe- nomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' When new interactions are discovered, and de- scribed in a scientific publication, state-of-the-art machine reading engines such as REACH [22], TRIPS [23], and EVEX [24] can extract them, together with other relevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' These automated readers are able to extract tens of thousands of biological entity interactions from hundreds of papers in a few hours, and produce a ma- chine-readable, structured output [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Despite this abun- dance of available interactions, there is still no efficient way to assemble them into accurate models that correctly reflect the system under investigation and the same biolog- ical context and recapitulate the observed experimental be- havior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Recently, a few tools, such as Path2Models [25] and IN- DRA [26, 27], have been created to help modelers collect biological interactions, assemble a model, and perform xxxx-xxxx/0x/$xx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='00 © 200x IEEE Published by the IEEE Computer Society ———————————————— A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Butchy is with the Department of Bioengineering, University of Pitts- burgh, Pittsburgh, PA 15213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' E-mail: adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='butchy@pitt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Telmer is with the Department of Biological Sciences, Carnegie Mellon University, Pittsburgh, PA 15213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' E-mail: ctelmer@cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Miskov-Zivanov is with the Departments of Electrical and Computer Engineering, Bioengineering, and Computational Biology, University of Pittsburgh, Pittsburgh, PA 15213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' E-mail: nmzivanov@pitt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' C 2 IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, MANUSCRIPT ID simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' These tools assemble quantitative and quali- tative models using available pathway information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' how- ever, the quality of the assembled models is dependent upon the modeling approach, and the granularity of the information they are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' These techniques rely on accu- rate information, and their performance suffers when the interaction information is incomplete, from a different bio- logical context, or erroneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Other methods have been proposed to automatically expand, test, and select the best model, with respect to a given performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' These approaches integrate stochastic model simulations with statistical model checking only [28], or also incorporating Markov clustering [29], or genetic algorithm [30], and therefore have different strengths and weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The Markov clustering approach to model extension is well suited for the combinatorial explosion in the number of possible model extensions while the genetic algorithm ap- proach is overwhelmed by large number of extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Markov clustering prioritizes strongly connected compo- nents at the expense of interactions involving nodes of low degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The genetic algorithm explores the effect of single extensions distributed throughout the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In this work, we examine the complexities inherent to automatic model assembly and extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' We use two novel algorithms, Breadth First Addition (BFA) and Depth First Addition (DFA), which utilize the same principles as the breadth-first search and depth-first search algorithms in network studies [31] to illustrate the key limitations of iterative model assembly and extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In contrast to pre- vious work [28-30], these methods not only represent a new approach to bottom-up model assembly but are also used to demonstrate the existence of key biological prop- erties which hinder automated modeling of biological sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' We demonstrate these properties using both syn- thetic networks, Erdös-Rényi random networks (ER) [32] and Barabási-Albert scale-free networks (BA) [33], as well as two published expert curated and validated models, a T cell large granular lymphocyte (TLGL) leukemia model [34], and a model of naïve Tcell differentiation (Tcell) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' By using different network structures, we are able to more comprehensively explore automated model assembly and identify the main difficulties with the BFA and DFA ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 2 METHODS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='1 Discrete Models and Simulations The underlying structure of models that we study here is a network 𝐺(𝑉, 𝐸), where 𝑉 is a set of nodes (model ele- ments), and 𝐸 is a set of directed edges (regulatory influ- ences between elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' A few toy examples of such net- works are shown in Figure 1 (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Model elements usually represent proteins, genes, chemicals, or biological pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' For each model element 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' ∈ 𝑉 (𝑖 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 𝑁, where 𝑁 = |𝑉|), we define an update rule 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' = 𝑓"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (𝑣#, 𝑣$, … , 𝑣%), which can either be a constant (input nodes in network 𝐺) or it can depend on a subset of elements from 𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In the lat- ter case, for each element 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' this subset is often referred to as an influence set for 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' and it consists of its positive (acti- vating) and negative (inhibiting) regulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Positive regulators of 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=" comprise set 𝑉&'( !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' and are represented with regular arrowheads in Figure 1 (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Negative regulators of 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' comprise set 𝑉)*+ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' and are represented with blunt arrow- heads in Figure 1 (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The high throughput retrieval of interaction infor- mation from literature typically only includes knowledge of the sign of influence (positive or negative) and rarely ad- ditional information about relationships between regula- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In such cases, logic functions and elements with two levels, 0 (low) and 1 (high), have been found most suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' To broaden the application beyond just Boolean functions to other cases where interactions were enriched either through manual curation or more specific information re- trieval, we will assume that each element 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' can have 𝐿!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' number of discrete levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' While the choice of function does not affect the main algorithms described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='2, in order to simulate models, and closely approximate different functions, including Boolean, we adopted the common approach that computes a (weighted) sum of reg- ulator values to determine element update values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The general form of this function is: 𝑔"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' = 𝑓"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (𝑣#, 𝑣$, … , 𝑣%) = ∑ 𝑤,𝑣, ""∈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='#$% !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' − ∑ 𝑤/𝑣/ "&∈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=" '() !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (1) The weighting factors 𝑤, and 𝑤/ can be used to account for different influence strengths for regulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' To remain within boundaries of the allowed levels for element 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='. 𝐿!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' − 1), the function 𝑔"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' is then used to determine a suitable increment/decrement for 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', 𝛿"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' = 𝑓(𝑔"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' ), such that: 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',)*12 = 8 0 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' + 𝛿"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' ≤ 0 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' + 𝛿"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 0 < 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' + 𝛿"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' < 𝐿!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' − 1 𝐿!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' − 1 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' + 𝛿"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' ≥ 𝐿!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' − 1 (2) Together, the set of model elements 𝑉, element influences forming the set 𝐸, and the set of element update rules 𝐹, comprise an Executable Model, ℳ(𝑉, 𝐸, 𝐹), a model that in- cludes all the necessary information for simulation and dy- namic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' We use the Discrete, Stochastic, Heterogeneous simula- tor (DiSH) [36] which allows for simulations of discrete models with various types of update functions, and has several different simulation schemes, that can be either de- terministic or stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' For the analysis we conducted here, we used the USB-RSQ simulation scheme in DiSH (uniform, step-based, random-order, sequential update scheme, described in detail in [36]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' It has been shown pre- viously [36, 37] that, by taking into account the random- ness in timing of signaling events, the USB-RSQ simulation scheme is able to recapitulate the network dynamics within cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' DiSH simulates the models starting from an initial state 𝒒ℳ,4 = A𝑠"*,4, 𝑠"+,4, … , 𝑠",,4C (assigned before simula- tions), where 𝑠"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',4 denotes the state value of element 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' at time point 𝑡 = 0, and for a pre-defined number of time steps, 𝑇 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', when the steady state is reached).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Each such simulation run, 𝑟, yields for every model element 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' ∈ 𝑉, a trajectory of values, 𝒔"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 5 = A𝑠"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',# 5 , 𝑠"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',$ 5 , … 𝑠"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',6 5 C, where 𝑠"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',2 5 is the state value of element 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' at time point 𝑡 (𝑡 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' , 𝑇) within run 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Due to the randomness of the update scheme, element trajectories may vary across multiple runs that BUTCHY ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' : TITLE 3 start with the same initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Therefore, for the same time step 𝑡, following the approach from [36], we compute the mean of values 𝑠"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',2 5 across different runs, to obtain av- erage trajectories for all elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' More formally, we com- pute an average element trajectory of element 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' as: 𝒔H"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' = 1 𝑅 J 𝒔"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 5 7 58# = 1 𝑅 JA𝑠"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',# 5 , 𝑠"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',$ 5 , … 𝑠"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',6 5 C 7 58# = A𝑠̅"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',#, 𝑠̅"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',$, … 𝑠̅"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',6C (3) where 𝑅 is the overall number of conducted simulation runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' For example, in Figure 1 (B), we illustrate simulation trajectories for elements of the toy models in Figure 1 (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' We denote average model state for model ℳ(𝑉, 𝐸, 𝐹) at time step 𝑡 as a vector of average element states at time step 𝑡: 𝒒ℳ,2 9"+ = A𝑠̅"*,2, 𝑠̅"+,2, … , 𝑠̅",,2C (4) We define model behavior resulting from a specific initial model state 𝒒ℳ,4 = (S#, S$, … , S%) as: 𝑸ℳ = A𝒒ℳ,4, 𝒒ℳ,# 9"+, … , 𝒒ℳ,6 9"+C (5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='2 Extension method inputs We define here inputs used by extension methods and by our evaluation methodology: Baseline Model, Golden Model, and Candidate Knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Existing models of a system of interest are often lever- aged and contextualized for a specific purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The Base- line Model is the existing, high confidence model before updating with extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' As a special case, we can also assume that the Baseline Model is an empty network with no nodes or edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The Golden Model is assumed to contain all relevant knowledge about the system, including accurate element relationships and update functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The Candidate Knowledge is a set of directed edges, including their source and target nodes, which are candidates for ad- dition to the Baseline Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Given the Golden Model knowledge, through simula- tions, for different initial states representing different con- ditions and scenarios, we can obtain Golden Model behav- ior, 𝑸:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', as in Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 𝑸:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' represents the true expected behavior of the system being modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' As part of 𝑸:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', we also obtain the average Golden Model state at the final sim- ulation time step 𝑇 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', steady state), 𝒒:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',6 9"+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The above definition of Golden Model is important for the rest of our discussion since Golden Model is used as an input to our evaluation methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' However, in real sce- narios, the Golden Model is usually not known in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Instead, the goal of model assembly and extension algo- rithms is to discover the Golden Model, while only the real system behavior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', measured state values for system components, may be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The system state data can be used to form the target behavior 𝑸N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Ideally, the Golden Model behavior is identical to the target behavior, 𝑸:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' = 𝑸N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The target state at time 𝑇 is part of the target behavior and is denoted as 𝒒O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' As will be detailed in the following sub-sections, exten- sion algorithms start with the Baseline Model for which 𝑸<;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' ≠ 𝑸N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Next, they add selected edges from the Candi- date Knowledge to create new models, called Candidate Models, which are then iteratively updated and simulated to obtain 𝑸=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' in each iteration, and to ultimately find a model that most closely reproduces the target behavior 𝑸N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' A toy example illustrating directed cyclic network models explored in this work and the flow of the proposed meth- odology for evaluating extension algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (A) (top) An example Golden Model used in evaluation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (middle) Example input graphs, Candidate Knowledge, and Baseline Model, used in extension methods ([28-30] and this work);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (bottom) An example Candidate Model recommended by extension methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (B) Average element trajectories obtained from stochastic simulation for the three example models (Golden, Baseline, and Candidate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (C) An example iterative procedure that uses the Total Model Error (TME) metric to evaluate each intermediate Candidate Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Golden Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Iterative Extension* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Golden Model Behavior ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='TME ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Edge Removal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Candidate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Baseline Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Baseline Model Behavior ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Knowledge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='(Baseline) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Iteration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Where: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='a = TME of Baseline Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Model Extension ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='b = TME of Candidate Model 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='c = TME of Candidate Model 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Candidate Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Candidate Model Behavior ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='d = TME of Candidate Model 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='e = TME of Candidate Model 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='f = TME of Candidate Model 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='→ = Path of minimizing TME ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='→ = Explored TME paths ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='F4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' MANUSCRIPT ID 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='3 Model evaluation metric Given two models, ℳ# and ℳ$, if they have the same ele- ment sets, 𝑉ℳ* ≡ 𝑉ℳ+ ≡ 𝑉 (𝑁 = |𝑉|), and if we simulate them starting from the same initial state, 𝒒ℳ*,4 = 𝒒ℳ+,4 = A𝑠"*,4, 𝑠"+,4, … , 𝑠",,4C, to obtain their behaviors, 𝑸ℳ* and 𝑸ℳ+, respectively, we can compute the difference between the two model behaviors, Δ2A𝑸ℳ*, 𝑸ℳ+C at any simulation time step 𝑡 as: Δ2A𝑸ℳ*, 𝑸ℳ+C = ∑ S𝑠̅"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',2 ℳ* − 𝑠̅"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',2 ℳ+S % !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='8# (6) In other words, Δ2 finds the absolute difference between an element’s average state in time step 𝑡, in model ℳ# (𝑠̅"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',2 ℳ*) and in model ℳ$ (𝑠̅"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',2 ℳ+) and sums these differences across all model elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' From (6), we derive the Total Model Error (TME) metric, as Δ6, when 𝑡 = 𝑇, between a Candidate Model behavior 𝑸=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' and known target behavior 𝑸N: TMEA𝑸=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', 𝑸NC = Δ6A𝑸=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', 𝑸NC = ∑ S𝑠̅"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',6 =;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' − 𝑠̂"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',6S % !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='8# (7) Or, in the case when a Golden Model is used: TME(𝑸=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', 𝑸:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=') = Δ6(𝑸=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', 𝑸:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=') = ∑ S𝑠̅>!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',6 =;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' − 𝑠̅>!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',6 :;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='S % !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='8# (8) Besides the above defined Δ2, other types of functions could be used to compute the difference between two mod- els, such as the squared error, or more statistic-based eval- uation methods like the Chi-squared test to compare the Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The Breadth and Depth First Addition (BFA and DFA, respectively) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Top: The pseudocode for the two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Bottom: An example illustrating the Candidate Knowledge and Baseline Model inputs and steps for BFA and DFA algorithms: (A, D) The inputs to the BFA and DFA algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (B) In the BFA extension process, the Baseline Model is extended with single interactions from Candidate Knowledge and the TME is calculated for each Candidate Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The Candidate Model with the lowest TME is selected and becomes the Baseline Model for the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (E) In the DFA extension process, the Baseline Model is extended with a single interaction from Candidate Knowledge and the TME is calculated to determine if the Candidate Model has a lower TME than the Baseline Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' As soon as the TME decreases, that edge of Candidate Knowledge is incorporated into the Candidate Model, and it becomes the Baseline Model for the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (C, F) For both algorithms, the process is repeated with the remaining Candidate Knowledge until all edges are added back, the TME reaches zero, or there are no edges that reduce the TME below its current lowest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Algorithm: Breadth First Addition (BFA) Algorithm: Depth First Addition (DFA) Input: baseline model (MBM), list of edges (ENEw), TME of the baseline model Input: baseline model, list of edges, expected performance of the golden (TMEBM), expected performance of the golden model (QGM) model, current TME Output: extended baseline model that minimizes the TME Output: extended baseline model that minimizes the TME 1: while (TMEBM!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='= 0) and (ENEw !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='= [) 1: EADDED = FALSE 2: Initialize scores = [] 2: for edge in ENEw: 3: for edge in ENEw: 3: 4: McM = a candidate model is created by adding the edge to the MBM 4: simulate McM 5: simulate McM 5: TME(QGM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='QcM ) use the TME function to compare 6: TME(QGM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='QcM ) use the TME function to compare the 6: the candidate model to the expected performance 7: 7: candidate model to the expected performance of the golden model of the golden model 8: 8: Append TMEcM to the scores list if TMEcM < TMEBM: 9: 9: end for McM = a candidate model is created by 10: 10: find index = min(scores) adding the edge to the MBM 11: 11: TMEcM = scores(index) MBM = McM 12: 12: if TMEcM < TMEBM: ENEw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='delete(edge) 13: 13: McM = a candidate model is created by adding 14: 14: the edge to the MBM EADDED = TRUE 15: 15: MBM = McM exit for loop 16: 16: 17: ENEw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='delete(index) 17: end if 18: TMEBM = TMEcM 18: end for 19: else 19: if (EADDED =- FALSE) 20: return MBM 20: return MBM 21: end if 21: end if 22: end while 22: : end while 23: return MBM 23: return MBMCandidate Baseline Model Candidate Baseline Model A Knowledge TME = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 Knowledge TME = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 BFA DFA B D Inputs Inputs 2) 2) 3) 3) B E First First Extension Extension Round Round Extension 1 Extension 2 Extension 3 Extension 1 TME = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 TME = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 TME = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 TME = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 c : Second Second Extension Extension Round Round Extension 2&1 Extension 2&3 Extension 1&2 Extension 1&3 TME = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 TME = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 TME = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 TME = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0BUTCHY ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' : TITLE 5 distribution of model states at time step 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' We use the ab- solute difference of the model’s end state (𝑡 = 𝑇) for a few reasons: it would not exaggerate the effect of large differ- ences (as would be observed in the squared error);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' it is less computationally expensive than the Chi-squared test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' and it more accurately matches how computational biologists compare computational model simulations against sparse biological measurements, where the full time-course of the model elements is often unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='4 Methodology for evaluating model extension In this work, we are interested in evaluating automated model extension, that is, the limitations of automatically extending the Baseline Model with behavior 𝑸<;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' to achieve the target or Golden Model behavior 𝑸:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='. There- fore, in our studies we assume that the Golden Model is known, and to obtain Baseline Models we use the proce- dure illustrated in Figure 1 (A) and described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' For a given Golden Model, we create multiple Baseline Models by removing edges from the Golden Model, in order to disrupt its behavior and to determine whether the ex- tension algorithms are able to recover the Golden Model from a range of Baseline Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The removed edges form the Candidate Knowledge sets (Figure 1 (A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The extension algorithms are given the Baseline Model and the Candi- date Knowledge and tasked with extending the Baseline Model using edges from the Candidate Knowledge, to cre- ate Candidate Models (Figure 1 (A)) and reproduce the Golden Model behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Using the DiSH simulator, we simulate the Golden, Baseline, and Candidate Models to observe how elements of each model behave over time, and to obtain model be- haviors 𝑸:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', 𝑸<;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', 𝑸=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', respectively (Figure 1 (B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The goal of this procedure is to find Candidate Model(s) with be- havior similar to the Golden Model behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' By tracking TME (Equation 8) across consecutive extension iterations, we can add Candidate Knowledge to the Baseline Model to form new Candidate Models and determine whether these new models perform more closely to the Golden Model (Equation 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' If the TME decreases, the Candidate Model is considered an improvement to the Baseline Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' If the TME increases, the Candidate Model is considered worse than the model from previous iteration, and the Candidate Knowledge incorporated is removed from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' At each iteration, all Candidate Knowledge is added one interaction at a time and the TME is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Candidate Knowledge with the largest de- crease of TME is incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='5 The Breadth First and Depth First Algorithms In this analysis, we employ two algorithms to illustrate two different philosophies in automated assembly and exten- sion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' namely (i) incorporating the least amount of infor- mation necessary into the model that best improves the model and (ii) incorporating the most amount of infor- mation into the model as long as it relates to and improves the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' These algorithms are called the: (i) Breadth First Addition (BFA) algorithm that compares all potential ad- ditions against each other to only add the best supported information at any one time, and the (ii) Depth First Addi- tion (DFA) algorithm that incorporates any new infor- mation that improves the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The pseudocode for the two algorithms is shown in Figure 2 (top) and we depict example demonstrations for both algorithms in Figure 2 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The Breadth First Addition (BFA) algorithm starts by evaluating the contribution of each new edge to decreasing TME, that is, it simulates the model that consists of the original Baseline Model and a selected new edge, and then computes TME of that extended model according to Eq 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Next, it permanently incorporates the new edge that leads to the largest decrease in the original TME, and then it re- peats the steps with this new extended model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', similar to what was done with the original model, it evaluates ad- dition of the remaining edges to this new model by com- puting their TME values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' This process is repeated until at least one of the following conditions is satisfied: (i) the ex- tended model matches the expected end values of the Golden Model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (ii) there are no more edges to evaluate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (iii) no edge can be added to the Baseline Model without in- creasing TME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The pseudocode and the toy example for the BFA algorithm are shown in Error!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Reference source not found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The Depth First Addition (DFA) algorithm, similar to Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Network structure illustration, standard graph attributes, and node degree distribution histograms for different net- work types: Erdos-Renyi random networks, Barabasi-Albert scale-free networks, and two human-curated published biological networks, TLGL and Tcell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Erdos-Renyi Barabasi-Albert Published Published Network Type Network Network TLGL Model Tcell Model Network Structure Number of Nodes 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 87 80 Number of Edges 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='5 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='9 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 171 122 Model Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='019 Model Average Degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='05 Undirected Model Clustering 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 Undirected Model Diameter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 Number of Models Used 50 50 1 1 Node Degree Distribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 89 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 89 10+6 IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, MANUSCRIPT ID the BFA algorithm, starts with evaluation of edges by com- puting their contribution to decreasing TME of the Base- line Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Different from BFA, as soon as it finds an edge which leads to a TME lower than the current TME, it adds that edge to the Baseline Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' These steps are then re- peated using the new extended model and the remaining edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Same as for the BFA algorithm, the DFA algorithm stops when at least one of the three conditions above, (i)- (iii) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The pseudocode and the toy example for the DFA algorithm are shown in Error!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Reference source n ot found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 3 RESULTS We describe here our experimental setup, including the set of benchmarks that we created (Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='2), and we follow with a discussion of the outcomes of our study (Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='1 Benchmarks: Synthetic and Curated Models In this analysis, we explore how the BFA and DFA algo- rithms affect automated assembly and extension of two types of synthetic networks and two manually curated published biological signaling pathway networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The Erdos-Renyi (ER) network type is considered a ran- dom graph and does not share many similarities to biolog- ical networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The Barabasi-Albert (BA) network type is a scale-free network that has many shared characteristics with biological networks (most notably their node-degree distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Since we generated the ER and BA networks in a random manner, we created 50 models for each net- work type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' We employed the python package, NetworkX [38] to create all synthetic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The last two networks we used in our studies are the human-curated biological model of T cell large granular lymphocyte (TLGL) leukemia [34] and the biological model of naïve T cell differentiation (Tcell) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The TLGL model has been used previously [39, 40] to perform struc- tural and dynamic analysis in order to identify potential therapeutic targets, while the Tcell model was created to explore the control circuitry of naïve T cell differentiation [41][42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In Figure 3, we show example networks illustrating dif- ferent structure of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' We also list several de- scriptive statistics for networks to demonstrate the similar- ities and differences between these network types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Model Density is the fraction of edges present over all possible edges between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Model Average Degree is the sum of each node’s degree across all model nodes (with degree be- ing the number of edges that are incident to the node), di- vided by the number of nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Undirected Model Clustering [43] is a measure of the degree to which nodes in a graph tend to cluster together in groups of local triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Undirected Model Diameter is the maximum dis- tance from any node in the network to any other node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In the last row in Figure 3, we provide histograms of the Node Degree Distribution metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In the case of ER and BA net- works, the histograms show average values for 50 gener- ated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='2 Experimental Setup For the purposes of the evaluation discussed here, we assume that each model element 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' ∈ 𝑉 (𝑖 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 𝑁, where 𝑁 = |𝑉|), can be in one of the three states, OFF (value 0), LOW activity (value 1), and HIGH activity (value 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' This assumption makes the synthetic networks comparable to the published biological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' We randomly initialized the synthetic networks (as they are not based on human- curated or biological knowledge) while we initialized the Tcell [35] and TLGL [34] models based on the values listed in their corresponding publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' As nodes and edges are added back into the model, we assume that the initial state value of each model element 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', is 𝑠"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=',# 5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' For each created model, we conducted 𝑅 = 100 simulation runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' For synthetic models, we simulated ER and BA models each with T = 2,500 time steps, while we simulated human cu- rated models—TLGL and Tcell— for T = 5,000 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The simulation length was governed by how long each net- work type required to reach a steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='3 Network structure and baseline information complicate model assembly For each Golden Model, we used five different removal probabilities 𝑝5*?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' \'"9@ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='75, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='00] to ran- domly select edges for removal from the Golden Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Edges that were removed formed the Candidate Knowledge and the remaining edges formed the Baseline Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' When 𝑝5*?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' \'"9@ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='00, the Baseline Model is empty (no edges) and both the BFA and DFA algorithms will at- tempt to reassemble the biological networks with only Candidate Knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In all conducted studies (𝑝5*?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' \'"9@ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='75, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='00]), both the BFA and DFA algo- rithms were given the exact same Baseline Models and Candidate Knowledge and tasked to reconstruct the Golden Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The recall—or ratio of edges returned to the Baseline Model out of all removed edges—is shown in Fig- ure 4 for each network type (Erdos-Renyi - blue, Barabasi- Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Recall distributions for all explored scenarios, for each network type (Erdos-Renyi - blue, Barabasi-Albert - red, TLGL - green, Tcell – purple) and at different edge removal probability (𝑝5*?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' \'"9@ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='75, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='00]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (A) BFA algorithm results and (B) DFA algorithm results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 口 ER 口 BA TLGL Tcell A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='8 recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 10 25 50 75 100 Premoval B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='8 recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='2 美 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 10 25 50 75 100 PremovalBUTCHY ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' : TITLE 7 Albert - red, TLGL - green, Tcell - purple) and each algo- rithm (BFA – part A, DFA – part B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In general, network type drastically affects recall rates, and for the most part, each network’s recall trends down with higher 𝑝5*?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='\'"9@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' This makes intuitive sense as the more edges that are removed from each network, the more information there is to add back, and therefore the recall has a larger denominator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', the size of the Candidate Knowledge set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Even with many missing edges, both BFA and DFA can still converge on local minima as long as each edge reduces TME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Both ER and Tcell network types corre- spond to higher rates of recall than in BA and TLGL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' As both BFA and DFA add edges back based on each edge’s effect on TME, this points to ER and Tcell networks having more edges which tangibly reduce TME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' BA networks are noted for their hub and spoke structure, with a small num- ber of highly connected nodes, and a large number of sparsely connected nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' These networks are known for their redundancy, with the removal of an edge often com- pensated for by the rest of the network, the behavior that is observed in our results (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='4 Model performance is difficult to encapsulate into one metric to optimize We also examined the relationship between the selected 𝑝5*?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' \'"9@ and TME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' We expected the TME to be proportional to the amount of the information removed from the model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', the number of edges in the Candidate Knowledge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' To explore the effect of network structure on automated as- sembly and extension, we evaluated the starting TME of each Baseline Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' For each Baseline Model of each net- work type, we calculated the actual percentage of edges re- moved based on the 𝑝5*?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='\'"9@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' This percentage was termed the “Percent Removed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' For each network type, we plotted the Percent Removed from the Golden Model and the TME before extension started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Next, starting with a Golden Model of each network type, we removed every combina- tion of two edges and calculated the TME of the resultant Baseline Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The results of these two analyses are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' We observed from our analysis that TME is not propor- tional to missing information and that the contribution of different edges to the model’s TME can vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' At higher lev- els of Percent Removed, the relationship to TME is not lin- ear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' This points to the fact that even with only a few edges missing, a model can have quite high TME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' We found that while TME does generally increase with more information removed, this increase is not directly proportional or con- sistent with information removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' TME functions as a sim- plified error function that approximates the Baseline Mod- els deviation from Golden Model behavior but does not completely reflect how much information is missing from the Baseline Model or indicate how much information the algorithm must add back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Additionally, network type has a large influence on the TME response to missing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Networks like the Barabasi-Albert networks appear more robust to infor- mation removal, with no single edge resulting in large changes in TME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' This same behavior is not observed in the Erdos-Renyi or human curated models where only a few edges can strongly affect TME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Indeed, returning to Figure 4, it appears that BA networks are some of the hardest to assemble and extend with automated methods relying on error evaluation, due to each edge only contributing a little to TME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' A more comprehensive error function would re- quire more information about the Golden Model’s network Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (A) Percent Removed plotted against TME for the ER, BA, TLGL, and Tcell network types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' (B) The effect of the removal of pairs of edges from the network, first edge index indicated by the x-axis value, second edge index indicated by the y-axis value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The TME values are represented with shades of blue, from the minimum observed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', no error, TME=0, shown in white) to the maximum observed (TME=50, shown in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Solid blue lines show the importance of particular edges to model perfor- mance and TME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' ER BA TLGL Tcell 60 607 60 601 A 40 40J 40 40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' : 20 20} 20 20 : 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' of 0+ 01 0 50 100 0 50 100 0 50 100 0 50 100 Percent Removed Percent Removed Percent Removed Percent Removed 82 72- 50 B 92 162 72- 82 142- 62 0 62 Second Edge Removed Second Edge Removed 72- 40 122 52 62- 52- 42 30 82 42- 32- 62 32 20 22 22 42 12 A2 22 12- 10 2 2-4 21 2122232425262 7282 122232425262728292 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='、 22 42 62 82102122142162 2 12 22 62 72 Do First Edge Removed First Edge Removed First Edge Removed First Edge Removed8 IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, MANUSCRIPT ID structure and dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' however, this proves elusive as the more information about the Golden Model there is, the easier this problem becomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='5 Initialization values play a small but important role in network assembly Finally, when adding Candidate Knowledge back into the model, if a new node is introduced into the Baseline Model, there is no information surrounding how it should be initialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In Figure 6 we show the effects of different initialization assumptions when adding Candidate Knowledge back into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Each network type was extended with BFA and DFA algorithms using one of five different initialization schemes: initializing new model el- ements with a fixed value (0, 1 or 2), initializing the model with the correct initialization used in the Golden Model, and randomly assigning an initial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In general, initial- ization does not play a large role in automated assembly or extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In Figure 6, there appears to be little difference between initialization types for the ER and BA network types, and the TLGL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Although the human curated models (TLGL and Tcell) do diverge slightly from this trend, this is much more prominent for the Tcell model, which is an outlier, with automated model assembly and extension suffering due to the focused nature of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' This is not to say that initialization is a problem that can be disregarded in model assembly, rather it is to be considered after the correct structure of the model has been identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' This is particularly true in the case of logical models where initialization can impact downstream model elements de- pending upon nature of the logic functions that are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' For example, initializing to 0 a model element involved in many “AND” operations will affect downstream model el- ements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' As described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='1, we used summation functions in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' This choice likely made the role of initialization less important, as the inclusion of a new edge (and thus, a new regulator for some element in the model) would not impact the effect of other regulators in such a substantial way as would be present with logic update functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Taken together, the discussion in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='5 and Figures 4-6 demonstrate the key difficulties to automated model assembly and extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Several methods exist which create such automated pipelines but do not focus on how they incorporate biological information into executa- ble models [27, 44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' To date, only a few methods have been proposed to automatically assemble and extend mod- els, while also evaluating the available information and its impact on the created executable model [28-30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Still, even these methods do not fully assess the structural and dy- namic impacts of adding new biological information to an executable model, and therefore do not address the com- plexities to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 4 CONCLUSION In this paper, we have presented an automated assembly and extension pipeline to depict the types and magnitudes of the problems facing computational and system biolo- gists as they work to solve automated model assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Through the largest assembly and extension analysis of synthetic and human-curated models to date, we have characterized the complexities of the automated model as- sembly problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Our findings demonstrate that iterative model assembly, devoid of context, lacking starting struc- tural information in the form of a baseline model, and without robust dynamic information describing the golden model’s behavior, is intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' More often, model assem- bly creates models which perform similarly in dynamics, but do not represent the full information of a full “Golden” network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In this paper, we have demonstrated that particular fo- cus must be paid to a model’s structure and baseline infor- mation, as these can complicate model assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In pick- ing a metric to optimize during model assembly, we have illustrated that a single metric more often serves to sim- plify the golden model, rather than recapitulate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Lastly, we have shown that initializing model elements only play Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The ten networks for each network type were disassembled and then reassembled (either through BFA or DFA) under different initialization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In “0” new model elements are initialized with a starting value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Similarly, “1” and “2” follow similar schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' “Golden” initializes the model element as it would be seen in the Golden Model, while “Random” randomly initializes the model element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In each assembly, the number of edges added back were recorded and used to calculate each assembly method’s recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' ER BA TLGL Tcell 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='0 BFA DFA BFA BFA DFA BFA DFA 0 1 □2 Golden RandomBUTCHY ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' : TITLE 9 a small role in network assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' In future work, we plan to further investigate the effect of network type, additional parametrization of update functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=', timing effects), methods to determine initial state for simulations, and other error functions on the qual- ity of recommended Candidate Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' We will also ex- plore the effect of erroneous Candidate Knowledge on ex- tension methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' ACKNOWLEDGMENT NMZ is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' This work was funded in part by DARPA award W911NF-17-1-0135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' The authors would like to thank Kai-Wen Liang for his instrumental work in the implementation of the BFA algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Gyori, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Bachman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Subramanian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Muhlich, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Galescu, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Sorger, "From word models to executable models of signaling networks using automated assembly," Molecular systems biology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' 11, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Adam A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Butchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Adam is a PhD candi- date in the Bioengineering Department at the University of Pittsburgh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Adam completed a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' in Chemical Engineer- ing and a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' in Biochemistry at Villa- nova University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' He is working on Dis- crete Modeling of Macrophage Activa- tion and its role in the cancer microenvi- ronment and lung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Cheryl A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Telmer Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Telmer is a Re- search Biologist at Carnegie Mellon Uni- versity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Cheryl and Natasa began work- ing together as iGEM advisors in 2013 and have expanded their collaborations through the DARPA Big Mechanism and World Modelers programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Biologists are constantly trying new tools that have the potential to improve our understand- ing of complex systems, and the standardized representation and computational modeling approaches being developed by the Melody Lab are a great contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Natasa Miskov-Zivanov Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Miskov-Zi- vanov is an Assistant Professor of Elec- trical and Computer Engineering, Bioen- gineering, and Computational and Sys- tems Biology at the University of Pitts- burgh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' She received a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' degree in electrical engineering and computer sci- ence from University of Novi Sad, Serbia and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' degrees in electri- cal and computer engineering from Carnegie Mellon University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Be- fore joining University of Pittsburgh as a faculty, she spent several years as a postdoctoral researcher in Computational and Systems Bi- ology at the University of Pittsburgh, and as research scientist and instructor in Computer Science and in Electrical and Computer Engi- neering at Carnegie Mellon University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} +page_content=' Miskov-Zivanov’s research interests include hybrid, knowledge-driven and data-driven, model recommendation and reasoning for complex systems with applica- tions in systems and synthetic biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FIT4oBgHgl3EQf7St_/content/2301.11397v1.pdf'} diff --git a/4dFQT4oBgHgl3EQf4Ta7/content/tmp_files/2301.13431v1.pdf.txt b/4dFQT4oBgHgl3EQf4Ta7/content/tmp_files/2301.13431v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..521bdb983bdf0ed4252b02ae531faf7c8e2f63a5 --- /dev/null +++ b/4dFQT4oBgHgl3EQf4Ta7/content/tmp_files/2301.13431v1.pdf.txt @@ -0,0 +1,2622 @@ +Breaking Out of the Ivory Tower: +A Large-scale Analysis of Patent Citations to HCI Research +Hancheng Cao +Computer Science +Stanford University +California, United States +hanchcao@stanford.edu +Yujie Lu +Computer Science +University of California, Santa +Barbara +California, United States +yujielu@ucsb.edu +Yuting Deng +School of Computer Science +Carnegie Mellon University +Pennsylvania, United States +yutingde@andrew.cmu.edu +Daniel A. McFarland +School of Education +Stanford University +California, United States +dmcfarla@stanford.edu +Michael S. Bernstein∗ +Computer Science +Stanford University +California, United States +msb@cs.stanford.edu +ABSTRACT +What is the impact of human-computer interaction research on +industry? While it is impossible to track all research impact path- +ways, the growing literature on translational research impact mea- +surement offers patent citations as one measure of how industry +recognizes and draws on research in its inventions. In this paper, +we perform a large-scale measurement study primarily of 70,000 +patent citations to premier HCI research venues, tracing how HCI +research are cited in United States patents over the last 30 years. We +observe that 20.1% of papers from these venues, including 60–80% +of papers at UIST and 13% of papers in a broader dataset of SIGCHI- +sponsored venues overall, are cited by patents—far greater than +premier venues in science overall (9.7%) and NLP (11%). However, +the time lag between a patent and its paper citations is long (10.5 +years) and getting longer, suggesting that HCI research and practice +may not be efficiently connected. +CCS CONCEPTS +• Human-centered computing → Empirical studies in HCI. +KEYWORDS +Industry impact, technology transfer, translational science, patent, +citation analysis +ACM Reference Format: +Hancheng Cao, Yujie Lu, Yuting Deng, Daniel A. McFarland, and Michael S. +Bernstein. 2023. Breaking Out of the Ivory Tower: A Large-scale Analysis +of Patent Citations to HCI Research. In Proceedings of ACM Conference +(Conference’17). ACM, New York, NY, USA, 24 pages. https://doi.org/10. +1145/nnnnnnn.nnnnnnn +∗Corresponding author. +Conference’17, July 2017, Washington, DC, USA +2023. ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00 +https://doi.org/10.1145/nnnnnnn.nnnnnnn +1 +INTRODUCTION +What is the impact of human-computer interaction research beyond +academia? Does HCI research diffuse into the industry1, contribut- +ing to technological inventions and products? Are most its insights +ignored by the industry? As an applied field of study intended to be +closely relevant to application — where a considerable proportion +of our research community’s contributions are functional proto- +types and design implications for practitioners — the answers to +these questions are critical to evaluating our translational success. +There have been rich discussions regarding the industry impact of +HCI research since the early years of the field, and the relationship +between research and practice in HCI has long been a focal subject +in both research papers [18, 19] and conference panels [9, 15, 22]. +The literature remains unclear on the field’s level of success +in achieving this impact. One line of the literature suggests high +barriers: that HCI research has remained distant from industry +impact, and that “HCI researchers and HCI practitioners work in +relatively separate spheres of influence” [22]. This line of work +also argues there is a considerable research-practice gap, one that +is “real and frustrating” [60] and likely the result of a long list of +barriers [18, 75]. However, another line of literature argues that the +field achieves considerable success, that “HCI is at the vanguard +of innovation and has repeatedly influenced industry” [32] and +that “there is no question that research in the area of user interface +software tools has had an enormous impact on the current practice +of software development” [57]. +These threads of work are not necessarily incompatible—high +barriers do not rule out the existence of successes that overcome +these barriers—but the field’s overall status remains unclear: how +far have we come, and how far do we have to go? One approach to- +ward resolving this debate is to pursue new methods for measuring +HCI’s impact. Prior work has developed rich in-depth qualitative ev- +idence ranging from personal technology transfer experience [22] +to interviews with multiple stakeholders involved in the translation +process [16]. Yet as the HCI community grows and both well-known +1In this paper, we use ‘industry’ to refer to non-research efforts that aim at practical +impacts, e.g. patents, products, design practices, which usually target a broad audience +than academic researchers. Thus, in this paper, industry labs whose primary focus is +to publish research papers are considered academia rather than industry. +arXiv:2301.13431v1 [cs.HC] 31 Jan 2023 + +Conference’17, July 2017, Washington, DC, USA +Hancheng Cao et al. +successes and painful failures become easier to point to, it becomes +more and more urgent that we also assess broader patterns. +To fill this gap, we draw on methods from the growing measure- +ment literature on innovation in translational sciences [1, 7, 45, 50, +77], where patent citations to research have been regarded as a +valuable proxy of the impact that science has on industrial practice. +While patent citation to research citation does not directly guar- +antee industry impact, it reveals one potential pathway through +which industrial inventors are aware of and recognize research ar- +ticles: a necessary but not sufficient step towards industry impact.2 +Work using this approach has revealed the relevance of research +and practice across science [1], mapped the translation landscape +in bio-medicine [45, 50], and demonstrated that referencing science +in the invention is associated with greater practical value [34]. +Leveraging the modern analysis approaches from this line of +work [51, 52], we report the first large-scale quantitative analysis +of how HCI research is (and is not) being cited by patents. In do- +ing so, we focus on one possible route of industry impact through +HCI research: patents. There are many types of contributions in +HCI—design patterns, behavioral results and theory among many +others—and a patent lens focuses us only on styles of contribution +that are considered prior art for patents, often systems and inter- +action contributions. Specifically, we draw on data from Microsoft +Academic Graph, Semantic Scholar, the United States Patent and +Trademark Office (USPTO), and linkages between them [51, 52]. +This dataset enables us to study research papers from four premier +venues in HCI, including CHI, CSCW, UIST, and UbiComp, and +then replicate across all 20 SIGCHI sponsored venues that appear in +Microsoft Academic Graph, tracing how those research papers are +cited in patent documents from the 1980s through 2018. We study +the institutes involved in the process, leverage citation analysis to +measure the number and proportion of papers cited by patents over +time and measure the length of time it takes before a paper is recog- +nized by patents. We further conduct textual analysis to understand +the topics that are likely to be cited in patents, and compare how +patent-cited research differs from its non-patent cited counterparts. +We observe that: (1) HCI research has been cited extensively +by patents — overall 20% of papers from CHI, CSCW, UIST and +UbiComp, and 13.4% of SIGCHI sponsored venues, are patent-cited, +including a surprising 60-80% of UIST papers over a twenty year +period, higher than 1.5% of science overall and 7.7% of biomedicine; +(2) The patent-paper time lag is long (on average 10.5 years) and +is getting longer, such that citations from academic HCI research +have dropped off by the time a paper receives patent attention; +and (3) Within HCI research, there is substantial heterogeneity in +patent citations across topics, for example, interaction and input +techniques research are especially likely to be referenced by patents +while theory, social and experience design research are not. This +analysis provides the first quantitative survey of the HCI technol- +ogy transfer landscape. While acknowledging potential limitations +of patent citation as a method, we conclude that HCI has had a +considerable impact on industry and is finding more relevance to +practice than most disciplines in science. Yet, it takes a long time for +2More discussion and reflection on the usage of patent citation to science to study +industry impact of research in Section 3.1 and Section 5.3 +innovations in academia to be recognized and taken up by industry, +corroborating the “long nose” theory on HCI innovation [12, 32]. +The contributions of this paper are as follows: +• We introduce measuring patent citations to science as a novel +method to study research-practice relationships in HCI. This +provides quantitative evidence that complements qualitative +evidence in existing HCI literature. We release our analyzed +dataset to enable future analysis.3 +• We present the first large-scale, empirical study measuring +the translational, longitudinal landscape of HCI research +from paper to patent inventions with comparisons to other +fields. This allows us to better understand and evaluate how +HCI as an applied field is or is not finding connections to +practice. +• Our work contributes to reflections and recommendations +for the HCI community to better foster a translational envi- +ronment and recognize impacts beyond academia. +2 +BACKGROUND AND RELATED WORK +In this section, we position our work in the literature on industry +impact, the HCI research-practice divide, and bibliometric analysis +in HCI. +2.1 +Industry impact +Industry impact are often achieved through technology transfer, +which refers to the transmission of knowledge generated by an +individual, the university, government agencies, or any institution +capable of generating knowledge, to another person or organiza- +tion, in an attempt to transform inventions and scientific outcomes +into new products and services that benefit society [55]. Govern- +ment and funding agencies (e.g., in the United States, NSF and NIH) +increasingly seek to nurture “translational research” to facilitate +industry impact from basic research so as to generate greater ap- +plied value and promote technology advances [76, 79], and prior +research has shown inventions that refer to high-quality research +are more likely to be great inventions of value [34, 61]. +Prior research has sought to identify when, where, and how sci- +entific research influences industry invention [3, 7, 17, 45]. There, +patent citations to science have been widely used as a proxy for +studying technology transfer from research to practice despite +noises, as it is one of the only available large-scale records of the +knowledge flow from research to practice that demonstrate the ini- +tial awareness and recognition of research in industrial inventions. +For instance, Tijssen [68] revealed through patent-paper citations +how Dutch-authored research papers influence inventions. Like- +wise, Ahmadpoor and Jones [1] studied 4.8 million US patents and +how they link to 32 million research articles, finding that over half +of patents cite back to a research article and that patents and papers +are on average 2–4 degrees separated from the other domain, pro- +viding some insight into the interplay between patents and prior +research. Jefferson et al. [37], Manjunath et al. [50] used patent cita- +tions to science data, measuring and reporting statistics describing +how research in biomedicine turns into inventions. Liaw et al. [46] +proposed a method to rank academic journals that utilizes non- +patent references in patent documents to evaluate their practical +3Available dataset at: https://doi.org/10.7910/DVN/QM8S1G. + +A Large-scale Analysis of Patent Citations to HCI Research +Conference’17, July 2017, Washington, DC, USA +impact. Other works used patent citation to science to study the +strategy of inventors (e.g. deep search vs. wider scope search) and +how the strategy relates to technology impacts and organization +performance [2, 25, 28, 38]. To facilitate further studies on how +inventions rely on basic science, Marx and Fuegi [51, 52] linked +and disambiguated patent citations to science linking the USPTO +dataset and Microsoft Academic Graph.4 +We build off this rich social science literature by studying indus- +try impacts of HCI research through leveraging and extending their +methods[32]. +2.2 +From HCI research to practice +HCI is a field that emphasizes the design and the use of computer +technology, especially interfaces between people and computers. +HCI research implement, demonstrate and test new technologies +through prototyping and end-user feedback [47], and most HCI +work includes ‘design implications’ sections aiming to translate +their research insights to more practical outcomes. The applied +nature of HCI lead to the community’s long-standing interest in +industry impact, with many publications and panel discussions at +conferences aimed at facilitating better technology transfer [15, +22, 39]. One line of the literature primarily focus on the many +barriers HCI faced in translating research insights to industrial +practice [18, 22], while another line of literature speaks to the +considerable impact that HCI research has had or could have on +the industry [32, 57, 65]. +Many papers argue that despite the insights that HCI research +can offer to practitioners, HCI research findings are rarely used in in- +dustry [18]: that there has been an “immense” research practice gap +in practice that is “real and frustrating” [60], that “HCI researchers +and HCI practitioners work in relatively separate spheres of influ- +ence” [22], and that “attendees at venues like ACM CHI often lament +that no HCI research ever goes into product” [32]. Colusso et al. +[18] interviewed design practitioners so as to understand why they +do not use academic research and why and how they use other re- +sources in their works, presenting a detailed catalog of barriers that +inhibit academic resources usage in industry, such as the content +being hard to read, hard to find, and not actionable. Chilana et al. +[16] stated the distinct goals of HCI research and product may result +in a research-practice gap, that the users who are the major focus +of the user-centered design approach in HCI research are generally +not the buyers of HCI products, and that to make a research-to- +product transition one has to switch from being user-centered to +adoption-centered. Furthermore, prior work [22, 75] suggested that +HCI researchers usually lack the knowledge, resources, connections, +experience, interest, or time to pursue technology transfer. Other +work has shown similar results demonstrating a research practice +gap in HCI [10, 27]. +Prior research has discussed potential approaches to address the +research-practice gap. For instance, Velt et al. [69] identified two +key dimensions of the research-practice gap – general theory vs. +particular artifacts, and academic HCI research vs. professional UX +design practice – and discussed the benefits of translation led by +researchers, by practitioners, or co-produced by both as bound- +ary objects. Colusso et al. [19] proposed a continuum translational +4We leverage this particular dataset in our analysis. +science model for HCI that consists of three steps: basic research, ap- +plied research, and design practice. Shneiderman [65] wrote a book +proposing principles to better blend science, engineering and design +to achieve innovations and breakthroughs. Other work discusses +the challenges and lessons learned from the specific translation of +HCI research to practice [62, 63]. +Meanwhile, another line of work argues that HCI research could +have considerable impact on industrial practice despite the barriers. +Harrison argues that “HCI is at the vanguard of innovation and +has repeatedly influenced industry [...] HCI research has a much +greater impact in identifying opportunities in the first place, es- +tablishing the science and methods, building a shared vision, and +developing a pipeline of human talent” [32]. Likewise, Myers et al. +[57] wrote “There is no question that research in the area of user +interface software tools has had an enormous impact on the cur- +rent practice of software development. Virtually all applications +today are built using window managers, toolkits, and interface +builders that have their roots in the research of the 70’s, 80’s, and +90’s”. Shneiderman’s work [66] further stated that “The remarkably +rapid dissemination of HCI research has brought profound changes +that enrich people’s lives”, but also providing a tire-tracks diagram +showing how HCI research on subjects such as hypertext, direct +manipulation, etc. turned into product innovations by industry. +Similarly, product innovations over the years mirror the early ideas +of canon HCI visions [11, 74]. Other research detailed successful +cases of tech transfer, such as the translation of the multi-touch +interface from research into the Apple iPhone and Microsoft Sur- +face, while highlighting a long time lag between initial research +and commercialization, which can be 20 years or more [12, 32, 66]. +This prior work guides us to the following research questions: +RQ1: What is the impact of HCI research on patents? How much +HCI research is cited in patents? +RQ2: When is the impact of HCI research on patents? How long +does that impact take? +RQ3: Where is the impact of HCI research on patents? Which +topics of research are especially likely or unlikely to diffuse? +RQ4: Who is involved in the process of recognizing HCI research +on patents? Which institutions produce such work, and which +consume it? +The rich qualitative insights derived from case studies, field- +work, interviews, and personal experience, open an opportunity +for complementary work that engages in quantitative, longitudinal +analysis that directly measures how HCI research gets recognized +in industry inventions and technologies. We believe that such a +viewpoint might systematically detail the translation landscape of +HCI as a field. +2.3 +Bibliometrics and HCI +As an important area of computing and information science, HCI +has featured several projects (e.g., [40, 49]) that quantitatively un- +derstand the structure and evolution of the field through the study +of writing and citation patterns, known as bibliometrics [26]. +One commonly used bibliometric method is an analysis of a large- +scale citation network, which leverages the increasingly available +citation data from publishers such as Web of Science and Microsoft +Academic Graph and their associated metadata of the scientific + +Conference’17, July 2017, Washington, DC, USA +Hancheng Cao et al. +publications (e.g. institutes, authors), and even textual analysis (e.g. +topic modeling, keyword extraction) of the scientific publications, +so as to gain insights on patterns behind the diffusion of scientific +ideas [26, 70], research productivity [48, 72], and identify potential +ethical and social issues in science [35, 41]. For instance, Koumaditis +and Hussain [42] leveraged citation data from 962 HCI publications +and reveal that HCI research can be categorized into major themes +of design, data management, user interaction, psychology, and +cognition, and they identified more recent trends in HCI in the +workplace, sensors, and wearables. Likewise, Kaye [40] reported +“some statistical analyses of CHI”, including author counts, gender +analysis, and representations of repeat authors so as to motivate dis- +cussions on the preferred state of CHI. Bartneck and Hu [5] reveal +that only a small percent of countries account for the majority of +CHI proceedings, and present a ranking of countries and organiza- +tions based on their H-index of CHI proceedings. Correia et al. [21] +used 1713 CSCW publications and characterized top CSCW papers, +citation patterns, prominent institutes as well as frequent topics, +highlighting the fact that CSCW is influenced primarily by a few +highly recognized scientists and papers. The authors further quanti- +tatively explored the relationship between collaboration types and +citations, paper frequency, etc [20]. Similar types of analysis have +also been done on more regional HCI conferences [4, 30, 56, 59] as +well as studying subcommunities in HCI [49, 71, 73]. +Visual analytics is another approach used to help understand +HCI’s evolution. For instance, Lee et al. [43] proposed a system +PaperLens to reveal trends, connections, and activity of 23 years +of the CHI conference proceedings. Matejka et al. [54] proposed +an interactive visualization that highlights family trees of CHI and +UIST papers. Henry et al. [33] presented a visual exploration of +four HCI conferences. They showed that the years when a given +conference was most selective are not correlated with those that +produced its most highly referenced articles and that influential +authors have distinct patterns of collaboration. +To the best of our knowledge, there have been no analyses lever- +aging quantitative methods to study recognition of HCI research +beyond academia as we present in this article. In contrast with +prior work, we leverage large-scale patent citations to quantify the +impact of HCI research in practice. +3 +METHOD +In this section, we describe the method we used to study the impact +of HCI research papers in practice using patent citations to science. +3.1 +Patent citations as a pathway to study +industry impact of research papers +We leverage patent citations to research as a proxy to study the +influence of HCI research on industrial practice at scale. While +patent citation to research citation does not directly mean industry +impact, it reveals one important potential pathway from research +to practice where industrial inventions become aware of and recog- +nize research articles, which is often a necessary but not sufficient +step towards producing industry impact. Alongside with studying +other forms of influence, such as design processes (e.g., usability +testing, heuristic evaluation), design patterns, open source software +(e.g., d3, Vega), patent citations to science could help us piece to- +gether the translational landscape in HCI. This method is widely +used in the innovation literature (e.g., [1, 25, 28, 38, 50, 51]). Patent +citations to research are considered valuable signals indicating the +influence of research on the industry, signals that “reflect genuine +links between science and technology.” [68], and “appear to be a +substantive if a noisy indicator of the role of specific, prior scientific +advances” [1]. While citations between research articles capture +research influence [26], patent-to-research citations capture “how +basic research influences commercialization and thus provides a +complementary measure of impact” [50]. Such data has been used +extensively to measure knowledge spillovers from academia and +government to industry [1, 23, 51]. +The rationale behind the validity of this approach is that in +patented inventions, inventors are obliged to disclose any “prior +art” related to their invention, i.e., all information known to that +individual to be material to patentability”,5 including materials that +the inventors leveraged in the invention process, or other similar +material to the focal invention in order to distinguish it. The prior +art includes both references to prior patents, and references to non- +patent literature, such as academic articles. Patent citation is an +important part of a patent, as missing prior art (either prior patents +or non-patent literature), could have potential legal issues. Apart +from citations provided by inventors, patent examiners who review +patents for approval or rejections also add references they think +are of relevance to ensure the legitimacy of the patent. +Prior work has validated this method. Nagaoka et al. [58] sur- +veyed 843 inventors finding patent citations to science are indeed +important linkages to science, despite possible errors of over- and +under-inclusion. Callaert et al. [13] interviewed 36 inventors and re- +port 44% of patent citations to science are considered as “important” +or “very important”, and another 34% are “background” citations. +Based on the rich literature in this space, we conclude that patent +citation to science can be used as a reliable data source to measure +the recognition of HCI research efforts in inventions, thus provid- +ing a valuable proxy of HCI research impact in the industry. Of +course, there is no perfect appoach for studying industry impact: +we discuss and reflect on the limitations of our method in detail in +Section 5.3, and it is especially important to bear in mind there are +multiple translational gaps in HCI research [19], and we are only +studying one important step in the process with regard to patent, +where certain types of contribution such as theory are likely to be +under evaluated through this dimension. +Empirically, we find support for the validity of using patent +citations to research as a proxy of impact in industry. We manu- +ally check patent reference lists of a number of patents. As shown +in Figure 1, the highly-cited patent by Apple Inc. “Mode-based +graphical user interfaces for touch sensitive input devices” (cited +1,898 times),6 cites closely related research papers in CHI on multi- +touch, such as “A Multi-Touch Three Dimensional Touch-sensitive +Tablet", which is the case of technology transfer discussed by Bux- +ton [12]. The even more well-cited Apple Inc. patent (cited 4,018 +times) “Method and apparatus for integrating manual input” 7 also +made reference to several relevant HCI papers. These cases motivate +5https://www.uspto.gov/web/offices/pac/mpep/mpep-2000.pdf +6https://patents.google.com/patent/US8239784B2/en +7https://patents.google.com/patent/US6323846B1/en + +A Large-scale Analysis of Patent Citations to HCI Research +Conference’17, July 2017, Washington, DC, USA +us to leverage patent citations as a signal indicating the invention’s +recognition of research. +3.2 +Dataset +To study how HCI papers are recognized by patents, we required +a citation graph from patent to research, and the metadata (e.g., +author name, affiliation, publication year, title, venue) from both +the paper side and patent side. The data preparation pipeline is +composed of three steps: 1) Prepare metadata of papers and patents, +and the citation graph from patents to research, 2) Select papers +from the venues of interest and clean the data, and 3) Link the clean +metadata based on the citation graph. This pipeline could be applied +to other research communities, or other venues within SIGCHI, by +selecting other venues of interest. +Patent citation to science that connects USPTO to Microsoft +Academic Graph. To capture references from patents to HCI re- +search papers, we drew on a public dataset [52, 53]. This dataset +is a state-of-the-art approach to connect each patent reference in +USPTO (1947-2020) to academic papers (1800-2020) from Microsoft +Academic Graph through matching unstructured front-page and +in-text references in patents to published papers using a disam- +biguation matching method, resulting in 22 million patent citations +to research papers (known as Patent Citation Science dataset).8 +In their papers, the dataset creators verified the quality of their +datasets through manual checking and error analysis. We captured +the reference type (e.g., from applicant, from examiner, unknown), +whether the reference appears in-text or on front page, the time +between paper publication and the citing patent application, and +whether a patent citation is a self-citation to a research paper by +one of the patent authors. A paper to patent pair is considered +self-cited when there is an overlap between the inventors of the +patent and the authors of the cited scientific papers. +Microsoft Academic Graph Metadata. The Microsoft Academic +Graph is a heterogeneous graph that provides scientific publication +records, citation relationships, the information of authors, insti- +tutions, journals, conferences, and fields of study. We leveraged +the public Microsoft Academic Graph dataset provided at Zenodo +Reliance on Science project site9 so as to extract information with +regard to academic publications, e.g., title, author, author affiliation, +and year. +USPTO metadata. We leveraged US patent data from the United +States Patent and Trademark Office (USPTO)10 to represent tech- +nological inventions. Patents have similar fields as academic publi- +cations, e.g., title, abstract, inventor, assignee, and year. +Semantic Scholar (abstract, citation). The abstract informa- +tion of the paper and their academic influence (e.g., number of +published papers, citation count) are missing or hard to process in +the original Microsoft Academic Graph metadata.11 To further ex- +pand data information about authors, papers, citations, and venues, +8Specifically, we used the patent-to-article citations of Version v37 (Jul 19, 2022) at +Zenodo: http://relianceonscience.org +9http://relianceonscience.org +10https://patentsview.org/download/data-download-tables +11https://docs.microsoft.com/en-us/academic-services/graph/resources-faq +we utilize the Semantic Scholar Academic Graph API,12 which fills +in this data. +The details of the data we utilize can be found in Appendix A. +3.3 +Data Preprocessing +Venue selection. In our analysis, we primarily considered four +impactful Human-Computer Interaction (HCI) venues: the ACM +CHI Conference on Human Factors in Computing Systems (CHI), +ACM Conference On Computer-Supported Cooperative Work And +Social Computing (CSCW), ACM Symposium on User Interface Soft- +ware and Technology (UIST), and International Joint Conference on +Pervasive and Ubiquitous Computing (UbiComp).13 For a broader +footprint of HCI research, we created a second dataset of SIGCHI +sponsored venues14 — a total of all 20 SIGCHI sponsored venues15 +that appear in the Microsoft Academic Graph, which covers not +only large, premier venues such as CHI, but also smaller, more +specialized venues such as MobileHCI and CHI PLAY. We used this +second set as more representative of the overall field of HCI, to +further validate our findings and compare with overall patterns +reported in other fields of science in a fairer way16. +Data Cleaning. We further conducted data cleaning on the four +chosen venues by looking up papers in Semantic Scholar rather +than Microsoft Academic Graph. We found that Microsoft Aca- +demic Graph metadata sometimes wrongly classify venues such as +“Brazilian Symposium on Human Factors in Computing Systems” +as “CHI”. To solve this issue, we filtered out irrelevant papers by +manually checking the full name of the venue column from Seman- +tic Scholar, which proves to be of better quality. We then applied +this filtering process to all the paper and patent citations to science +files by joining over the paper id. +Data Linking. In order to better combine the paper and patent +information for analysis, we linked patent data, Microsoft Academic +Graph data and Semantic scholar data via the Patent Citation Sci- +ence dataset.17 The joined data after 2019 has incomplete or little +coverage, thus we focus our analysis on HCI research papers and +patents that cite HCI papers before 2019. +Final Data Statistics. Our final data for analysis includes 23,432 +papers from the four chosen venues, with 16,014 from CHI, 3,084 +from CSCW, 1,746 from UIST, and 2,588 from UbiComp across 1980 +to 2018. Within these papers, we captured 69,900 citation records +from patent to science, with 42,676 from CHI, 5,900 from CSCW, +17,040 from UIST, and 4,284 from UbiComp, which are associated +with 30,660 patents. The broader SIGCHI sponsored venue data +include 57,385 papers in total (41% are papers from the four premier +12https://www.semanticscholar.org/product/api +13Starting 2017, the UbiComp conference main technical tracks consist of papers +published in Proceedings of the ACM on Interactive, Mobile, Wearable and Ubiquitous +Technologies (IMWUT), which we captured in our data. +14https://sigchi.org/conferences/upcoming-conferences/ +15Details of the venues in Appendix B +16Note that in this paper we primarily report findings on the four chose venues rather +than SIGCHI sponsored venues overall. We elect to focus on these four venues as a +practical matter, as we have spent considerable manual efforts in cleaning data related +to the four chosen venues to ensure data quality, as indicated in “Data Cleaning" +section, which makes our analysis more likely to reflect actual trends in these venues. +17Confusingly to HCI researchers, this is known as the “Patent Citation Science” +(PCS) dataset. We joined information from the patent side using the field patentid to +information from the paper side using the field magid. + +Conference’17, July 2017, Washington, DC, USA +Hancheng Cao et al. +(a) Patent US8239784 frontpage with abstract, inventors, assignee etc. +(b) Part of the citation list of Patent US8239784. +Figure 1: Patents are obliged to cite prior art, including prior patents and non-patent literature (e.g. research articles). Here, a +patent by Apple Inc., “Mode-based Graphical User Interfaces for Touch Sensitive Input Devices” [36], has citation to relevant +HCI papers, including “ActiveClick: Tactile Feedback for Touch Panels”, “A Multi-Touch Three Dimensional Touch-sensitive +Tablet”, a mis-named citation to Ken Hinckley (“Kinkley et al.”), and many other references to HCI research. + +(12) United States Patent +(10) Patent No.: +US 8.239.784 B2 +Hotelling et al. +(45) Date of Patent: +Aug. 7, 2012 +(54) +MODE-BASEDGRAPHICALUSER +(56) +References Cited +INTERFACES FOR TOUCH SENSITIVE +INPUT DEVICES +U.S.PATENT DOCUMENTS +3,333,160A +7/1967 +Gorski +(75) +Inventors: Steve Hotelling, San Jose, CA (US); +3,541,541A +11/1970 +Englebart +Brian Q.Huppi, San Francisco, CA +3,609,695A +9/1971 +Pirkle +3,662,105A +5/1972 +Hurst et al. +178/18 +(US):JoshuaA.Strickon.SanJose,CA +3,748.751 A +7/1973 +Breglia et al. +(US):DuncanRobertKerr.San +3,757,322A +9/1973 +Barkan et al. +Francisco,CA(US):BasOrding.San +3,798,370 A +3/1974 +Hurst +178/18 +Francsico, CA (US); Imran Chaudhri. +3,825.730A +7/1974 Worthington, Jr. et al. +San Francisco, CA (US); Greg Christie, +3,846,826 A +11/1974 Mueller +4,014,000A +3/1977 Uno et al. +SanJose.CA(US):JonathanP.Ive.San +4,017,848A +4/1977 +Francisco, CA (US) +Tannas, Jr. +4,146,924 A +3/1979 Birk et al. +(Continued) +(73) +Assignee: Apple Inc., Cupertino, CA (US) +(*) +Notice: +Subjecttoanydisclaimer,thetermofthis +FOREIGNPATENTDOCUMENTS +patent is extended or adjusted under 35 +CA +1243096 +10/1988 +U.S.C. 154(b) by 936 days. +(Continued) +(57) +ABSTRACT +A user interface method is disclosed. The method includes +detecting a touch and then determining a user interface mode +when a touch is detected. The method further includes acti- +vating one or more GUI elements based on the user interface +mode and in response to the detected touch.EVB Elektronik TSOP6238 IR Receiver Modules for Infrared +Remote Control Systems dated Jan. 2004 1-pg. +Fisher et al., Repetitive Motion Disorders: The Design of Optimal +Rate-Rest Profiles," Human Factors, 35(2):283-304 (Jun. 1993) +Fukumoto, et al., "ActiveClick: Tactile Feedback for Touch Panels, +in CHI 2001 Summary, pp. 121-122, 2001. +Fukumoto and Yoshinobu Tonomura, "Body Coupled Fingering: +Wireless Wearable Keyboard,' CHI 97, pp. 147-154 (Mar. 1997). +Hardy, Fingerworks" Mar. 7, 20o02; BBC World on Line. +Hillier and Gerald J. Lieberman, Introduction to Operations +Research (1986). +International Search Rep0rt dated Mar. 3, 2006 (PCT/US 05/03325) +Jacob et al., "Integrality and Separability of Input Devices," ACM +Transactions on Computer-Human Interaction, 1:3-26 (Mar. 1994) +ings, pp. 223-230, 1999. +Kionx "KXP84 Series Summary Data Sheet" copyright 2005,dated +Oct. 21, 2005, 4-pgs. +Lee et al., A Multi-Touch Three Dimensional Touch-Sensitive Tab- +let, in CHI '85 Proceedings, pp. 121-128, 2000. +Lee, “A Fast Multiple-Touch-Sensitive Input Device," Master's The. +sis, University of Toronto (1984). +Matsushita et al., "HoloWall: Designing a Finger, Hand, Body and +Object Sensitive Wal1,’ in Proceedings of UIST '97, Oct. 1997A Large-scale Analysis of Patent Citations to HCI Research +Conference’17, July 2017, Washington, DC, USA +venues), 83,793 citation records (51% are citations made to the four +premier venues), and are associated with 36,024 patents in total +(85% patents cited papers from the four premier venues). +Note that for all chosen venues, our data includes not only main +conference papers but also extended abstracts, posters and other +forms of publications. We did not attempt to filter and focus our +analysis only on main conference papers, given the difficulty to +classify and challenge fuzzy matching based on venue name (e.g. +in our dataset, many posters are not explicitly labeled as poster +publications and are hard to differentiate from main conference +papers). +We release our dataset at: https://doi.org/10.7910/DVN/QM8S1G. +4 +RESULTS +4.1 +RQ1: What is the impact of HCI research +on patents? +We first study the quantity of HCI papers that are later recognized +by patents and present a table of top papers cited by patents. +Proportion of papers that get cited by patents. To assess the +extent of HCI research being recognized in patents, we first cal- +culated the aggregated proportion of the number of HCI papers +at our four premier HCI venues, and SIGCHI sponsored venues +overall, that were cited by patents. We found 20.1% of papers in the +four venues, and 13.4% of papers from SIGCHI sponsored venues +overall, are recognized by patents. This rate is much higher than +the proportion of science cited by patents overall (approximately +1.5% [51]), and the prominent journal paper patent rate (9.7% across +multiple scientific fields [8]). The rate is also much higher than that +of bio-medicine in general, a field that has a rich tradition empha- +sizing translational science, which is at 7.7% [50]. We replicated our +analysis on premier venues in other areas of Computer Science by +comparing the premier HCI venue patent rate (20.1%) with premier +venue patent rate of other subfields, finding that AI patent rates +(as measured through AAAI and IJCAI, two of the largest and pre- +mier AI conferences) are 5%, Natural Language Processing patent +rates (as measured through ACL, EMNLP, and NAACL, three of the +largest and premier NLP conferences) are 11%, and Computer Vision +patent rates (as measured through CVPR, ECCV, and ICCV, three of +the largest and premier computer vision conferences) are 25%. Two- +proportion z tests further confirm the significance of the difference +in percentages with 𝑧 = 51.1, 23.9, -13.1, (𝑝 < .001) when compar- +ing premier HCI venue patent rate with patent rates of premier +venues in AI, Natural Language Processing and Computer Vision. +Taken together, these results suggest that HCI’s impact through +patent citations is higher than science overall, biomedicine, AI, and +NLP, and roughly at par with Computer Vision, an area of intense +industry interest. +Are research citations in patents truly central to the patents, or +are they thrown in just to satistfy a patent examiner? To answer +this question, we leverage a distinction between in-text citations +and front page citations in patents. This distinction allows us to +more directly measure the impact of HCI research in patents. In- +text patent citation to science, as suggested by prior work [8, 52], +are more likely to “capture the scientific articles upon which the +scientists truly relied upon for inspiration” and “have the potential +to more accurately represent the sources of scientific inspiration +upon which the inventors actually drew in the invention process" +since they “tend to be supplied by the inventors themselves”, in +contrast to “legally binding” front page citations which “tend to be +carefully reviewed (and sometimes added) by patent attorneys” [52]. +We find 4.1% papers in our chosen four venues have been cited in- +text by patents, whereas the proportion of patent in-text citation to +science is 2.3% for SIGCHI sponsored venues and 1.4% for science +overall. This result further replicates our finding that HCI research +appears to have real impact, surprisingly even moreso than many +other fields. +Investigating temporal patterns, we plot the total number of HCI +research papers in each of the four venue published over years, +shown in red in Figure 2. HCI research has grown rapidly over the +past 38 years for all four venues, especially at CHI: from 74 papers +in 1982 to 1200 in 2018. This growth is particularly pronounced +within the last 10 years. We then counted the total number of HCI +papers cited by patents by the publication year of the paper and +calculated the ratio between the number of HCI papers cited and +the total number of HCI papers accepted in a particular year by +each venue (blue line in Figure 2). The citation ratios start climbing +especially starting around 1990 and persist since then (Figure 2),18 +with several conferences observing a third to a half of their papers +cited by patents. At UIST in particular, the patent citation ratio +reaches 60% - 80% from 1990 - 2010. +The citation ratio decreased after 2015. One possible explanation +is the time lag between patent and paper is long, e.g., it might take +a decade for a paper to start gathering patent citations, and papers +since 2015 are still too young by this metric. This time lag will be +further discussed in Section 4.2. In other words, the data are right +censored, i.e., more recent papers have not been fully recognized +by patents captured in our dataset. As such, we expect a higher +proportion of HCI papers overall will be referenced by patents +eventually. +Increasing citations to HCI research in patents. A total of +30,660 patents cite research in the four chosen venue, and 36024 +patents cite research from SIGCHI sponsored venues overall. This +raw volume began increasing after 2000 (Figure 3, and has more +than quintupled since 2000 at CHI from around 175 patents per +year in 2000 to over 1000 per year in 2014). However, the number +of patents plateaus and even decreases a bit in more recent years, +e.g. patents begin citing less and less CSCW research starting in +2014. This could be a result of changes on the demand side, e.g., the +industry is less interested in novel social computing applications, +or on the supply side, e.g., HCI publishing more papers that are not +intended to be as industry-relevant. More evidence is needed to +derive the mechanisms behind this result, beyond the scope of our +current work. +Top cited papers by patents in HCI. We further examined the +HCI papers that were cited the most by patents by each venue +(Table 1). Papers highly cited by patents also tend to be highly cited +by research. The papers most highly cited by patents are primarily +18We removed years where conferences did not meet from our analysis and smoothed +the curve, e.g. CSCW was only held every other year until 2010. + +Conference’17, July 2017, Washington, DC, USA +Hancheng Cao et al. +Figure 2: Left: the number of papers published by each conference per year (red) and the number of papers published in that +year that were later cited by at least one patent (blue), at ACM CHI, CSCW, UbiComp, and UIST. Right: a substantial proportion +of HCI papers are recognized by patents, e.g. 60% - 80% UIST papers are recognized by patents 1990 - 2010. +systems work, e.g., building a new system or proposing a new de- +sign. This result parallels with the earlier observation that UIST has +the highest rate of papers cited by patents since UIST is particularly +targeted at new interfaces, software, and technologies. Most papers +in this list were published prior to 2005; however, the majority of +the patents that cited HCI papers come after 2005, indicating again +the potential long time lag between paper publication and patent +reference in Section 4.2. +Highly-cited papers in academia are more likely to be recog- +nized by patents. Moreover, we investigated how academic impact + +CHI +CHI +100 +1200 +Published + Percent of published papers later cited by patents +Published and later cited by patents +80 +S1000 +per +(%) +800 +60 +Percent ( +of +600 +ber +40 +400 +wnN +20 +200 +0 +0- +2010 +1980 +1985 +1990 +1995 +2000 +2005 +2015 +1985 +1990 +1995 +2000 +2005 +2010 +2015 +1980 +Year +Year +CSCW +CSCW +100 +Published +Percent of published papers later cited by patents +300 +Published and later cited by patents +80 +(%) +60 +Percent ( +40 +20 +0 +0°1980 +1995 +2015 +1995 +1990 +2000 +2005 +2010 +1985 +1990 +2000 +2005 +2010 +2015 +1980 +1985 +Year +Year +UIST +UIST +100 +150 +Percent of published papers later cited by patents +Published +Published and later cited by patents +80 +(%) +100 +60 +Percent +75 +Number +40 +50 +20 +25 +0 +0·1980 +1980 +1985 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +2000 +2005 +2010 +2015 +1985 +1995 +Year +Year +UbiComp +UbiComp +100 +500 +Published +Percent of published papers later cited by patents +Published and later cited by patents +80 +"400 +(%) +60 +Percent +of +40 +20 +0 +1985 +1980 +1990 +1995 +2000 +2005 +2010 +2015 +1985 +1995 +2005 +2010 +2015 +1990 +2000 +Year +YearA Large-scale Analysis of Patent Citations to HCI Research +Conference’17, July 2017, Washington, DC, USA +Figure 3: Left: over 1000 patents are citing CHI paper each year after 2014. The number of patents citing HCI research began +rising after 2000 and more than quintupled since then. Right: the number of patents citing SIGCHI sponsored venues follow +similar trend, as a large proportion (85%) made references to the four premier venues. +relates to patent impacts, measured by the paper’s number of cita- +tions from other academic papers (academic citation count) and the +number of citations from patents (patent citation count). Figure 4 +shows the academic citation count for both papers recognized by +patents and papers not recognized by patents over time. Patent- +cited papers have higher paper citations (average academic citation +count 117.1) than non-patent-cited papers (average academic ci- +tation count 27.9), a difference that is significant via an unpaired +t-test (𝑝 < .001), Cohen’s D=0.58. +We further conducted zero-inflated negative binomial regres- +sion19 over patent citation and paper citation count in CHI, CSCW, +UIST, and UbiComp and get regression coefficient of 0.0233, 0.0172, +0.0316, and 0.0175 respectively (𝑝 < .001). The coefficient indicates +that highly-cited papers in academia are indeed more likely to be +cited by patents. Such a relationship is especially salient at UIST. +4.2 +RQ2: When is the impact of HCI research +on patents? +How long does it take for patents to recognize papers? To examine +this question, we investigated the time lag between patent and +paper. +The time lag between patent and paper is long and getting +longer. To measure how long it takes for an HCI paper to be rec- +ognized by patents, for each patent, we investigated the time lag +between the issue date of the patent and the publication date of +all papers it cited from our four chosen venues. We measured the +lag from the patent backward rather than from the paper forward +because we cannot know whether a paper will receive a citation +but has not yet—but we can know how far back a patent’s citations +reach. +19Zero-inflated negative binomial regression is ideal for modelling count-based de- +pendent variables with zeroes, which corresponds to our data where a significant +proportion of HCI papers get no patent citation. +In the four premier HCI venues, the average patent-paper lag is +10.5 years (𝜎 = 6.8 years), indicating that patents on average refer- +ence HCI research papers published 10.5 years before the patent +filing date but there is significant variance over the time lag. +We then studied how the time lag varies over time by aggregat- +ing the patent-paper time lag at the individual patent levels. As +Figure 5a) shows, the median difference between the time the cited +paper is published and the time the paper is cited by the patent, is +becoming larger from 1989 to 2014 for all the venues from about +around 5 years to around 10 − 15 years. However, since 2014, this +trend bifurcates among different venues: the time lag for CSCW in- +creases to over 15 years and Ubicomp decreases to about 10 years in +2017. We also noticed that all venues have nearly indistinguishable +trends except Ubicomp, which has about 3 years of time lag lower +than other venues. In recent years, CSCW takes the longest time to +be recognized by patents, while UIST and UbiComp take a shorter +time, which could be explained by the fact that more system-driven +works are likely to diffuse more quickly into practice. +We also examined the time lag between the patent and its most +recent cited paper (Figure 5b), testing how recent the freshest re- +search is that patents draw on. These general trends are consistent +with the median time lag. Again, the difference between the time its +most recent cited paper was published and the time it is patented +also becomes larger from 1989 to 2011 for all the venues, from less +than 5 years to around 10 years. This increase gradually slowed +down, leading to a slight decrease in more recent years. +The patent citation also involves different sources, some are +added by the applicants/inventors, while others are added by patent +examiners. The dataset we used also provides a breakdown of refer- +ence types, including applicant/inventor added, the examiner added, +other, and unknown types. References added by patent examiners +are generally more recent (average time lag: 6 years) than what the +inventor added (average 11.8 years), although similar trends of long +time lags and increasing time lags are still observed. + +Patents Citing HCl Research +CHI +Number of patents +1000 +CSCW +UIST +800 +UbiComp +600 +400 +200 +0 +1990 +1995 +2000 +2005 +2010 +2015 +YearPatents Citing HCl Research +2000 +SIGCHI +patents +1500 +of +1000 +Number +500 +0 +1990 +1995 +2000 +2005 +2010 +2015 +YearConference’17, July 2017, Washington, DC, USA +Hancheng Cao et al. +Title +Patent Citations +Paper Citations +Year Published +CHI +A multi-touch three dimensional touch-sensitive tablet +708 +231 +1985 +PaperLink: a technique for hyperlinking from real paper to electronic content +200 +134 +1997 +Bringing order to the Web: automatically categorizing search results +196 +486 +2000 +A study in two-handed input +175 +544 +1986 +Generalized fisheye views +175 +2180 +1986 +SmartSkin: an infrastructure for freehand manipulation on interactive surfaces +166 +770 +2002 +AppLens and launchTile: two designs for one-handed thumb use on small devices +159 +133 +2005 +Active click: tactile feedback for touch panels +156 +195 +2001 +Finding others online: reputation systems for social online spaces +153 +100 +2002 +Applying electric field sensing to human-computer interfaces +142 +272 +1995 +CSCW +GroupLens: an open architecture for collaborative filtering of netnews +185 +5771 +1994 +WebSplitter: a unified XML framework for multi-device collaborative Web browsing +166 +186 +2000 +Blogging as a social activity, or, would you let 900 million people read your diary? +121 +584 +2004 +MMConf: an infrastructure for building shared multimedia applications +106 +313 +1990 +An experiment in integrated multimedia conferencing +103 +157 +1986 +Collaboration using multiple PDAs connected to a PC +94 +391 +1998 +Interaction and outeraction: instant messaging in action +90 +1225 +2000 +Providing presence cues to telephone users +83 +177 +2000 +Design of a multi-media vehicle for social browsing +72 +331 +1988 +Distributed multiparty desktop conferencing system: MERMAID +69 +153 +1990 +UIST +Sensing techniques for mobile interaction +254 +592 +2000 +The world through the computer: computer augmented interaction with real-world environments +227 +487 +1995 +HoloWall: designing a finger, hand, body, and object sensitive wall +197 +243 +1997 +A survey of design issues in spatial input +166 +417 +1994 +Tilting operations for small screen interfaces +158 +412 +1996 +Multi-finger and whole hand gestural interaction techniques for multi-user tabletop displays +156 +527 +2003 +DiamondTouch: a multi-user touch technology +153 +1336 +2001 +The document lens +135 +416 +1993 +The DigitalDesk calculator: tangible manipulation on a desk top display +132 +324 +1991 +Pad++: a zooming graphical interface for exploring alternate interface physics +131 +754 +1994 +UbiComp +Validated caloric expenditure estimation using a single body-worn sensor +113 +83 +2009 +InfoScope: Link from Real World to Digital Information Space +67 +34 +2001 +Self-Mapping in 802.11 Location Systems +63 +130 +2005 +The NearMe Wireless Proximity Server +62 +162 +2004 +Predestination: Inferring Destinations from Partial Trajectories +51 +498 +2006 +UbiTable: Impromptu Face-to-Face Collaboration on Horizontal Interactive Surfaces +40 +261 +2003 +Accurate GSM Indoor Localization +37 +537 +2005 +Very Low-Cost Sensing and Communication Using Bidirectional LEDs +34 +157 +2003 +Particle Filters for Location Estimation in Ubiquitous Computing: A Case Study +33 +254 +2004 +PowerLine Positioning: A Practical Sub-Room-Level Indoor Location System for Domestic Use +31 +152 +2006 +Table 1: Top CHI, CSCW, UIST, and UbiComp papers cited by patents. The majority of them are highly-cited papers in academia +whose major contribution is a system. +All results here indicate that patents mostly cite old research, +and are citing increasingly older research, which holds true across +venues and reference types. This conclusion is largely identical to +what is found in science in general [52]. We replicated our analysis +on other areas of Computer Science in a similar way as in Sec +4.1, finding that the time lag between patent and their referenced +papers for AI, Natural Language Processing, and Computer Vision +are 17 years, 13 years, and 10 years respectively, suggesting similar +patterns across subfields in Computer Science. +HCI research has moved on by the time a paper receives +patent attention. Has the HCI community left an idea behind +by the time industry gets interested? Concerns circulate that HCI +has a reputation for trend following and jumping to new shiny ar- +eas every few years [12, 32]. Are patent-cited papers still receiving +academic interest by the time it starts receiving patent citations? +To answer this question, for all papers from the four chosen venues +that eventually get cited by patents in our dataset, we compare +(a) the time lag between the publication year of the paper and the +issue year of the first patent that cites the research paper (first + +A Large-scale Analysis of Patent Citations to HCI Research +Conference’17, July 2017, Washington, DC, USA +Figure 4: Papers cited by patents receive more academic citations in HCI. +Figure 5: The time lag between patent and paper is long and getting longer across venues. +patent citation lag), and (b) the time lag between the publication +year of the paper and the paper’s “peak citation year” when the +research paper gets the most academic citations (peak citation lag). +Peak citation lag averages 5.74 years in our dataset, compared +with 7.48 years for first patent citation lag.20 A paired t-test confirms +20The first patent citation lag is lower than patent backward citation lag reported +earlier (10.5 years) due to right censoring, i.e. recent patent-cited papers are biased +towards short lags since those with long lags have not yet been observed in the dataset. +Peak citation lag have similar issues. If we allow paper enough time to accrue patent +citations, e.g. focus the analysis on papers published before 2000 (cutoff year), we get +an average first patent citation lag of 10.4 years (thus replicating the prior results) +that the difference between these two lags are significant 𝑡(3740) = +18.3 (𝑝 < .001), Cohen’s D=0.38. This result supports the concern +that HCI’s focus shifts to other topics by the time industry take up +an idea. +Self-cite tends to be faster. One exception to this temporal +pattern is that self-citation patents have a shorter patent-paper time +lag. Since 2008, the time lag for the non-self-cite patents increased +and peak citation lag of 7.5 years. We varied the cutoff year, and found on average +first patent citation lag is always longer than peak citation lag which suggests the +robustness of our finding. + +Non patent-cited +Patent-citedThe median time lag of the paper +cited by patents in year X +20 +CHI +(Year) +CSCW +15 +UIST +Time difference ( +UbiComp +10 +5 +0 +1990 +1995 +2000 +2005 +2010 +2015 +YearThe time lag of the most recent paper +cited by patents in year X +20 +CHI +(Year) +CSCW +15 +UIST +Time difference ( +UbiComp +10 +5 +0 +1990 +1995 +2000 +2005 +2010 +2015 +YearConference’17, July 2017, Washington, DC, USA +Hancheng Cao et al. +rapidly and was above 14.6 years in 2018, while the self-cite patents +remain below 6.3 years, which suggests that papers transferred +faster by authors themselves into patents compared with those +transferred by others. +4.3 +RQ3: Where is the impact of HCI research +on patents? +Which HCI research topics are the focus of industry activity? To +answer this question, we compare non-patent-cited HCI papers +to patent-cited HCI papers in the four chosen venues via Latent +Dirichlet Allocation (LDA), a classic method of topic modeling [6]. +LDA automatically discovers topics within documents, where each +topic is represented as a probability distribution of words. Each +document can also be represented as a probability distribution over +different topics. +We concatenated each paper title with its abstract (if available) to +represent its contents. Similarly, we concatenated each patent title +with its abstract (if available) to represent the patent’s contents. +We then tokenized the text corpora into unigrams and bigrams, +filtered out terms that appear fewer than 5 times in the corpus, +removed stop words in English, and then ran LDA modeling. We +varied the number of topics and align on seven topics resulting +in the highest quality topics. Figure 6 reports the result. Through +checking representative documents and word clusters with HCI +experts, we titled each topic: topic 0 is related to patent terms, the +topic is 1 on modalities, topic 2 is system interaction, topic 3 is on +evaluations, topic 4 is on theory, topic 5 is on social and experience +design, and topic 6 is on input techniques. +We then computed the topic distributions for each document +(paper or patent) in our corpus, then aggregated topic distributions +of all documents within a specific year that belong to a certain doc- +ument category (patents, patent cited papers, or non-patent cited +papers) so as to get an estimated number of documents that belong +to a particular topic for that document category for a particular year. +In the first row of Figure 7, we plotted the topic distribution for +patent-cited HCI papers (left), non-patent cited HCI papers (middle), +and patents (right), i.e., how many papers belong to topic X in year +Y. The second row of Figure 7 normalizes this topic distribution, i.e., +what is the proportion of topic X in year Y for a specific document +category, to better illustrates the distribution pattern. +As can be observed from Figure 7, system interaction has domi- +nated the patent-cited HCI papers over time, indicating that system- +oriented research has been of considerable importance in patent- +cited HCI research. From 1980 to 2000, about 40% patent-cited HCI +paper are system interaction related. After 2000, the percentage +of system interaction decreased to about 20% but began expand- +ing again in 2015. We also observed that input techniques have +expanded significantly over time and reached nearly 20% after 2015. +Evaluations have also grown in general and contributed about 20% +of all patent-cited HCI papers. +In comparison, the topic distribution of non-patent cited papers +shows a very different pattern. The results mirror the methodolog- +ical plurality of HCI, where not all contribution types have an +industry impact. Theory work is highly visible in non-patent cited +HCI papers over time, though the proportion is gradually decreas- +ing from about 40% before 2000 to about 20% in 2018. Social and +experience design has grown significantly from nearly 0 percent in +1980 to about 20% in 2018, indicating behavior-oriented research +has been of considerable importance in non-patent-cited HCI pa- +pers. Evaluations and system interaction contributed to about half +of all non-patent-cited HCI papers in 1980, but this percentage has +decreased to about 30% in 2018. Through unpaired t-test, we further +verify there exist statistically significant differences between topic +distributions in patent-cited papers and non-patent cited papers: +there is a higher proportion of theory (𝑝 < .001), social & experi- +ence design (𝑝 < .05) work, and lower proportion of system inter- +action (𝑝 < .001), modalities (𝑝 < .001) work in non-patent-cited +HCI papers compared to patent-cited counterparts. We emphasize +that this is not a negative outcome for theory, behavioral, and other +research that does not produce artifacts, as they have an impact +through other channels, or could influence patent in an indirect +way [19]. +Additionally, the variation of the patents’ topic distribution over +time is not consistent with that of papers. Since 1990, patent topics +have been dominated by input techniques,21 which first expand +from 1990 to 1993, then slightly shrink from 1993 to 2010 and +expand again since 2010. In 2018, about 40% of patents that cite +HCI research papers are input techniques. We also observed this +growth in patent-cited HCI papers, but not this significant. +4.4 +RQ4: Who is involved in the process of +recognizing HCI research on patents? +Last, we investigate through the four premier HCI venues which +institutions are most likely to develop patents that recognize HCI +research, and which institutions conduct HCI research that are most +cited by patents. Such analysis is important because it identifies +the role of different stakeholders within the technology translation +landscape [19]. +Apple, Microsoft, IBM, but no longer Xerox: top institutes +citing HCI research. We examined who are the top patent as- +signees (the entity that has the property right to the patent, e.g. +firm) that cite HCI research. The top patent assignees have been +dominated by companies: Apple, Microsoft, and International Busi- +ness Machines Corporation (IBM) are the top three companies +that were granted the highest number of HCI-citing patents in the +dataset. Other rise and fall over time. See appendix C for more +details. +PARC, CMU, MIT: top institutes that publish patent-cited +research. We assessed the institutes that published the most patent- +cited HCI papers across the years. As Figure 8 shows, contrary to +the fact that top patent assignees have been dominated by indus- +tries, top institutes that published patent-cited HCI papers have +been a combination of universities and companies. Top universi- +ties include Carnegie Mellon University, Massachusetts Institute of +Technology, University of California, and University of Washington. +Top companies that published patent-cited HCI papers include Xe- +rox Palo Alto Research Center and Microsoft. The ratio of patents +cited among all HCI papers significantly dropped from nearly half +before 2005 to less than 30% for most institutes after 2005, due to +21We exclude analysis of topic - ‘patent terms’ as the topic is generic language use in +patents. + +A Large-scale Analysis of Patent Citations to HCI Research +Conference’17, July 2017, Washington, DC, USA +Figure 6: Topics were identified through a Latent Dirichlet Allocation (LDA) analysis of the combined paper-patent corpus. +Figure 7: The first row shows the breakdowns of papers across 7 topics in HCI over time. The second row depicts the per- +centage of each topic in terms of paper number. Three columns depict "topic distribution of patent-cited HCI papers", "topic +distribution of non-patent cited HCI papers" and "topic distribution of patents" respectively. System Interaction dominates +the patent-cited HCI papers while Theory dominates the non-patent cited HCI papers and Input Techniques dominate patents +over time. +the fact that the total number of HCI papers grew significantly and +the right censoring issue. +Overall, 35.5% of Microsoft’s papers, 31.0% of IBM’s, and 65.1% +of Xerox’s were cited by patents. In comparison, universities have +a lower rate of papers cited by patents, e.g. 25.2%, 15.3%, 26.9% of +papers were recognized among Carnegie Mellon University, the +University of California system, and MIT respectively. This indi- +cates that among institutes publishing the most HCI papers, the + +Topic O: Patent Terms +Topic 1: Modalities +Topic 2: System Interaction +datum. system +support +video +environment +tool +use +object +associate +'displaybase +visualcharacter +user +design +user +device +audio +voicetext +application +provide +information receive +base +time +virtual +system +include +first +user +interactive +document +speech +Topic 6:Input Techniques +interface +display +include +content method +present +interaction +position + target +word +surface +user +input +touch +sensor +display +method +control +gesture +device +Topic 3: Evaluations +Topic 4: Theory +Topic 5: Social & Experience Design +first image object +study +time +human +system +design +base +design +user performance +work +medium +study online +use +result +group +study +technology +child +behavior method +community +taskactivity +support +social +people +systemdatum +practice +support +paper, technology +experience +use +hci +challenge +gameparticipant +model +analysis +process +play +research +playerPatent terms +Modalities +System Interaction +Evaluations +Theory +Social&experience design +Input techniquesPatent terms +Modalities +System Interaction +Evaluations +Theory +Social&experience design +Input techniquesPatent terms +Modalities +System Interaction +Evaluations +Theory +Social&experience design +Input techniquesConference’17, July 2017, Washington, DC, USA +Hancheng Cao et al. +Figure 8: Institutions publishing the most patent-cited research. +papers from the industry have a higher proportion of papers rec- +ognized by patents. However, the difference between industry and +universities becomes smaller when removing self-citing patents. +Self-citation. We also explored the degree of self-citation. We +find that 13.9% of patents self-cite the inventor’s own research. +Although the number of self-citing patents is growing, the percent- +age of self-citations in all HCI patent citations is decreasing from +around 20% to 5% in recent years. This suggests that while the HCI +field is expanding, the number of researchers directly referring to +their own research in patents is not growing at the same rate. Most +of the self-citations also come from industry, with Microsoft and Xe- +rox constituting 34.8% and 11.2% of total self-citations. Self-citation +from academia is much less common. +Summary of conclusions: Through our analysis, we find that +HCI research has had a significant impact on patents, with an in- +creasing number of patents recognizing research in CHI, CSCW, +UIST, and UbiComp. Patents are more likely to refer to systems- +oriented and highly-cited research in academia. However, the time +lag between patent and paper is long (>10 years) and getting longer, +suggesting HCI research and practice may be inefficiently con- +nected. We further verify the robustness of our main findings +through two additional analyses, which we report in Appendix +D. +5 +DISCUSSION +In this section, we discuss the implications of our findings: +5.1 +The patent-research relevance landscape in +HCI +By combining the findings from our large-scale analyses with that +of prior qualitative evidence established by literature (e.g. case +studies [15], personal experience, [22] and interviews [19]), we can +now offer a more comprehensive picture of the HCI translation +landscape. +The impact of HCI research on patents: Our work largely +corroborates literature arguing for the considerable impact of HCI +research on practice [12, 32, 57]. In our analysis, among HCI re- +search papers in CHI, CSCW, UIST, and UbiComp, 20.1% of all papers +have been referenced by patents, and 13.4% for SIGCHI sponsored +venues overall. This is a rate far higher than science in general +(1.5% [51]) and prominent journals across multiple scientific fields +(9.7% [8]). The rate is also higher than bio-medicine, a field that +has a more systematic technology translation system and a richer +tradition of studying technology translation, whose proportion is +7.7% [50].22 HCI research diffuses into the industry at a similar +rate as Computer Vision (25%) and at a higher rate than NLP (11%), +both areas of substantial industry funding and interest. Note our +estimate is a lower bound: given the long time lag of patent-paper +citations, recent papers may have not fully expressed their impact +yet (right censoring). When only considering earlier years that do +not suffer much from right censoring issues (e.g. prior to 2005), we +see roughly 30%-50% of papers published in those years have been +cited by patents. For UIST, the proportion is even higher, close to +80% for many years. +22Bio-medicine papers from US institutes only—a filter we did not apply for our study +of HCI—have a proportion of 23.3% [50], which is roughly the same as HCI research. + +Patent cited +Non patent citedPatent cited +Non patent citedPatent cited +Non patent citedPatent cited +Non patent citedA Large-scale Analysis of Patent Citations to HCI Research +Conference’17, July 2017, Washington, DC, USA +Issues with the current HCI translation into patents: As +argued by Bill Buxton in ‘the long nose of innovation’ [12], the +bulk of innovation takes place over a long period: the mouse was +first built in 1965 by William English and Doug Engelbart, but was +only popularized in the 1990s when Microsoft released a large-scale +commercial mouse; multitouch was published in 1985, but took 22 +years to become a product. Our analysis further demonstrates that +even the initial step of having research recognized in a patent, which +may be well before there is an actual product, takes considerable +time. In fact, the ubiquity of long time delay between research and +practice, and thus lack of immediate impact on the industry after +the publication of a research paper, could be one underlying reason +why many papers on HCI translation argue that HCI lacks practical +impact [18, 22, 63]. Furthermore, our analysis demonstrates that +the time lag between patent and research is getting longer over +time, indicating that the translation process in HCI may become +more inefficient over time. This result is in line with a general +trend across science (average over time: 14.4 years), where they +report an average patent citation to science time lag of about 8 +years in the 1990s, rising to about 15 years in 2018 [52]. The specific +reason for the (increasing) time lag would need further work. We +also show that the HCI community often leaves an idea behind +by the time industry gets interested, as a paper’s peak citation +lag is generally shorter than the paper’s first patent citation lag. +The result indicates that with a long time lag, HCI research has +moved on and is exploring new emerging technologies that are +not yet reliable enough, cheap enough, power-efficient enough, or +accurate enough for the industry yet. The observation supports +the observation that HCI research often plays “the time machine +game”,23 where it fast forwards into the future by acquiring early +versions of emerging technology (e.g., VR, AR, multi-touch, AI) and +exploring the interactive applications of that technology. Unless +HCI is directly working on reducing those barriers to industry entry +for that technology, HCI research cannot directly accelerate the +time lag: it is simply painting a compelling vision of the future +before that future arrives. +5.2 +How could the HCI community do better to +facilitate technology transfer and +industrial impact? +Encourage communications and collaborations across academia +and industry. Through our analysis, we have found that even +though research articles from both academia and industry are rec- +ognized by patents, the proportion of papers in academia recognized +by patents is much lower. While the result could be that industry +research papers by themselves are more applied than research pa- +pers from academia, or that industry has more internal incentives +to have their research patented24, this could also be a sign that prac- +titioners are not fully aware of some application-oriented advances +in academia, and that information diffusion between academia and +practice is inefficient [12]. +23A term attributed to Jeff Pierce, formerly a research manager at IBM Research and +faculty member at Georgia Tech. +24Microsoft Research, for example, would award decorative “patent cubes” to re- +searchers for each new patent they co-authored, which researchers would often stack +into decorative pyramids and display in their offices +Our work thus echoes calls for a more inclusive and translation- +friendly environment [9, 15, 18, 19]: that both academia and in- +dustry should 1) better recognize the importance of technology +translation rather than considering translation irrelevant, 2) estab- +lish more communication and collaboration channels to engage +people, e.g. SIGGRAPH-style Emerging Tech festivals where aca- +demic researchers show their published HCI work to an applied +audience and encourage researchers in serving as advising role +in the industry, and 3) involve more HCI materials in Computer +Science curriculum at universities to get ‘future practitioners’ more +familiar with HCI research ideas, and thus prepare them as trans- +lational developers who are more likely to bridge academia and +industry [60] +Encourage self-driven technology transfer. Self-driven tech- +nology transfer (e.g. patents recognizing one’s own paper) gen- +erally happens much faster than technology transfer in general. +Intuitively, the self-driven transfer would not encounter many of +the same communication and information diffusion barriers. Self- +driven technology transfer could also potentially solve many of the +‘recognition’ issues in the translational process as discussed in prior +works [32]. However, as shown in our analysis, though the amount +of self-driven technology transfer in HCI is going up over time, it +is not on par with the rate of increase for research articles. While +not all researchers should actively engage in technology transfer, +there could be more steps to be taken to encourage self-driven tech- +nology transfer from the academic side so that translation could +happen more efficiently, e.g. through better supporting and recog- +nizing attempts to self-translate one’s own research by providing +legal apparatuses and funding support. Meanwhile, we want to +emphasize while there are benefits of self-driven transfer, it may +currently not distribute opportunities equally. For instance, in the +life sciences [24], women faculty members patent at about 40% of +the rate of men. It would be important to identify and mitigate these +potential issues so as to ensure an inclusive technology transfer +environment. Relatedly, as suggested by prior work [19], there exist +multiple translational gaps in HCI, and basic researchers should +also be encouraged to engage more with applied researchers and +do more system work, which would eventually help translate HCI +research insights into industry impacts. +Recognizing translational work in HCI. More broadly, our +work echoes prior work on the need of recognizing translational +efforts in HCI. For instance, when allocating funding or considering +researcher promotion, their impacts in the industry could be taken +into consideration as a separate metric aside from impacts within +academia. Our work points to a potential way to quantify one +important pathway towards HCI research’s impacts on the industry, +through analyzing patent-to-science citation data. +Impact signals. Prior approaches to quantifying research im- +pact mostly focus on impact within academia through bibliomet- +ric analysis. However, no quantitative metric fully captures the +complexities of our world. Could the h-index be fruitfully comple- +mented with other information? (a “patent relevance” p-index?) +While our analysis show impacts in academia and impacts in patents +correlate, we also find papers with high patent citations do not nec- +essarily have high paper citations: in one extreme case, the most + +Conference’17, July 2017, Washington, DC, USA +Hancheng Cao et al. +patent-cited paper in our dataset, “A multi-touch three-dimensional +touch-sensitive tablet” [44], is more popular in the patent world +than in academia. If evaluations primarily consider the academic +impacts of such research work, the work’s value may have been +underestimated. As one potential pathway to industry impacts that +are relatively easy to scale, patents provide a potential signal to +more holistically evaluate research. +Of course, patent relevance, or practice relevance in general25, +is not the solitary metric of scientific value, and research and re- +searchers should not be judged based on a single metric, e.g. to +receive funding or get a promotion. Thus, our work should not +be interpreted as stating that non-patent cited research represents +any sort of failure. There are many, many examples of influential +HCI research that is not patented (or even patentable). For instance, +our work shows that system building or application-oriented HCI +research is more likely to find relevance in patents rather than +design-oriented or behavioral research. The result is not an indica- +tion that applied-oriented research is more valuable: there could +be the indirect influence of other types of works on application- +oriented research, e.g. applied research getting inspiration from +behavior work, as suggested by the translational science model +in HCI [19] – which we seek to address in future work, and 2) +it is equally important to maintain a diversity of research ideas, +which has proven to facilitate greater innovation for science in +general [35]. If the measurement of this impact is desirable, we will +require new methods, such as multi-hop influence over citation +network [1], linguistic concept diffusion [14], from the paper to the +public or media [77, 78]. +5.3 +Limitations and Future Work +Patent citations to research are only a proxy signal of industry +impact, which is a hard-to-quantify concept otherwise. It is only one, +among many (e.g. open source software, design patterns), potential +pathway to industry impacts. First, not all patents will turn into +products or practices, so they may not be actual “industry impact” +instances (false positives). There could be many other factors, such +as assignee strategy and resources, that could influence the process. +Even if a patent does end up as a product, most of the time the +patent will not be valuable or impactful, with 97% of all patents +never recouping the cost of filing them26. However, the fact that +inventors decide to go through the long and expensive process of +filing a patent to protect their intellectual property does indicate +they are considering their invention having at least some potential +to be of relevance to the practice domain, which could be regarded +as an intended act aiming at industry impact or technology transfer. +Second, industry impact could happen even if there is no patent- +ing process involved (false negative), which is not uncommon in +software [29]: startups will launch products without patents from +time to time, which is quite different from the innovation landscape +of more traditional fields; design processes (e.g., usability testing, +heuristic evaluation), design patterns, and open source software +(e.g., d3, Vega Lite) also have significant industry impact that is +not reflected though patents. As such, our analysis of using patent +25Though arguably it’s much harder to quantify other forms of practice relevance, e.g. +how research influence design patterns and open source software +26https://www.forbes.com/sites/stephenkey/2017/11/13/in-todays-market-do- +patents-even-matter/ +citation to HCI research papers could be different from the actual +translation landscape: the patent dataset could introduce both false +positives and negatives, e.g., even if a patent cites a HCI paper, it +may never be taken up in practice as product, and an actual product +that gets influenced by HCI research that is unpatentable will not +be observed and measured through our current approach. +Despite all the shortcomings of patent citation to science, the +availability and scale of the dataset make it a rare lens in the in- +novation literature to enable conclusions on the research-practice +relationship at scale [50–52]. In our work, in addition to building on +these methods from the innovation literature, we tied our analysis +to qualitative evidence discussed in prior works so as to validate +our findings. +In future work, we plan to 1) involve more qualitative evidence +(e.g. interviewing inventors’ motivation behind citing HCI research) +to further validate our findings, and 2) take more steps to quantify +how HCI research turns into valuable inventions, e.g. by using +patent citations to other patents as a proxy of patent value, which +correlates well with other metrics of patent value, e.g. whether they +are renewed to a full term, and whether they get licensed [31, 64]. +Our work also currently mostly focuses on measuring industry +relevance at the paper level, which may not necessarily be the +principal unit of knowledge: for example, several papers on the +same idea can get cited by patents. While we have made preliminary +attempts to analyze the topics prevalent to patents, patent-cited +research papers, and non-patent cited research papers, future work +could better study at the concept level what specific research ideas +are transferred into research, either through keywords provided +by the author (which is unfortunately not available in our current +dataset), or natural language processing based approach such as +phrase mining [14], which may help track transfer of innovations +at a more fine-grained level. +Other limitations include: (1) our dataset is focused on United +States patents, which limits our cultural context and generalizability, +though arguably a significant proportion of inventors/organizations +using (and pushing) HCI research in practice are US-based [67]; +(2) while discussing in a descriptive way in our paper with findings +on the role of academic impacts (section 4.1), topic (section 4.3), +and institute/actors (section 4.4) in relating to patent impact, we do +not have causal evidence/analysis on the causal mechanisms what +cause some papers to have more industry relevance, which is an +important topic we seek to address in future work, and (3) if there +are recent trends in the last 5-10 years that have changed these +patterns, it is still too recent to see their impact. +6 +CONCLUSIONS +In this work, drawing inspiration from the innovation literature, we +quantitatively study one important pathway from HCI research to +industry impact by conducting a large-scale analysis of how patent +documents from USPTO refer to research articles in CHI, CSCW, +UIST, UbiComp and other SIGCHI sponsored venues. We contribute +to the literature by measuring to what extent HCI research has +been featured in patent citations, with a high proportion of papers +referenced in patents. Patents are more likely to refer to systems- +oriented and highly-cited research in HCI. However, we also reveal +potential translation issues: HCI research and practice may not be + +A Large-scale Analysis of Patent Citations to HCI Research +Conference’17, July 2017, Washington, DC, USA +efficiently coupled, since the time lag between paper and patent +is long and getting longer. Our work not only demonstrates the +potential of using patent citation data to science as a powerful tool +to study the industry impact of HCI research, but also points to +suggestions for the HCI community to better facilitate translation +from research to practice. +ACKNOWLEDGMENTS +The authors thank Yian Yin for helpful suggestions on polishing +the work, and Mary Czerwinski, Bongshin Lee, Lucy Lu Wang, +James Zou, Shumin Zhai and many others for insightful discus- +sions. Hancheng Cao was supported by Stanford Interdisciplinary +Graduate Fellowship. +REFERENCES +[1] Mohammad Ahmadpoor and Benjamin F Jones. 2017. The dual frontier: Patented +inventions and prior scientific advance. Science 357, 6351 (2017), 583–587. +[2] Sam Arts and Lee Fleming. 2018. Paradise of novelty—or loss of human capital? +Exploring new fields and inventive output. Organization Science 29, 6 (2018), +1074–1092. +[3] Thomas E Backer. 1991. Knowledge utilization: The third wave. 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To capture the information required by patent +citation to science, we utilize a public dataset available over Zen- +odo.27 We leverage the patent-to-article citations of Version v37 (Jul +19, 2022), including _pcs_mag_doi_pmid.tsv and papercitations.tsv. +For _pcs_mag_doi_pmid.tsv, we mainly focus on the fields reftype, +diff_month, selfciteconf_avg. We focus on fields citingpaperid +and citedpaperid in papercitations.tsv, which we used to join with +Microsoft Academic Graph Metadata. +Microsoft Academic Graph Metadata. Microsoft Academic +Graph Metadata is also available over Zenodo.28 The data files we +utilize include authoridname_normalized.tsv, conferenceidname.tsv, +paperauthoridaffiliationname.tsv, paperauthororder.tsv, paperconfer- +enceid.tsv and paperyear.tsv. +USPTO Metadata. We acquire USPTO metadata from PatentsView.29 +We utilize datafiles assignee, inventor, patent, patent_assignee, and +patent_inventor. +Semantic Scholar. We request Semantic Scholar API30 with re- +search article IDs retrieved from Microsoft Academic Graph Meta- +data for extra paper information. The fields we queried include +title, abstract, venue, year, referenceCount, citationCount, +authors, as well as name, affiliations, paperCount, and +citationCount associated with each author. +we retrieved all the above data in Aug 2022. +B +SIGCHI SPONSORED VENUES +The 20 SIGCHI venues that we include in our analysis are: Hu- +man Factors in Computing Systems (CHI), User Interface Software +and Technology (UIST), Ubiquitous Computing (UbiComp), Con- +ference on Computer Supported Cooperative Work (CSCW), Con- +ference on Tangible and Embedded Interaction (TEI), Symposium +on Eye Tracking Research & Application (ETRA), International +Conference on Supporting Group Work (GROUP), Conference on +Intelligent User Interfaces (IUI), Creativity and Cognition (C&C), +Interaction Design and Children (IDC), International Conference +on User Modeling, Adaptation, and Personalization (UMAP), Sym- +posium on Engineering Interactive Computing System (EICS), Con- +ference on Automotive User Interfaces and Interactive Vehicular +Applications (AutomotiveUI), Conference on Human-Robot Interac- +tion (HRI), International Conference on Computational Collective +Intelligence (CI), Conference on Recommender Systems (RecSys), +Annual Symposium on Computer-Human Interaction in Play (CHI +PLAY), International Conference on Multimodal Interaction (ICMI), +Symposium on Spatial User Interaction (SUI), Symposium on Vir- +tual Reality Software and Technology (VRST). +27http://relianceonscience.org +28http://relianceonscience.org +29https://patentsview.org/download/data-download-tables +30https://api.semanticscholar.org/api-docs/graph#tag/Paper-Data/operation/get_ +graph_get_paper_references +In total, there are 57,385 papers where 13.4% of them (7678 pa- +pers) have been cited by patents in our dataset. +C +TOP PATENT ASSIGNEES OVER TIME +We show top patent assignees over time in Fig 9. +D +ADDITIONAL ANALYSIS ON +NON-SELF-CITING PATENTS AND +NON-RESEARCHER PATENTS +We provide two additional analyses using a subset of four pre- +mier venues to further verify the robustness of our findings. To rule +out the possibility that the impacts of HCI research on patents is a +result of self-cite, or driven primarily by HCI researchers – thus one +may argue the impact of HCI research in industry is actually limited +– we run the same analysis using 1) patents that do not include +self-cite to one’s own research papers (“non-self-citing patents”)), +which is 26, 382 (86.04% of original patents), and 2) patents that +are invented by people who have never published any CHI, CSCW, +UIST or UbiComp research papers ( (“non-researcher” patents), +which we operationalized through excluding patents where inven- +tor last name have appeared in author lists of papers from the four +venues we focused on.31 This results in 5, 251 (17.12% of original +patents) of “non-researcher” patents. We find consistent patterns in +our main analysis where a high proportion of HCI research papers +are cited by patents, and there is a long time lag between patent +and paper. More specific results are as follows: +Proportion of papers that get cited by patents. The propor- +tion of papers cited by non-self-citing patents is plotted in Figure 10 +and the ratio rises and persists since 1990 at over 30%. At UIST in +particular, the patent citation ratio reaches 60% - 80% from 1990 - +2010. This suggests that non-self-citing patents, similar to our main +result, recognize a considerable number of HCI research papers. +Identical trends can be observed for non-researcher patents, as +shown in Figure 13. +Increasing citations to HCI research in patents. Figure 11 +shows the number of non-self-citing patents that cite HCI research +over time. It can be observed that non-self-citing patents first in- +crease in 2000 and then peak around 2014, ranging from 200 to 1000 +across different venues. This agrees with the overall trend reported +in the main paper. +Identical trends can be observed for non-researcher patents, as +shown in Figure 14. +Time lag between patent and paper is long and getting longer. +The temporal trend of the measured time lag between the issue date +of non-self-citing patents and the publication date of HCI papers +they cited are plotted in Figure 15a. Similar to the trend reported +in the main results (Figure 5), the median time lag increased from +1989 to 2014 for all the venues from about around 5 years to around +10−15 years while since 2014, this trend bifurcates among different +venues. The time lag between the patent and its most recent cited +paper (Figure 15b ) is also examined, showing identical trends. +Identical trends can be observed for non-researcher patents, as +shown in Figure 12. +31This set of patents is a smaller set than actual “non-researcher” patents. The primary +objective is to ensure a set of patents with inventors who, for sure, have never published +papers in the four academic venues we studied without tedious author disambiguation. + +Conference’17, July 2017, Washington, DC, USA +Hancheng Cao et al. +Figure 9: Top patent assignees that cite HCI research over time. + +A Large-scale Analysis of Patent Citations to HCI Research +Conference’17, July 2017, Washington, DC, USA +Figure 10: (Non-self-cite) Left: the number of papers published by each conference per year (red) and the number of papers +published in that year that were later cited by at least one patent (blue), at ACM CHI, CSCW, UbiComp, and UIST. + +CHI +CHI +100 +1200 +Published +Percent of published papers later cited by patents +Published and later cited by patents +80 +S1000 +per +(%) +800 +60 +Percent ( +of +600 +ber +40 +400 +wnN +20 +200 +0 +0- +2010 +1980 +1985 +1990 +1995 +2000 +2005 +2015 +1985 +1990 +1995 +2000 +2005 +2010 +2015 +1980 +Year +Year +CSCW +CSCW +100 +Published +Percent of published papers later cited by patents +300 +Published and later cited by patents +80 +(%) +60 +Percent ( +40 +20 +0 +1995 +2015 +1995 +1990 +2000 +2005 +2010 +1985 +1990 +2000 +2005 +2010 +2015 +1980 +1985 +Year +Year +UIST +UIST +100 +Percent of published papers later cited by patents +150 +Published +Published and later cited by patents +80 +(%) +100 +60 +Percent +75 +Number +40 +50 +20 +25 +0 +0°1980 +1980 +1985 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +2000 +2005 +2010 +2015 +1985 +1995 +Year +Year +UbiComp +UbiComp +100 +500 +Published +Percent of published papers later cited by patents +Published and later cited by patents +80 +"400 +(%) +60 +Percent +of +40 +20 +0 +1985 +1980 +1990 +1995 +2000 +2005 +2010 +2015 +1985 +1995 +2005 +2010 +2015 +1990 +2000 +Year +YearConference’17, July 2017, Washington, DC, USA +Hancheng Cao et al. +Figure 11: (Non-self-cite) The number of patents that cite HCI papers over time. +Figure 12: (Non-self-cite) The time lag between patent and paper is long and getting longer for different types of citations and +venues. + +Patents Citing HCl Research +CHI +1000 +Number of patents +CSCW +UIST +800 +UbiComp +600 +400 +200 +0 +1990 +1995 +2000 +2005 +2010 +2015 +YearThe time lag of the most recent paper +cited by patents in year X +20 +CHI +(Year) +CSCW +15 +UIST +Time difference ( +UbiComp +10 +5 +0 +1990 +1995 +2000 +2005 +2010 +2015 +YearThe median time lag of the paper +cited by patents in year X +20 +CHI +(Year) +CSCW +15 +UIST +Time difference ( +UbiComp +10 +5 +0 +1990 +1995 +2000 +2005 +2010 +2015 +YearA Large-scale Analysis of Patent Citations to HCI Research +Conference’17, July 2017, Washington, DC, USA +Figure 13: (Non-researcher) Left: the number of papers published by each conference per year (red) and the number of papers +published in that year that were later cited by at least one patent (blue), at ACM CHI, CSCW, UbiComp, and UIST. + +CHI +CHI +100 +1200 +Published + Percent of published papers later cited by patents +Published and later cited by patents +80 +S1000 +per +(%) +800 +60 +Percent ( +of +600 +ber +40 +400 +wnN +20 +200 +0 +0- +1990 +1995 +2010 +2015 +1980 +1985 +2000 +2005 +1985 +1990 +1995 +2000 +2005 +2010 +1980 +2015 +Year +Year +CSCW +CSCW +100 +Published + Percent of published papers later cited by patents +300 +Published and later cited by patents +80 +(%) +60 +Percent ( +Number +40 +100 +20 +0 +0°1980 +1985 +1990 +1995 +2005 +2010 +2015 +2000 +1985 +1990 +2000 +2005 +2010 +1980 +1995 +2015 +Year +Year +UIST +UIST +100 +150 +Percent of published papers later cited by patents +Published +Published and later cited by patents +80 +(%) +100 +60 +Percent ( +75 +Number +40 +50 +20 +25 +0 +0°1980 +1980 +1985 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +2000 +2005 +2010 +2015 +1985 +1995 +Year +Year +UbiComp +UbiComp +100 +500 +Published +Percent of published papers later cited by patents +Published and later cited by patents +80 +"400 +(%) +60 +Percent +of +40 +20 +0 +1985 +2015 +1980 +1990 +1995 +2000 +2005 +2010 +1985 +1995 +2005 +2010 +1990 +2000 +2015 +Year +YearConference’17, July 2017, Washington, DC, USA +Hancheng Cao et al. +Figure 14: (Non-researcher) The number of patents that cite HCI papers over time. +Figure 15: (Non-researcher) The time lag between patent and paper is long and getting longer for different types of citations +and venues. + +Patents Citing HCl Research +400 +CHI +Number of patents +CSCW +UIST +300 +UbiComp +200 +100 +0 +1990 +1995 +2000 +2005 +2010 +2015 +YearThe time lag of the most recent paper +cited by patents in year X +20 +CHI +(Year) +CSCW +15 +UIST +Time difference ( +UbiComp +10 +5 +0 +1990 +1995 +2000 +2005 +2010 +2015 +YearThe median time lag of the paper +cited by patents in year X +20 +CHI +(Year) +CSCW +15 +UIST +Time difference ( +UbiComp +10 +5 +0 +1990 +1995 +2000 +2005 +2010 +2015 +Year \ No newline at end of file diff --git a/4dFQT4oBgHgl3EQf4Ta7/content/tmp_files/load_file.txt b/4dFQT4oBgHgl3EQf4Ta7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6269401120180c052dd02be1c76e73828a5edd17 --- /dev/null +++ b/4dFQT4oBgHgl3EQf4Ta7/content/tmp_files/load_file.txt @@ -0,0 +1,1990 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf,len=1989 +page_content='Breaking Out of the Ivory Tower: A Large-scale Analysis of Patent Citations to HCI Research Hancheng Cao Computer Science Stanford University California, United States hanchcao@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='edu Yujie Lu Computer Science University of California, Santa Barbara California, United States yujielu@ucsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='edu Yuting Deng School of Computer Science Carnegie Mellon University Pennsylvania, United States yutingde@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='edu Daniel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' McFarland School of Education Stanford University California, United States dmcfarla@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='edu Michael S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Bernstein∗ Computer Science Stanford University California, United States msb@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='edu ABSTRACT What is the impact of human-computer interaction research on industry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' While it is impossible to track all research impact path- ways, the growing literature on translational research impact mea- surement offers patent citations as one measure of how industry recognizes and draws on research in its inventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In this paper, we perform a large-scale measurement study primarily of 70,000 patent citations to premier HCI research venues, tracing how HCI research are cited in United States patents over the last 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We observe that 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1% of papers from these venues, including 60–80% of papers at UIST and 13% of papers in a broader dataset of SIGCHI- sponsored venues overall, are cited by patents—far greater than premier venues in science overall (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='7%) and NLP (11%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' However, the time lag between a patent and its paper citations is long (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='5 years) and getting longer, suggesting that HCI research and practice may not be efficiently connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' CCS CONCEPTS Human-centered computing → Empirical studies in HCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' KEYWORDS Industry impact, technology transfer, translational science, patent, citation analysis ACM Reference Format: Hancheng Cao, Yujie Lu, Yuting Deng, Daniel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' McFarland, and Michael S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Bernstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Breaking Out of the Ivory Tower: A Large-scale Analysis of Patent Citations to HCI Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In Proceedings of ACM Conference (Conference’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' ACM, New York, NY, USA, 24 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='nnnnnnn ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Conference’17, July 2017, Washington, DC, USA 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' ACM ISBN 978-x-xxxx-xxxx-x/YY/MM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='nnnnnnn 1 INTRODUCTION What is the impact of human-computer interaction research beyond academia?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Does HCI research diffuse into the industry1, contribut- ing to technological inventions and products?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Are most its insights ignored by the industry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' As an applied field of study intended to be closely relevant to application — where a considerable proportion of our research community’s contributions are functional proto- types and design implications for practitioners — the answers to these questions are critical to evaluating our translational success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' There have been rich discussions regarding the industry impact of HCI research since the early years of the field, and the relationship between research and practice in HCI has long been a focal subject in both research papers [18, 19] and conference panels [9, 15, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The literature remains unclear on the field’s level of success in achieving this impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' One line of the literature suggests high barriers: that HCI research has remained distant from industry impact, and that “HCI researchers and HCI practitioners work in relatively separate spheres of influence” [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This line of work also argues there is a considerable research-practice gap, one that is “real and frustrating” [60] and likely the result of a long list of barriers [18, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' However, another line of literature argues that the field achieves considerable success, that “HCI is at the vanguard of innovation and has repeatedly influenced industry” [32] and that “there is no question that research in the area of user interface software tools has had an enormous impact on the current practice of software development” [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' These threads of work are not necessarily incompatible—high barriers do not rule out the existence of successes that overcome these barriers—but the field’s overall status remains unclear: how far have we come, and how far do we have to go?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' One approach to- ward resolving this debate is to pursue new methods for measuring HCI’s impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Prior work has developed rich in-depth qualitative ev- idence ranging from personal technology transfer experience [22] to interviews with multiple stakeholders involved in the translation process [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Yet as the HCI community grows and both well-known 1In this paper, we use ‘industry’ to refer to non-research efforts that aim at practical impacts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' patents, products, design practices, which usually target a broad audience than academic researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Thus, in this paper, industry labs whose primary focus is to publish research papers are considered academia rather than industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='13431v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='HC] 31 Jan 2023 Conference’17, July 2017, Washington, DC, USA Hancheng Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' successes and painful failures become easier to point to, it becomes more and more urgent that we also assess broader patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' To fill this gap, we draw on methods from the growing measure- ment literature on innovation in translational sciences [1, 7, 45, 50, 77], where patent citations to research have been regarded as a valuable proxy of the impact that science has on industrial practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' While patent citation to research citation does not directly guar- antee industry impact, it reveals one potential pathway through which industrial inventors are aware of and recognize research ar- ticles: a necessary but not sufficient step towards industry impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2 Work using this approach has revealed the relevance of research and practice across science [1], mapped the translation landscape in bio-medicine [45, 50], and demonstrated that referencing science in the invention is associated with greater practical value [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Leveraging the modern analysis approaches from this line of work [51, 52], we report the first large-scale quantitative analysis of how HCI research is (and is not) being cited by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In do- ing so, we focus on one possible route of industry impact through HCI research: patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' There are many types of contributions in HCI—design patterns, behavioral results and theory among many others—and a patent lens focuses us only on styles of contribution that are considered prior art for patents, often systems and inter- action contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Specifically, we draw on data from Microsoft Academic Graph, Semantic Scholar, the United States Patent and Trademark Office (USPTO), and linkages between them [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This dataset enables us to study research papers from four premier venues in HCI, including CHI, CSCW, UIST, and UbiComp, and then replicate across all 20 SIGCHI sponsored venues that appear in Microsoft Academic Graph, tracing how those research papers are cited in patent documents from the 1980s through 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We study the institutes involved in the process, leverage citation analysis to measure the number and proportion of papers cited by patents over time and measure the length of time it takes before a paper is recog- nized by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We further conduct textual analysis to understand the topics that are likely to be cited in patents, and compare how patent-cited research differs from its non-patent cited counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We observe that: (1) HCI research has been cited extensively by patents — overall 20% of papers from CHI, CSCW, UIST and UbiComp, and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='4% of SIGCHI sponsored venues, are patent-cited, including a surprising 60-80% of UIST papers over a twenty year period, higher than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='5% of science overall and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='7% of biomedicine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' (2) The patent-paper time lag is long (on average 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='5 years) and is getting longer, such that citations from academic HCI research have dropped off by the time a paper receives patent attention;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' and (3) Within HCI research, there is substantial heterogeneity in patent citations across topics, for example, interaction and input techniques research are especially likely to be referenced by patents while theory, social and experience design research are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This analysis provides the first quantitative survey of the HCI technol- ogy transfer landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' While acknowledging potential limitations of patent citation as a method, we conclude that HCI has had a considerable impact on industry and is finding more relevance to practice than most disciplines in science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Yet, it takes a long time for 2More discussion and reflection on the usage of patent citation to science to study industry impact of research in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1 and Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='3 innovations in academia to be recognized and taken up by industry, corroborating the “long nose” theory on HCI innovation [12, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The contributions of this paper are as follows: We introduce measuring patent citations to science as a novel method to study research-practice relationships in HCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This provides quantitative evidence that complements qualitative evidence in existing HCI literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We release our analyzed dataset to enable future analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='3 We present the first large-scale, empirical study measuring the translational, longitudinal landscape of HCI research from paper to patent inventions with comparisons to other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This allows us to better understand and evaluate how HCI as an applied field is or is not finding connections to practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Our work contributes to reflections and recommendations for the HCI community to better foster a translational envi- ronment and recognize impacts beyond academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 2 BACKGROUND AND RELATED WORK In this section, we position our work in the literature on industry impact, the HCI research-practice divide, and bibliometric analysis in HCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1 Industry impact Industry impact are often achieved through technology transfer, which refers to the transmission of knowledge generated by an individual, the university, government agencies, or any institution capable of generating knowledge, to another person or organiza- tion, in an attempt to transform inventions and scientific outcomes into new products and services that benefit society [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Govern- ment and funding agencies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', in the United States, NSF and NIH) increasingly seek to nurture “translational research” to facilitate industry impact from basic research so as to generate greater ap- plied value and promote technology advances [76, 79], and prior research has shown inventions that refer to high-quality research are more likely to be great inventions of value [34, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Prior research has sought to identify when, where, and how sci- entific research influences industry invention [3, 7, 17, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' There, patent citations to science have been widely used as a proxy for studying technology transfer from research to practice despite noises, as it is one of the only available large-scale records of the knowledge flow from research to practice that demonstrate the ini- tial awareness and recognition of research in industrial inventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' For instance, Tijssen [68] revealed through patent-paper citations how Dutch-authored research papers influence inventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Like- wise, Ahmadpoor and Jones [1] studied 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='8 million US patents and how they link to 32 million research articles, finding that over half of patents cite back to a research article and that patents and papers are on average 2–4 degrees separated from the other domain, pro- viding some insight into the interplay between patents and prior research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Jefferson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [37], Manjunath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [50] used patent cita- tions to science data, measuring and reporting statistics describing how research in biomedicine turns into inventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Liaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [46] proposed a method to rank academic journals that utilizes non- patent references in patent documents to evaluate their practical 3Available dataset at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='7910/DVN/QM8S1G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' A Large-scale Analysis of Patent Citations to HCI Research Conference’17, July 2017, Washington, DC, USA impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Other works used patent citation to science to study the strategy of inventors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' deep search vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' wider scope search) and how the strategy relates to technology impacts and organization performance [2, 25, 28, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' To facilitate further studies on how inventions rely on basic science, Marx and Fuegi [51, 52] linked and disambiguated patent citations to science linking the USPTO dataset and Microsoft Academic Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='4 We build off this rich social science literature by studying indus- try impacts of HCI research through leveraging and extending their methods[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2 From HCI research to practice HCI is a field that emphasizes the design and the use of computer technology, especially interfaces between people and computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' HCI research implement, demonstrate and test new technologies through prototyping and end-user feedback [47], and most HCI work includes ‘design implications’ sections aiming to translate their research insights to more practical outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The applied nature of HCI lead to the community’s long-standing interest in industry impact, with many publications and panel discussions at conferences aimed at facilitating better technology transfer [15, 22, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' One line of the literature primarily focus on the many barriers HCI faced in translating research insights to industrial practice [18, 22], while another line of literature speaks to the considerable impact that HCI research has had or could have on the industry [32, 57, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Many papers argue that despite the insights that HCI research can offer to practitioners, HCI research findings are rarely used in in- dustry [18]: that there has been an “immense” research practice gap in practice that is “real and frustrating” [60], that “HCI researchers and HCI practitioners work in relatively separate spheres of influ- ence” [22], and that “attendees at venues like ACM CHI often lament that no HCI research ever goes into product” [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Colusso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [18] interviewed design practitioners so as to understand why they do not use academic research and why and how they use other re- sources in their works, presenting a detailed catalog of barriers that inhibit academic resources usage in industry, such as the content being hard to read, hard to find, and not actionable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Chilana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [16] stated the distinct goals of HCI research and product may result in a research-practice gap, that the users who are the major focus of the user-centered design approach in HCI research are generally not the buyers of HCI products, and that to make a research-to- product transition one has to switch from being user-centered to adoption-centered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Furthermore, prior work [22, 75] suggested that HCI researchers usually lack the knowledge, resources, connections, experience, interest, or time to pursue technology transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Other work has shown similar results demonstrating a research practice gap in HCI [10, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Prior research has discussed potential approaches to address the research-practice gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' For instance, Velt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [69] identified two key dimensions of the research-practice gap – general theory vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' particular artifacts, and academic HCI research vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' professional UX design practice – and discussed the benefits of translation led by researchers, by practitioners, or co-produced by both as bound- ary objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Colusso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [19] proposed a continuum translational 4We leverage this particular dataset in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' science model for HCI that consists of three steps: basic research, ap- plied research, and design practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Shneiderman [65] wrote a book proposing principles to better blend science, engineering and design to achieve innovations and breakthroughs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Other work discusses the challenges and lessons learned from the specific translation of HCI research to practice [62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Meanwhile, another line of work argues that HCI research could have considerable impact on industrial practice despite the barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Harrison argues that “HCI is at the vanguard of innovation and has repeatedly influenced industry [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='] HCI research has a much greater impact in identifying opportunities in the first place, es- tablishing the science and methods, building a shared vision, and developing a pipeline of human talent” [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Likewise, Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [57] wrote “There is no question that research in the area of user interface software tools has had an enormous impact on the cur- rent practice of software development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Virtually all applications today are built using window managers, toolkits, and interface builders that have their roots in the research of the 70’s, 80’s, and 90’s”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Shneiderman’s work [66] further stated that “The remarkably rapid dissemination of HCI research has brought profound changes that enrich people’s lives”, but also providing a tire-tracks diagram showing how HCI research on subjects such as hypertext, direct manipulation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' turned into product innovations by industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Similarly, product innovations over the years mirror the early ideas of canon HCI visions [11, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Other research detailed successful cases of tech transfer, such as the translation of the multi-touch interface from research into the Apple iPhone and Microsoft Sur- face, while highlighting a long time lag between initial research and commercialization, which can be 20 years or more [12, 32, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This prior work guides us to the following research questions: RQ1: What is the impact of HCI research on patents?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' How much HCI research is cited in patents?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' RQ2: When is the impact of HCI research on patents?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' How long does that impact take?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' RQ3: Where is the impact of HCI research on patents?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Which topics of research are especially likely or unlikely to diffuse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' RQ4: Who is involved in the process of recognizing HCI research on patents?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Which institutions produce such work, and which consume it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The rich qualitative insights derived from case studies, field- work, interviews, and personal experience, open an opportunity for complementary work that engages in quantitative, longitudinal analysis that directly measures how HCI research gets recognized in industry inventions and technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We believe that such a viewpoint might systematically detail the translation landscape of HCI as a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='3 Bibliometrics and HCI As an important area of computing and information science, HCI has featured several projects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', [40, 49]) that quantitatively un- derstand the structure and evolution of the field through the study of writing and citation patterns, known as bibliometrics [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' One commonly used bibliometric method is an analysis of a large- scale citation network, which leverages the increasingly available citation data from publishers such as Web of Science and Microsoft Academic Graph and their associated metadata of the scientific Conference’17, July 2017, Washington, DC, USA Hancheng Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' publications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' institutes, authors), and even textual analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' topic modeling, keyword extraction) of the scientific publications, so as to gain insights on patterns behind the diffusion of scientific ideas [26, 70], research productivity [48, 72], and identify potential ethical and social issues in science [35, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' For instance, Koumaditis and Hussain [42] leveraged citation data from 962 HCI publications and reveal that HCI research can be categorized into major themes of design, data management, user interaction, psychology, and cognition, and they identified more recent trends in HCI in the workplace, sensors, and wearables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Likewise, Kaye [40] reported “some statistical analyses of CHI”, including author counts, gender analysis, and representations of repeat authors so as to motivate dis- cussions on the preferred state of CHI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Bartneck and Hu [5] reveal that only a small percent of countries account for the majority of CHI proceedings, and present a ranking of countries and organiza- tions based on their H-index of CHI proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Correia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [21] used 1713 CSCW publications and characterized top CSCW papers, citation patterns, prominent institutes as well as frequent topics, highlighting the fact that CSCW is influenced primarily by a few highly recognized scientists and papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The authors further quanti- tatively explored the relationship between collaboration types and citations, paper frequency, etc [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Similar types of analysis have also been done on more regional HCI conferences [4, 30, 56, 59] as well as studying subcommunities in HCI [49, 71, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Visual analytics is another approach used to help understand HCI’s evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' For instance, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [43] proposed a system PaperLens to reveal trends, connections, and activity of 23 years of the CHI conference proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Matejka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [54] proposed an interactive visualization that highlights family trees of CHI and UIST papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [33] presented a visual exploration of four HCI conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' They showed that the years when a given conference was most selective are not correlated with those that produced its most highly referenced articles and that influential authors have distinct patterns of collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' To the best of our knowledge, there have been no analyses lever- aging quantitative methods to study recognition of HCI research beyond academia as we present in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In contrast with prior work, we leverage large-scale patent citations to quantify the impact of HCI research in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 3 METHOD In this section, we describe the method we used to study the impact of HCI research papers in practice using patent citations to science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1 Patent citations as a pathway to study industry impact of research papers We leverage patent citations to research as a proxy to study the influence of HCI research on industrial practice at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' While patent citation to research citation does not directly mean industry impact, it reveals one important potential pathway from research to practice where industrial inventions become aware of and recog- nize research articles, which is often a necessary but not sufficient step towards producing industry impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Alongside with studying other forms of influence, such as design processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', usability testing, heuristic evaluation), design patterns, open source software (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', d3, Vega), patent citations to science could help us piece to- gether the translational landscape in HCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This method is widely used in the innovation literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', [1, 25, 28, 38, 50, 51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Patent citations to research are considered valuable signals indicating the influence of research on the industry, signals that “reflect genuine links between science and technology.” [68], and “appear to be a substantive if a noisy indicator of the role of specific, prior scientific advances” [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' While citations between research articles capture research influence [26], patent-to-research citations capture “how basic research influences commercialization and thus provides a complementary measure of impact” [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Such data has been used extensively to measure knowledge spillovers from academia and government to industry [1, 23, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The rationale behind the validity of this approach is that in patented inventions, inventors are obliged to disclose any “prior art” related to their invention, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', all information known to that individual to be material to patentability”,5 including materials that the inventors leveraged in the invention process, or other similar material to the focal invention in order to distinguish it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The prior art includes both references to prior patents, and references to non- patent literature, such as academic articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Patent citation is an important part of a patent, as missing prior art (either prior patents or non-patent literature), could have potential legal issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Apart from citations provided by inventors, patent examiners who review patents for approval or rejections also add references they think are of relevance to ensure the legitimacy of the patent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Prior work has validated this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Nagaoka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [58] sur- veyed 843 inventors finding patent citations to science are indeed important linkages to science, despite possible errors of over- and under-inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Callaert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [13] interviewed 36 inventors and re- port 44% of patent citations to science are considered as “important” or “very important”, and another 34% are “background” citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Based on the rich literature in this space, we conclude that patent citation to science can be used as a reliable data source to measure the recognition of HCI research efforts in inventions, thus provid- ing a valuable proxy of HCI research impact in the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Of course, there is no perfect appoach for studying industry impact: we discuss and reflect on the limitations of our method in detail in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='3, and it is especially important to bear in mind there are multiple translational gaps in HCI research [19], and we are only studying one important step in the process with regard to patent, where certain types of contribution such as theory are likely to be under evaluated through this dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Empirically, we find support for the validity of using patent citations to research as a proxy of impact in industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We manu- ally check patent reference lists of a number of patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' As shown in Figure 1, the highly-cited patent by Apple Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' “Mode-based graphical user interfaces for touch sensitive input devices” (cited 1,898 times),6 cites closely related research papers in CHI on multi- touch, such as “A Multi-Touch Three Dimensional Touch-sensitive Tablet", which is the case of technology transfer discussed by Bux- ton [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The even more well-cited Apple Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' patent (cited 4,018 times) “Method and apparatus for integrating manual input” 7 also made reference to several relevant HCI papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' These cases motivate 5https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='uspto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='gov/web/offices/pac/mpep/mpep-2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='pdf 6https://patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='com/patent/US8239784B2/en 7https://patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='com/patent/US6323846B1/en A Large-scale Analysis of Patent Citations to HCI Research Conference’17, July 2017, Washington, DC, USA us to leverage patent citations as a signal indicating the invention’s recognition of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2 Dataset To study how HCI papers are recognized by patents, we required a citation graph from patent to research, and the metadata (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', author name, affiliation, publication year, title, venue) from both the paper side and patent side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The data preparation pipeline is composed of three steps: 1) Prepare metadata of papers and patents, and the citation graph from patents to research, 2) Select papers from the venues of interest and clean the data, and 3) Link the clean metadata based on the citation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This pipeline could be applied to other research communities, or other venues within SIGCHI, by selecting other venues of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Patent citation to science that connects USPTO to Microsoft Academic Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' To capture references from patents to HCI re- search papers, we drew on a public dataset [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This dataset is a state-of-the-art approach to connect each patent reference in USPTO (1947-2020) to academic papers (1800-2020) from Microsoft Academic Graph through matching unstructured front-page and in-text references in patents to published papers using a disam- biguation matching method, resulting in 22 million patent citations to research papers (known as Patent Citation Science dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='8 In their papers, the dataset creators verified the quality of their datasets through manual checking and error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We captured the reference type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', from applicant, from examiner, unknown), whether the reference appears in-text or on front page, the time between paper publication and the citing patent application, and whether a patent citation is a self-citation to a research paper by one of the patent authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' A paper to patent pair is considered self-cited when there is an overlap between the inventors of the patent and the authors of the cited scientific papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Microsoft Academic Graph Metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The Microsoft Academic Graph is a heterogeneous graph that provides scientific publication records, citation relationships, the information of authors, insti- tutions, journals, conferences, and fields of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We leveraged the public Microsoft Academic Graph dataset provided at Zenodo Reliance on Science project site9 so as to extract information with regard to academic publications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', title, author, author affiliation, and year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' USPTO metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We leveraged US patent data from the United States Patent and Trademark Office (USPTO)10 to represent tech- nological inventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Patents have similar fields as academic publi- cations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', title, abstract, inventor, assignee, and year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Semantic Scholar (abstract, citation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The abstract informa- tion of the paper and their academic influence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', number of published papers, citation count) are missing or hard to process in the original Microsoft Academic Graph metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='11 To further ex- pand data information about authors, papers, citations, and venues, 8Specifically, we used the patent-to-article citations of Version v37 (Jul 19, 2022) at Zenodo: http://relianceonscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='org 9http://relianceonscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='org 10https://patentsview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='org/download/data-download-tables 11https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='com/en-us/academic-services/graph/resources-faq we utilize the Semantic Scholar Academic Graph API,12 which fills in this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The details of the data we utilize can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='3 Data Preprocessing Venue selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In our analysis, we primarily considered four impactful Human-Computer Interaction (HCI) venues: the ACM CHI Conference on Human Factors in Computing Systems (CHI), ACM Conference On Computer-Supported Cooperative Work And Social Computing (CSCW), ACM Symposium on User Interface Soft- ware and Technology (UIST), and International Joint Conference on Pervasive and Ubiquitous Computing (UbiComp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='13 For a broader footprint of HCI research, we created a second dataset of SIGCHI sponsored venues14 — a total of all 20 SIGCHI sponsored venues15 that appear in the Microsoft Academic Graph, which covers not only large, premier venues such as CHI, but also smaller, more specialized venues such as MobileHCI and CHI PLAY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We used this second set as more representative of the overall field of HCI, to further validate our findings and compare with overall patterns reported in other fields of science in a fairer way16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Data Cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We further conducted data cleaning on the four chosen venues by looking up papers in Semantic Scholar rather than Microsoft Academic Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We found that Microsoft Aca- demic Graph metadata sometimes wrongly classify venues such as “Brazilian Symposium on Human Factors in Computing Systems” as “CHI”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' To solve this issue, we filtered out irrelevant papers by manually checking the full name of the venue column from Seman- tic Scholar, which proves to be of better quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We then applied this filtering process to all the paper and patent citations to science files by joining over the paper id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Data Linking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In order to better combine the paper and patent information for analysis, we linked patent data, Microsoft Academic Graph data and Semantic scholar data via the Patent Citation Sci- ence dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='17 The joined data after 2019 has incomplete or little coverage, thus we focus our analysis on HCI research papers and patents that cite HCI papers before 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Final Data Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Our final data for analysis includes 23,432 papers from the four chosen venues, with 16,014 from CHI, 3,084 from CSCW, 1,746 from UIST, and 2,588 from UbiComp across 1980 to 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Within these papers, we captured 69,900 citation records from patent to science, with 42,676 from CHI, 5,900 from CSCW, 17,040 from UIST, and 4,284 from UbiComp, which are associated with 30,660 patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The broader SIGCHI sponsored venue data include 57,385 papers in total (41% are papers from the four premier 12https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='semanticscholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='org/product/api 13Starting 2017, the UbiComp conference main technical tracks consist of papers published in Proceedings of the ACM on Interactive, Mobile, Wearable and Ubiquitous Technologies (IMWUT), which we captured in our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 14https://sigchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='org/conferences/upcoming-conferences/ 15Details of the venues in Appendix B 16Note that in this paper we primarily report findings on the four chose venues rather than SIGCHI sponsored venues overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We elect to focus on these four venues as a practical matter, as we have spent considerable manual efforts in cleaning data related to the four chosen venues to ensure data quality, as indicated in “Data Cleaning" section, which makes our analysis more likely to reflect actual trends in these venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 17Confusingly to HCI researchers, this is known as the “Patent Citation Science” (PCS) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We joined information from the patent side using the field patentid to information from the paper side using the field magid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Conference’17, July 2017, Washington, DC, USA Hancheng Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' (a) Patent US8239784 frontpage with abstract, inventors, assignee etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' (b) Part of the citation list of Patent US8239784.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Figure 1: Patents are obliged to cite prior art, including prior patents and non-patent literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' research articles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Here, a patent by Apple Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', “Mode-based Graphical User Interfaces for Touch Sensitive Input Devices” [36], has citation to relevant HCI papers, including “ActiveClick: Tactile Feedback for Touch Panels”, “A Multi-Touch Three Dimensional Touch-sensitive Tablet”, a mis-named citation to Ken Hinckley (“Kinkley et al.”), and many other references to HCI research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' (12) United States Patent (10) Patent No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' : US 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='784 B2 Hotelling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' (45) Date of Patent: Aug.' 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+page_content=' 154(b) by 936 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' (Continued) (57) ABSTRACT A user interface method is disclosed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The method includes detecting a touch and then determining a user interface mode when a touch is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The method further includes acti- vating one or more GUI elements based on the user interface mode and in response to the detected touch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='EVB Elektronik TSOP6238 IR Receiver Modules for Infrared Remote Control Systems dated Jan.' metadata={'source': 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on Line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Hillier and Gerald J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Lieberman, Introduction to Operations Research (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' International Search Rep0rt dated Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 3, 2006 (PCT/US 05/03325) Jacob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', "Integrality and Separability of Input Devices," ACM Transactions on Computer-Human Interaction, 1:3-26 (Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 1994) ings, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 223-230, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Kionx "KXP84 Series Summary Data Sheet" copyright 2005,dated Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 21, 2005, 4-pgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=", A Multi-Touch Three Dimensional Touch-Sensitive Tab- let, in CHI '85 Proceedings, pp." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 121-128, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Lee, “A Fast Multiple-Touch-Sensitive Input Device," Master\'s The.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' sis, University of Toronto (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Matsushita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', "HoloWall: Designing a Finger, Hand, Body and Object Sensitive Wal1,’ in Proceedings of UIST \'97, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 1997A Large-scale Analysis of Patent Citations to HCI Research Conference’17, July 2017, Washington, DC, USA venues), 83,793 citation records (51% are citations made to the four premier venues), and are associated with 36,024 patents in total (85% patents cited papers from the four premier venues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Note that for all chosen venues, our data includes not only main conference papers but also extended abstracts, posters and other forms of publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We did not attempt to filter and focus our analysis only on main conference papers, given the difficulty to classify and challenge fuzzy matching based on venue name (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' in our dataset, many posters are not explicitly labeled as poster publications and are hard to differentiate from main conference papers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We release our dataset at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='7910/DVN/QM8S1G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 4 RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1 RQ1: What is the impact of HCI research on patents?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We first study the quantity of HCI papers that are later recognized by patents and present a table of top papers cited by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Proportion of papers that get cited by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' To assess the extent of HCI research being recognized in patents, we first cal- culated the aggregated proportion of the number of HCI papers at our four premier HCI venues, and SIGCHI sponsored venues overall, that were cited by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We found 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1% of papers in the four venues, and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='4% of papers from SIGCHI sponsored venues overall, are recognized by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This rate is much higher than the proportion of science cited by patents overall (approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='5% [51]), and the prominent journal paper patent rate (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='7% across multiple scientific fields [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The rate is also much higher than that of bio-medicine in general, a field that has a rich tradition empha- sizing translational science, which is at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='7% [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We replicated our analysis on premier venues in other areas of Computer Science by comparing the premier HCI venue patent rate (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1%) with premier venue patent rate of other subfields, finding that AI patent rates (as measured through AAAI and IJCAI, two of the largest and pre- mier AI conferences) are 5%, Natural Language Processing patent rates (as measured through ACL, EMNLP, and NAACL, three of the largest and premier NLP conferences) are 11%, and Computer Vision patent rates (as measured through CVPR, ECCV, and ICCV, three of the largest and premier computer vision conferences) are 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Two- proportion z tests further confirm the significance of the difference in percentages with 𝑧 = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='9, -13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1, (𝑝 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='001) when compar- ing premier HCI venue patent rate with patent rates of premier venues in AI, Natural Language Processing and Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Taken together, these results suggest that HCI’s impact through patent citations is higher than science overall, biomedicine, AI, and NLP, and roughly at par with Computer Vision, an area of intense industry interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Are research citations in patents truly central to the patents, or are they thrown in just to satistfy a patent examiner?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' To answer this question, we leverage a distinction between in-text citations and front page citations in patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This distinction allows us to more directly measure the impact of HCI research in patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In- text patent citation to science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' as suggested by prior work [8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 52],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' are more likely to “capture the scientific articles upon which the scientists truly relied upon for inspiration” and “have the potential to more accurately represent the sources of scientific inspiration upon which the inventors actually drew in the invention process" since they “tend to be supplied by the inventors themselves”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' in contrast to “legally binding” front page citations which “tend to be carefully reviewed (and sometimes added) by patent attorneys” [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We find 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1% papers in our chosen four venues have been cited in- text by patents, whereas the proportion of patent in-text citation to science is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='3% for SIGCHI sponsored venues and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='4% for science overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This result further replicates our finding that HCI research appears to have real impact, surprisingly even moreso than many other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Investigating temporal patterns, we plot the total number of HCI research papers in each of the four venue published over years, shown in red in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' HCI research has grown rapidly over the past 38 years for all four venues, especially at CHI: from 74 papers in 1982 to 1200 in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This growth is particularly pronounced within the last 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We then counted the total number of HCI papers cited by patents by the publication year of the paper and calculated the ratio between the number of HCI papers cited and the total number of HCI papers accepted in a particular year by each venue (blue line in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The citation ratios start climbing especially starting around 1990 and persist since then (Figure 2),18 with several conferences observing a third to a half of their papers cited by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' At UIST in particular, the patent citation ratio reaches 60% - 80% from 1990 - 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The citation ratio decreased after 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' One possible explanation is the time lag between patent and paper is long, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', it might take a decade for a paper to start gathering patent citations, and papers since 2015 are still too young by this metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This time lag will be further discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In other words, the data are right censored, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', more recent papers have not been fully recognized by patents captured in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' As such, we expect a higher proportion of HCI papers overall will be referenced by patents eventually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Increasing citations to HCI research in patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' A total of 30,660 patents cite research in the four chosen venue, and 36024 patents cite research from SIGCHI sponsored venues overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This raw volume began increasing after 2000 (Figure 3, and has more than quintupled since 2000 at CHI from around 175 patents per year in 2000 to over 1000 per year in 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' However, the number of patents plateaus and even decreases a bit in more recent years, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' patents begin citing less and less CSCW research starting in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This could be a result of changes on the demand side, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', the industry is less interested in novel social computing applications, or on the supply side, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', HCI publishing more papers that are not intended to be as industry-relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' More evidence is needed to derive the mechanisms behind this result, beyond the scope of our current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Top cited papers by patents in HCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We further examined the HCI papers that were cited the most by patents by each venue (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Papers highly cited by patents also tend to be highly cited by research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The papers most highly cited by patents are primarily 18We removed years where conferences did not meet from our analysis and smoothed the curve, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' CSCW was only held every other year until 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Conference’17, July 2017, Washington, DC, USA Hancheng Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Figure 2: Left: the number of papers published by each conference per year (red) and the number of papers published in that year that were later cited by at least one patent (blue), at ACM CHI, CSCW, UbiComp, and UIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Right: a substantial proportion of HCI papers are recognized by patents, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 60% - 80% UIST papers are recognized by patents 1990 - 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' systems work, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', building a new system or proposing a new de- sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This result parallels with the earlier observation that UIST has the highest rate of papers cited by patents since UIST is particularly targeted at new interfaces, software, and technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Most papers in this list were published prior to 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' however, the majority of the patents that cited HCI papers come after 2005, indicating again the potential long time lag between paper publication and patent reference in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Highly-cited papers in academia are more likely to be recog- nized by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' we investigated how academic impact ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='CHI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='CHI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='100 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2015 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1985 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1995 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='UbiComp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='UbiComp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Published ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Percent of published papers later cited by patents ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Published and later cited by patents ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='80 "' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Percent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1985 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1980 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1990 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1995 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2005 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2015 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1985 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1995 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2005 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2015 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1990 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='YearA Large-scale Analysis of Patent Citations to HCI Research ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Conference’17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' July 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' USA Figure 3: Left: over 1000 patents are citing CHI paper each year after 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The number of patents citing HCI research began rising after 2000 and more than quintupled since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Right: the number of patents citing SIGCHI sponsored venues follow similar trend, as a large proportion (85%) made references to the four premier venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' relates to patent impacts, measured by the paper’s number of cita- tions from other academic papers (academic citation count) and the number of citations from patents (patent citation count).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Figure 4 shows the academic citation count for both papers recognized by patents and papers not recognized by patents over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Patent- cited papers have higher paper citations (average academic citation count 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1) than non-patent-cited papers (average academic ci- tation count 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='9), a difference that is significant via an unpaired t-test (𝑝 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='001), Cohen’s D=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We further conducted zero-inflated negative binomial regres- sion19 over patent citation and paper citation count in CHI, CSCW, UIST, and UbiComp and get regression coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='0233, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='0172, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='0316, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='0175 respectively (𝑝 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The coefficient indicates that highly-cited papers in academia are indeed more likely to be cited by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Such a relationship is especially salient at UIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2 RQ2: When is the impact of HCI research on patents?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' How long does it take for patents to recognize papers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' To examine this question, we investigated the time lag between patent and paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The time lag between patent and paper is long and getting longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' To measure how long it takes for an HCI paper to be rec- ognized by patents, for each patent, we investigated the time lag between the issue date of the patent and the publication date of all papers it cited from our four chosen venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We measured the lag from the patent backward rather than from the paper forward because we cannot know whether a paper will receive a citation but has not yet—but we can know how far back a patent’s citations reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 19Zero-inflated negative binomial regression is ideal for modelling count-based de- pendent variables with zeroes, which corresponds to our data where a significant proportion of HCI papers get no patent citation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In the four premier HCI venues, the average patent-paper lag is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='5 years (𝜎 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='8 years), indicating that patents on average refer- ence HCI research papers published 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='5 years before the patent filing date but there is significant variance over the time lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We then studied how the time lag varies over time by aggregat- ing the patent-paper time lag at the individual patent levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' As Figure 5a) shows, the median difference between the time the cited paper is published and the time the paper is cited by the patent, is becoming larger from 1989 to 2014 for all the venues from about around 5 years to around 10 − 15 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' However, since 2014, this trend bifurcates among different venues: the time lag for CSCW in- creases to over 15 years and Ubicomp decreases to about 10 years in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We also noticed that all venues have nearly indistinguishable trends except Ubicomp, which has about 3 years of time lag lower than other venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In recent years, CSCW takes the longest time to be recognized by patents, while UIST and UbiComp take a shorter time, which could be explained by the fact that more system-driven works are likely to diffuse more quickly into practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We also examined the time lag between the patent and its most recent cited paper (Figure 5b), testing how recent the freshest re- search is that patents draw on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' These general trends are consistent with the median time lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Again, the difference between the time its most recent cited paper was published and the time it is patented also becomes larger from 1989 to 2011 for all the venues, from less than 5 years to around 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This increase gradually slowed down, leading to a slight decrease in more recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The patent citation also involves different sources, some are added by the applicants/inventors, while others are added by patent examiners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The dataset we used also provides a breakdown of refer- ence types, including applicant/inventor added, the examiner added, other, and unknown types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' References added by patent examiners are generally more recent (average time lag: 6 years) than what the inventor added (average 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='8 years), although similar trends of long time lags and increasing time lags are still observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Patents Citing HCl Research CHI Number of patents 1000 CSCW UIST 800 UbiComp 600 400 200 0 1990 1995 2000 2005 2010 2015 YearPatents Citing HCl Research 2000 SIGCHI patents 1500 of 1000 Number 500 0 1990 1995 2000 2005 2010 2015 YearConference’17, July 2017, Washington, DC, USA Hancheng Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Title ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Patent Citations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Paper Citations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Year Published ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='CHI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='A multi-touch three dimensional touch-sensitive tablet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='708 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='231 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1985 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='PaperLink: a technique for hyperlinking from real paper to electronic content ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='134 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1997 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Bringing order to the Web: automatically categorizing search results ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='196 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='486 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='A study in two-handed input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='175 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='544 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1986 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Generalized fisheye views ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='175 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2180 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Finding others online: reputation systems for social online spaces ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='153 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2002 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Applying electric field sensing to human-computer interfaces ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='142 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='324 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1991 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Pad++: a zooming graphical interface for exploring alternate interface physics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='131 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='754 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1994 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='UbiComp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Validated caloric expenditure estimation using a single body-worn sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='113 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='83 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2009 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='InfoScope: Link from Real World to Digital Information Space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Self-Mapping in 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='11 Location Systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='130 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2005 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='The NearMe Wireless Proximity Server ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='62 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='162 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2004 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Predestination: Inferring Destinations from Partial Trajectories ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='498 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2006 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='UbiTable: Impromptu Face-to-Face Collaboration on Horizontal Interactive Surfaces ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='261 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2003 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Accurate GSM Indoor Localization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='537 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2005 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Very Low-Cost Sensing and Communication Using Bidirectional LEDs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='157 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2003 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Particle Filters for Location Estimation in Ubiquitous Computing: A Case Study ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='254 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2004 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='PowerLine Positioning: A Practical Sub-Room-Level Indoor Location System for Domestic Use ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='152 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2006 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Table 1: Top CHI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' CSCW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' UIST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' and UbiComp papers cited by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The majority of them are highly-cited papers in academia whose major contribution is a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' All results here indicate that patents mostly cite old research, and are citing increasingly older research, which holds true across venues and reference types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This conclusion is largely identical to what is found in science in general [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We replicated our analysis on other areas of Computer Science in a similar way as in Sec 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1, finding that the time lag between patent and their referenced papers for AI, Natural Language Processing, and Computer Vision are 17 years, 13 years, and 10 years respectively, suggesting similar patterns across subfields in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' HCI research has moved on by the time a paper receives patent attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Has the HCI community left an idea behind by the time industry gets interested?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Concerns circulate that HCI has a reputation for trend following and jumping to new shiny ar- eas every few years [12, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Are patent-cited papers still receiving academic interest by the time it starts receiving patent citations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' To answer this question, for all papers from the four chosen venues that eventually get cited by patents in our dataset, we compare (a) the time lag between the publication year of the paper and the issue year of the first patent that cites the research paper (first A Large-scale Analysis of Patent Citations to HCI Research Conference’17, July 2017, Washington, DC, USA Figure 4: Papers cited by patents receive more academic citations in HCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Figure 5: The time lag between patent and paper is long and getting longer across venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' patent citation lag), and (b) the time lag between the publication year of the paper and the paper’s “peak citation year” when the research paper gets the most academic citations (peak citation lag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Peak citation lag averages 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='74 years in our dataset, compared with 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='48 years for first patent citation lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='20 A paired t-test confirms 20The first patent citation lag is lower than patent backward citation lag reported earlier (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='5 years) due to right censoring, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' recent patent-cited papers are biased towards short lags since those with long lags have not yet been observed in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Peak citation lag have similar issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' If we allow paper enough time to accrue patent citations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' focus the analysis on papers published before 2000 (cutoff year), we get an average first patent citation lag of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='4 years (thus replicating the prior results) that the difference between these two lags are significant 𝑡(3740) = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='3 (𝑝 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='001), Cohen’s D=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This result supports the concern that HCI’s focus shifts to other topics by the time industry take up an idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Self-cite tends to be faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' One exception to this temporal pattern is that self-citation patents have a shorter patent-paper time lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Since 2008, the time lag for the non-self-cite patents increased and peak citation lag of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='5 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We varied the cutoff year, and found on average first patent citation lag is always longer than peak citation lag which suggests the robustness of our finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Non patent-cited Patent-citedThe median time lag of the paper cited by patents in year X 20 CHI (Year) CSCW 15 UIST Time difference ( UbiComp 10 5 0 1990 1995 2000 2005 2010 2015 YearThe time lag of the most recent paper cited by patents in year X 20 CHI (Year) CSCW 15 UIST Time difference ( UbiComp 10 5 0 1990 1995 2000 2005 2010 2015 YearConference’17, July 2017, Washington, DC, USA Hancheng Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' rapidly and was above 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='6 years in 2018, while the self-cite patents remain below 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='3 years, which suggests that papers transferred faster by authors themselves into patents compared with those transferred by others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='3 RQ3: Where is the impact of HCI research on patents?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Which HCI research topics are the focus of industry activity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' To answer this question, we compare non-patent-cited HCI papers to patent-cited HCI papers in the four chosen venues via Latent Dirichlet Allocation (LDA), a classic method of topic modeling [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' LDA automatically discovers topics within documents, where each topic is represented as a probability distribution of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Each document can also be represented as a probability distribution over different topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We concatenated each paper title with its abstract (if available) to represent its contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Similarly, we concatenated each patent title with its abstract (if available) to represent the patent’s contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We then tokenized the text corpora into unigrams and bigrams, filtered out terms that appear fewer than 5 times in the corpus, removed stop words in English, and then ran LDA modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We varied the number of topics and align on seven topics resulting in the highest quality topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Figure 6 reports the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Through checking representative documents and word clusters with HCI experts, we titled each topic: topic 0 is related to patent terms, the topic is 1 on modalities, topic 2 is system interaction, topic 3 is on evaluations, topic 4 is on theory, topic 5 is on social and experience design, and topic 6 is on input techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We then computed the topic distributions for each document (paper or patent) in our corpus, then aggregated topic distributions of all documents within a specific year that belong to a certain doc- ument category (patents, patent cited papers, or non-patent cited papers) so as to get an estimated number of documents that belong to a particular topic for that document category for a particular year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In the first row of Figure 7, we plotted the topic distribution for patent-cited HCI papers (left), non-patent cited HCI papers (middle), and patents (right), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', how many papers belong to topic X in year Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The second row of Figure 7 normalizes this topic distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', what is the proportion of topic X in year Y for a specific document category, to better illustrates the distribution pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' As can be observed from Figure 7, system interaction has domi- nated the patent-cited HCI papers over time, indicating that system- oriented research has been of considerable importance in patent- cited HCI research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' From 1980 to 2000, about 40% patent-cited HCI paper are system interaction related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' After 2000, the percentage of system interaction decreased to about 20% but began expand- ing again in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We also observed that input techniques have expanded significantly over time and reached nearly 20% after 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Evaluations have also grown in general and contributed about 20% of all patent-cited HCI papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In comparison, the topic distribution of non-patent cited papers shows a very different pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The results mirror the methodolog- ical plurality of HCI, where not all contribution types have an industry impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Theory work is highly visible in non-patent cited HCI papers over time, though the proportion is gradually decreas- ing from about 40% before 2000 to about 20% in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Social and experience design has grown significantly from nearly 0 percent in 1980 to about 20% in 2018, indicating behavior-oriented research has been of considerable importance in non-patent-cited HCI pa- pers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Evaluations and system interaction contributed to about half of all non-patent-cited HCI papers in 1980, but this percentage has decreased to about 30% in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Through unpaired t-test, we further verify there exist statistically significant differences between topic distributions in patent-cited papers and non-patent cited papers: there is a higher proportion of theory (𝑝 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='001), social & experi- ence design (𝑝 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='05) work, and lower proportion of system inter- action (𝑝 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='001), modalities (𝑝 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='001) work in non-patent-cited HCI papers compared to patent-cited counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We emphasize that this is not a negative outcome for theory, behavioral, and other research that does not produce artifacts, as they have an impact through other channels, or could influence patent in an indirect way [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Additionally, the variation of the patents’ topic distribution over time is not consistent with that of papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Since 1990, patent topics have been dominated by input techniques,21 which first expand from 1990 to 1993, then slightly shrink from 1993 to 2010 and expand again since 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In 2018, about 40% of patents that cite HCI research papers are input techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We also observed this growth in patent-cited HCI papers, but not this significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='4 RQ4: Who is involved in the process of recognizing HCI research on patents?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Last, we investigate through the four premier HCI venues which institutions are most likely to develop patents that recognize HCI research, and which institutions conduct HCI research that are most cited by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Such analysis is important because it identifies the role of different stakeholders within the technology translation landscape [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Apple, Microsoft, IBM, but no longer Xerox: top institutes citing HCI research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We examined who are the top patent as- signees (the entity that has the property right to the patent, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' firm) that cite HCI research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The top patent assignees have been dominated by companies: Apple, Microsoft, and International Busi- ness Machines Corporation (IBM) are the top three companies that were granted the highest number of HCI-citing patents in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Other rise and fall over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' See appendix C for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' PARC, CMU, MIT: top institutes that publish patent-cited research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We assessed the institutes that published the most patent- cited HCI papers across the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' As Figure 8 shows, contrary to the fact that top patent assignees have been dominated by indus- tries, top institutes that published patent-cited HCI papers have been a combination of universities and companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Top universi- ties include Carnegie Mellon University, Massachusetts Institute of Technology, University of California, and University of Washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Top companies that published patent-cited HCI papers include Xe- rox Palo Alto Research Center and Microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The ratio of patents cited among all HCI papers significantly dropped from nearly half before 2005 to less than 30% for most institutes after 2005, due to 21We exclude analysis of topic - ‘patent terms’ as the topic is generic language use in patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' A Large-scale Analysis of Patent Citations to HCI Research Conference’17, July 2017, Washington, DC, USA Figure 6: Topics were identified through a Latent Dirichlet Allocation (LDA) analysis of the combined paper-patent corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Figure 7: The first row shows the breakdowns of papers across 7 topics in HCI over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The second row depicts the per- centage of each topic in terms of paper number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Three columns depict "topic distribution of patent-cited HCI papers", "topic distribution of non-patent cited HCI papers" and "topic distribution of patents" respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' System Interaction dominates the patent-cited HCI papers while Theory dominates the non-patent cited HCI papers and Input Techniques dominate patents over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' the fact that the total number of HCI papers grew significantly and the right censoring issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Overall, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='5% of Microsoft’s papers, 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='0% of IBM’s, and 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1% of Xerox’s were cited by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In comparison, universities have a lower rate of papers cited by patents, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2%, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='3%, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='9% of papers were recognized among Carnegie Mellon University, the University of California system, and MIT respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This indi- cates that among institutes publishing the most HCI papers, the Topic O: Patent Terms Topic 1: Modalities Topic 2: System Interaction datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='support ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='video ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='environment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='tool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='use ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='object ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='associate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content="'displaybase " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='visualcharacter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='design ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='device ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='audio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='voicetext ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='application ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='provide ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='information receive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='base ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='virtual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='include ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='interactive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='document ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='speech ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Topic 6:Input Techniques ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='interface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='display ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='include ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='content method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='present ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='interaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='position ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='word ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='surface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='touch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='display ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='gesture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='device ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Topic 3: Evaluations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Topic 4: Theory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Topic 5: Social & Experience Design ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='first image object ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='user performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='work ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='study online ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='use ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='result ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='study ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='technology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='child ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='behavior method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='community ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='taskactivity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='support ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='social ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='people ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='systemdatum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='practice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='support ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' technology experience use hci challenge gameparticipant model analysis process play research playerPatent terms Modalities System Interaction Evaluations Theory Social&experience design Input techniquesPatent terms Modalities System Interaction Evaluations Theory Social&experience design Input techniquesPatent terms Modalities System Interaction Evaluations Theory Social&experience design Input techniquesConference’17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' July 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' USA Hancheng Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Figure 8: Institutions publishing the most patent-cited research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' papers from the industry have a higher proportion of papers rec- ognized by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' However, the difference between industry and universities becomes smaller when removing self-citing patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Self-citation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We also explored the degree of self-citation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We find that 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='9% of patents self-cite the inventor’s own research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Although the number of self-citing patents is growing, the percent- age of self-citations in all HCI patent citations is decreasing from around 20% to 5% in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This suggests that while the HCI field is expanding, the number of researchers directly referring to their own research in patents is not growing at the same rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Most of the self-citations also come from industry, with Microsoft and Xe- rox constituting 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='8% and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2% of total self-citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Self-citation from academia is much less common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Summary of conclusions: Through our analysis, we find that HCI research has had a significant impact on patents, with an in- creasing number of patents recognizing research in CHI, CSCW, UIST, and UbiComp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Patents are more likely to refer to systems- oriented and highly-cited research in academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' However, the time lag between patent and paper is long (>10 years) and getting longer, suggesting HCI research and practice may be inefficiently con- nected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We further verify the robustness of our main findings through two additional analyses, which we report in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 5 DISCUSSION In this section, we discuss the implications of our findings: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1 The patent-research relevance landscape in HCI By combining the findings from our large-scale analyses with that of prior qualitative evidence established by literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' case studies [15], personal experience, [22] and interviews [19]), we can now offer a more comprehensive picture of the HCI translation landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The impact of HCI research on patents: Our work largely corroborates literature arguing for the considerable impact of HCI research on practice [12, 32, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In our analysis, among HCI re- search papers in CHI, CSCW, UIST, and UbiComp, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1% of all papers have been referenced by patents, and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='4% for SIGCHI sponsored venues overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This is a rate far higher than science in general (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='5% [51]) and prominent journals across multiple scientific fields (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='7% [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The rate is also higher than bio-medicine, a field that has a more systematic technology translation system and a richer tradition of studying technology translation, whose proportion is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='7% [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='22 HCI research diffuses into the industry at a similar rate as Computer Vision (25%) and at a higher rate than NLP (11%), both areas of substantial industry funding and interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Note our estimate is a lower bound: given the long time lag of patent-paper citations, recent papers may have not fully expressed their impact yet (right censoring).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' When only considering earlier years that do not suffer much from right censoring issues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' prior to 2005), we see roughly 30%-50% of papers published in those years have been cited by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' For UIST, the proportion is even higher, close to 80% for many years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 22Bio-medicine papers from US institutes only—a filter we did not apply for our study of HCI—have a proportion of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='3% [50], which is roughly the same as HCI research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Patent cited Non patent citedPatent cited Non patent citedPatent cited Non patent citedPatent cited Non patent citedA Large-scale Analysis of Patent Citations to HCI Research Conference’17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' July 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' USA Issues with the current HCI translation into patents: As argued by Bill Buxton in ‘the long nose of innovation’ [12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' the bulk of innovation takes place over a long period: the mouse was first built in 1965 by William English and Doug Engelbart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' but was only popularized in the 1990s when Microsoft released a large-scale commercial mouse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' multitouch was published in 1985, but took 22 years to become a product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Our analysis further demonstrates that even the initial step of having research recognized in a patent, which may be well before there is an actual product, takes considerable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In fact, the ubiquity of long time delay between research and practice, and thus lack of immediate impact on the industry after the publication of a research paper, could be one underlying reason why many papers on HCI translation argue that HCI lacks practical impact [18, 22, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Furthermore, our analysis demonstrates that the time lag between patent and research is getting longer over time, indicating that the translation process in HCI may become more inefficient over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This result is in line with a general trend across science (average over time: 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='4 years), where they report an average patent citation to science time lag of about 8 years in the 1990s, rising to about 15 years in 2018 [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The specific reason for the (increasing) time lag would need further work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We also show that the HCI community often leaves an idea behind by the time industry gets interested, as a paper’s peak citation lag is generally shorter than the paper’s first patent citation lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The result indicates that with a long time lag, HCI research has moved on and is exploring new emerging technologies that are not yet reliable enough, cheap enough, power-efficient enough, or accurate enough for the industry yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The observation supports the observation that HCI research often plays “the time machine game”,23 where it fast forwards into the future by acquiring early versions of emerging technology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', VR, AR, multi-touch, AI) and exploring the interactive applications of that technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Unless HCI is directly working on reducing those barriers to industry entry for that technology, HCI research cannot directly accelerate the time lag: it is simply painting a compelling vision of the future before that future arrives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2 How could the HCI community do better to facilitate technology transfer and industrial impact?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Encourage communications and collaborations across academia and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Through our analysis, we have found that even though research articles from both academia and industry are rec- ognized by patents, the proportion of papers in academia recognized by patents is much lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' While the result could be that industry research papers by themselves are more applied than research pa- pers from academia, or that industry has more internal incentives to have their research patented24, this could also be a sign that prac- titioners are not fully aware of some application-oriented advances in academia, and that information diffusion between academia and practice is inefficient [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 23A term attributed to Jeff Pierce, formerly a research manager at IBM Research and faculty member at Georgia Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 24Microsoft Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' would award decorative “patent cubes” to re- searchers for each new patent they co-authored,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' which researchers would often stack into decorative pyramids and display in their offices Our work thus echoes calls for a more inclusive and translation- friendly environment [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 19]: that both academia and in- dustry should 1) better recognize the importance of technology translation rather than considering translation irrelevant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 2) estab- lish more communication and collaboration channels to engage people,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' SIGGRAPH-style Emerging Tech festivals where aca- demic researchers show their published HCI work to an applied audience and encourage researchers in serving as advising role in the industry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' and 3) involve more HCI materials in Computer Science curriculum at universities to get ‘future practitioners’ more familiar with HCI research ideas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' and thus prepare them as trans- lational developers who are more likely to bridge academia and industry [60] Encourage self-driven technology transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Self-driven tech- nology transfer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' patents recognizing one’s own paper) gen- erally happens much faster than technology transfer in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Intuitively, the self-driven transfer would not encounter many of the same communication and information diffusion barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Self- driven technology transfer could also potentially solve many of the ‘recognition’ issues in the translational process as discussed in prior works [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' However, as shown in our analysis, though the amount of self-driven technology transfer in HCI is going up over time, it is not on par with the rate of increase for research articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' While not all researchers should actively engage in technology transfer, there could be more steps to be taken to encourage self-driven tech- nology transfer from the academic side so that translation could happen more efficiently, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' through better supporting and recog- nizing attempts to self-translate one’s own research by providing legal apparatuses and funding support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Meanwhile, we want to emphasize while there are benefits of self-driven transfer, it may currently not distribute opportunities equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' For instance, in the life sciences [24], women faculty members patent at about 40% of the rate of men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' It would be important to identify and mitigate these potential issues so as to ensure an inclusive technology transfer environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Relatedly, as suggested by prior work [19], there exist multiple translational gaps in HCI, and basic researchers should also be encouraged to engage more with applied researchers and do more system work, which would eventually help translate HCI research insights into industry impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Recognizing translational work in HCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' More broadly, our work echoes prior work on the need of recognizing translational efforts in HCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' For instance, when allocating funding or considering researcher promotion, their impacts in the industry could be taken into consideration as a separate metric aside from impacts within academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Our work points to a potential way to quantify one important pathway towards HCI research’s impacts on the industry, through analyzing patent-to-science citation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Impact signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Prior approaches to quantifying research im- pact mostly focus on impact within academia through bibliomet- ric analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' However, no quantitative metric fully captures the complexities of our world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Could the h-index be fruitfully comple- mented with other information?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' (a “patent relevance” p-index?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=') While our analysis show impacts in academia and impacts in patents correlate, we also find papers with high patent citations do not nec- essarily have high paper citations: in one extreme case, the most Conference’17, July 2017, Washington, DC, USA Hancheng Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' patent-cited paper in our dataset, “A multi-touch three-dimensional touch-sensitive tablet” [44], is more popular in the patent world than in academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' If evaluations primarily consider the academic impacts of such research work, the work’s value may have been underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' As one potential pathway to industry impacts that are relatively easy to scale, patents provide a potential signal to more holistically evaluate research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Of course, patent relevance, or practice relevance in general25, is not the solitary metric of scientific value, and research and re- searchers should not be judged based on a single metric, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' to receive funding or get a promotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Thus, our work should not be interpreted as stating that non-patent cited research represents any sort of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' There are many, many examples of influential HCI research that is not patented (or even patentable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' For instance, our work shows that system building or application-oriented HCI research is more likely to find relevance in patents rather than design-oriented or behavioral research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The result is not an indica- tion that applied-oriented research is more valuable: there could be the indirect influence of other types of works on application- oriented research, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' applied research getting inspiration from behavior work, as suggested by the translational science model in HCI [19] – which we seek to address in future work, and 2) it is equally important to maintain a diversity of research ideas, which has proven to facilitate greater innovation for science in general [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' If the measurement of this impact is desirable, we will require new methods, such as multi-hop influence over citation network [1], linguistic concept diffusion [14], from the paper to the public or media [77, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='3 Limitations and Future Work Patent citations to research are only a proxy signal of industry impact, which is a hard-to-quantify concept otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' It is only one, among many (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' open source software, design patterns), potential pathway to industry impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' First, not all patents will turn into products or practices, so they may not be actual “industry impact” instances (false positives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' There could be many other factors, such as assignee strategy and resources, that could influence the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Even if a patent does end up as a product, most of the time the patent will not be valuable or impactful, with 97% of all patents never recouping the cost of filing them26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' However, the fact that inventors decide to go through the long and expensive process of filing a patent to protect their intellectual property does indicate they are considering their invention having at least some potential to be of relevance to the practice domain, which could be regarded as an intended act aiming at industry impact or technology transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Second, industry impact could happen even if there is no patent- ing process involved (false negative), which is not uncommon in software [29]: startups will launch products without patents from time to time, which is quite different from the innovation landscape of more traditional fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' design processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', usability testing, heuristic evaluation), design patterns, and open source software (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', d3, Vega Lite) also have significant industry impact that is not reflected though patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' As such, our analysis of using patent 25Though arguably it’s much harder to quantify other forms of practice relevance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' how research influence design patterns and open source software 26https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='forbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='com/sites/stephenkey/2017/11/13/in-todays-market-do- patents-even-matter/ citation to HCI research papers could be different from the actual translation landscape: the patent dataset could introduce both false positives and negatives, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=', even if a patent cites a HCI paper, it may never be taken up in practice as product, and an actual product that gets influenced by HCI research that is unpatentable will not be observed and measured through our current approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Despite all the shortcomings of patent citation to science, the availability and scale of the dataset make it a rare lens in the in- novation literature to enable conclusions on the research-practice relationship at scale [50–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In our work, in addition to building on these methods from the innovation literature, we tied our analysis to qualitative evidence discussed in prior works so as to validate our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' In future work, we plan to 1) involve more qualitative evidence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' interviewing inventors’ motivation behind citing HCI research) to further validate our findings, and 2) take more steps to quantify how HCI research turns into valuable inventions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' by using patent citations to other patents as a proxy of patent value, which correlates well with other metrics of patent value, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' whether they are renewed to a full term, and whether they get licensed [31, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Our work also currently mostly focuses on measuring industry relevance at the paper level, which may not necessarily be the principal unit of knowledge: for example, several papers on the same idea can get cited by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' While we have made preliminary attempts to analyze the topics prevalent to patents,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' patent-cited research papers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' and non-patent cited research papers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' future work could better study at the concept level what specific research ideas are transferred into research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' either through keywords provided by the author (which is unfortunately not available in our current dataset),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' or natural language processing based approach such as phrase mining [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' which may help track transfer of innovations at a more fine-grained level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Other limitations include: (1) our dataset is focused on United States patents, which limits our cultural context and generalizability, though arguably a significant proportion of inventors/organizations using (and pushing) HCI research in practice are US-based [67];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' (2) while discussing in a descriptive way in our paper with findings on the role of academic impacts (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1), topic (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='3), and institute/actors (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='4) in relating to patent impact, we do not have causal evidence/analysis on the causal mechanisms what cause some papers to have more industry relevance, which is an important topic we seek to address in future work, and (3) if there are recent trends in the last 5-10 years that have changed these patterns, it is still too recent to see their impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 6 CONCLUSIONS In this work, drawing inspiration from the innovation literature, we quantitatively study one important pathway from HCI research to industry impact by conducting a large-scale analysis of how patent documents from USPTO refer to research articles in CHI, CSCW, UIST, UbiComp and other SIGCHI sponsored venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We contribute to the literature by measuring to what extent HCI research has been featured in patent citations, with a high proportion of papers referenced in patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Patents are more likely to refer to systems- oriented and highly-cited research in HCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' However, we also reveal potential translation issues: HCI research and practice may not be A Large-scale Analysis of Patent Citations to HCI Research Conference’17, July 2017, Washington, DC, USA efficiently coupled, since the time lag between paper and patent is long and getting longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Our work not only demonstrates the potential of using patent citation data to science as a powerful tool to study the industry impact of HCI research, but also points to suggestions for the HCI community to better facilitate translation from research to practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank Yian Yin for helpful suggestions on polishing the work, and Mary Czerwinski, Bongshin Lee, Lucy Lu Wang, James Zou, Shumin Zhai and many others for insightful discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Hancheng Cao 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The meaning of translational research and why it matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Jama 299, 2 (2008), 211–213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [77] Yian Yin, Yuxiao Dong, Kuansan Wang, Dashun Wang, and Benjamin F Jones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Public use and public funding of science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Nature human behaviour (2022), 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [78] Yian Yin, Jian Gao, Benjamin F Jones, and Dashun Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Coevolution of policy and science during the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Science 371, 6525 (2021), 128–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' [79] Elias Zerhouni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The NIH roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' A Large-scale Analysis of Patent Citations to HCI Research Conference’17, July 2017, Washington, DC, USA A DETAILS OF DATA ACQUISITION Here we provide details of the data acquisition procedure that generate our final analyzed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Patent citation to science that connects USPTO to Microsoft Academic Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' To capture the information required by patent citation to science, we utilize a public dataset available over Zen- odo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='27 We leverage the patent-to-article citations of Version v37 (Jul 19, 2022), including _pcs_mag_doi_pmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='tsv and papercitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='tsv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' For _pcs_mag_doi_pmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='tsv, we mainly focus on the fields reftype, diff_month, selfciteconf_avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We focus on fields citingpaperid and citedpaperid in papercitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='tsv, which we used to join with Microsoft Academic Graph Metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Microsoft Academic Graph Metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Microsoft Academic Graph Metadata is also available over Zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='28 The data files we utilize include authoridname_normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='tsv, conferenceidname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='tsv, paperauthoridaffiliationname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='tsv, paperauthororder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='tsv, paperconfer- enceid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='tsv and paperyear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='tsv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' USPTO Metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We acquire USPTO metadata from PatentsView.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='29 We utilize datafiles assignee, inventor, patent, patent_assignee, and patent_inventor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Semantic Scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We request Semantic Scholar API30 with re- search article IDs retrieved from Microsoft Academic Graph Meta- data for extra paper information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The fields we queried include title, abstract, venue, year, referenceCount, citationCount, authors, as well as name, affiliations, paperCount, and citationCount associated with each author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' we retrieved all the above data in Aug 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' B SIGCHI SPONSORED VENUES The 20 SIGCHI venues that we include in our analysis are: Hu- man Factors in Computing Systems (CHI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' User Interface Software and Technology (UIST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Ubiquitous Computing (UbiComp),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Con- ference on Computer Supported Cooperative Work (CSCW),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Con- ference on Tangible and Embedded Interaction (TEI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Symposium on Eye Tracking Research & Application (ETRA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' International Conference on Supporting Group Work (GROUP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Conference on Intelligent User Interfaces (IUI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Creativity and Cognition (C&C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Interaction Design and Children (IDC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' International Conference on User Modeling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Adaptation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' and Personalization (UMAP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Sym- posium on Engineering Interactive Computing System (EICS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Con- ference on Automotive User Interfaces and Interactive Vehicular Applications (AutomotiveUI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Conference on Human-Robot Interac- tion (HRI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' International Conference on Computational Collective Intelligence (CI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Conference on Recommender Systems (RecSys),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Annual Symposium on Computer-Human Interaction in Play (CHI PLAY),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' International Conference on Multimodal Interaction (ICMI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Symposium on Spatial User Interaction (SUI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Symposium on Vir- tual Reality Software and Technology (VRST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 27http://relianceonscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='org 28http://relianceonscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='org 29https://patentsview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='org/download/data-download-tables 30https://api.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='semanticscholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='org/api-docs/graph#tag/Paper-Data/operation/get_ graph_get_paper_references In total, there are 57,385 papers where 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='4% of them (7678 pa- pers) have been cited by patents in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' C TOP PATENT ASSIGNEES OVER TIME We show top patent assignees over time in Fig 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' D ADDITIONAL ANALYSIS ON NON-SELF-CITING PATENTS AND NON-RESEARCHER PATENTS We provide two additional analyses using a subset of four pre- mier venues to further verify the robustness of our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' To rule out the possibility that the impacts of HCI research on patents is a result of self-cite, or driven primarily by HCI researchers – thus one may argue the impact of HCI research in industry is actually limited – we run the same analysis using 1) patents that do not include self-cite to one’s own research papers (“non-self-citing patents”)), which is 26, 382 (86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='04% of original patents), and 2) patents that are invented by people who have never published any CHI, CSCW, UIST or UbiComp research papers ( (“non-researcher” patents), which we operationalized through excluding patents where inven- tor last name have appeared in author lists of papers from the four venues we focused on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='31 This results in 5, 251 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='12% of original patents) of “non-researcher” patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' We find consistent patterns in our main analysis where a high proportion of HCI research papers are cited by patents, and there is a long time lag between patent and paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' More specific results are as follows: Proportion of papers that get cited by patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The propor- tion of papers cited by non-self-citing patents is plotted in Figure 10 and the ratio rises and persists since 1990 at over 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' At UIST in particular, the patent citation ratio reaches 60% - 80% from 1990 - 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This suggests that non-self-citing patents, similar to our main result, recognize a considerable number of HCI research papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Identical trends can be observed for non-researcher patents, as shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Increasing citations to HCI research in patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Figure 11 shows the number of non-self-citing patents that cite HCI research over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' It can be observed that non-self-citing patents first in- crease in 2000 and then peak around 2014, ranging from 200 to 1000 across different venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' This agrees with the overall trend reported in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Identical trends can be observed for non-researcher patents, as shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Time lag between patent and paper is long and getting longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The temporal trend of the measured time lag between the issue date of non-self-citing patents and the publication date of HCI papers they cited are plotted in Figure 15a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Similar to the trend reported in the main results (Figure 5), the median time lag increased from 1989 to 2014 for all the venues from about around 5 years to around 10−15 years while since 2014, this trend bifurcates among different venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The time lag between the patent and its most recent cited paper (Figure 15b ) is also examined, showing identical trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Identical trends can be observed for non-researcher patents, as shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' 31This set of patents is a smaller set than actual “non-researcher” patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' The primary objective is to ensure a set of patents with inventors who, for sure, have never published papers in the four academic venues we studied without tedious author disambiguation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Conference’17, July 2017, Washington, DC, USA Hancheng Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Figure 9: Top patent assignees that cite HCI research over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' A Large-scale Analysis of Patent Citations to HCI Research Conference’17, July 2017, Washington, DC, USA Figure 10: (Non-self-cite) Left: the number of papers published by each conference per year (red) and the number of papers published in that year that were later cited by at least one patent (blue), at ACM CHI, CSCW, UbiComp, and UIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='CHI ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1985 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='UIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='UIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Percent of published papers later cited by patents ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Published ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Published and later cited by patents ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Percent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='25 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Percent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1985 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1980 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1990 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1995 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2005 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2015 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1985 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1995 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2005 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2015 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1990 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='YearConference’17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' July 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' USA Hancheng Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Figure 11: (Non-self-cite) The number of patents that cite HCI papers over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Figure 12: (Non-self-cite) The time lag between patent and paper is long and getting longer for different types of citations and venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Patents Citing HCl Research ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='CHI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='Number of patents ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='CSCW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='UIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='UbiComp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1990 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1995 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2005 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2015 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='YearThe time lag of the most recent paper ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='cited by patents in year X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} 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+page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1990 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='1995 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2005 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='2010 ' 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number of papers published in that year that were later cited by at least one patent (blue),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' at ACM CHI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' CSCW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' UbiComp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' and UIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content='CHI ' metadata={'source': 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14: (Non-researcher) The number of patents that cite HCI papers over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Figure 15: (Non-researcher) The time lag between patent and paper is long and getting longer for different types of citations and venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} +page_content=' Patents Citing HCl Research 400 CHI Number of patents CSCW UIST 300 UbiComp 200 100 0 1990 1995 2000 2005 2010 2015 YearThe time lag of the most recent paper cited by patents in year X 20 CHI (Year) CSCW 15 UIST Time difference ( UbiComp 10 5 0 1990 1995 2000 2005 2010 2015 YearThe median time lag of the paper cited by patents in year X 20 CHI (Year) CSCW 15 UIST Time difference ( UbiComp 10 5 0 1990 1995 2000 2005 2010 2015 Year' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFQT4oBgHgl3EQf4Ta7/content/2301.13431v1.pdf'} diff --git a/5NE3T4oBgHgl3EQfQgli/content/tmp_files/2301.04413v1.pdf.txt b/5NE3T4oBgHgl3EQfQgli/content/tmp_files/2301.04413v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c49d1495df8540eff0d70e3b0b3907c8499c4c28 --- /dev/null +++ b/5NE3T4oBgHgl3EQfQgli/content/tmp_files/2301.04413v1.pdf.txt @@ -0,0 +1,1246 @@ +arXiv:2301.04413v1 [cs.IR] 11 Jan 2023 +CoSPLADE: Contextualizing SPLADE for +Conversational Information Retrieval +Nam Le Hai1[0000−0002−9020−8790], Thomas Gerald2, Thibault Formal1,3, +Jian-Yun Nie4, Benjamin Piwowarski1[0000−0001−6792−3262], and Laure +Soulier1,2[0000−0001−9827−7400] +1 Sorbonne Université, CNRS, ISIR, F-75005 Paris, France +first.last @sorbonne-universite.fr +2 Université Paris-Saclay, CNRS, LISN, 91405 Orsay France first.last @lisn.fr +3 Naver Labs Europe, Meylan, France first.last @naverlabs.com +4 University of Montreal, Montreal, Canada nie@iro.umontreal.ca +Abstract. Conversational search is a difficult task as it aims at retriev- +ing documents based not only on the current user query but also on the +full conversation history. Most of the previous methods have focused on +a multi-stage ranking approach relying on query reformulation, a criti- +cal intermediate step that might lead to a sub-optimal retrieval. Other +approaches have tried to use a fully neural IR first-stage, but are ei- +ther zero-shot or rely on full learning-to-rank based on a dataset with +pseudo-labels. In this work, leveraging the CANARD dataset, we propose +an innovative lightweight learning technique to train a first-stage ranker +based on SPLADE. By relying on SPLADE sparse representations, we +show that, when combined with a second-stage ranker based on T5Mono, +the results are competitive on the TREC CAsT 2020 and 2021 tracks. +Keywords: information retrieval · conversational search · first-stage +ranking. +1 +Introduction +With the introduction of conversational assistants like Siri, Alexa or Cortana, +conversational Information Retrieval, a variant of adhoc IR, has emerged as +an important research domain [4,6]. In conversational IR, a search is conducted +within a session, and the user’s information need is expressed through a sequence +of queries, similarly to natural conversations – thus introducing complex inter- +dependencies between queries and responses. +Not surprisingly, neural IR models have been shown to perform the best on +conversational IR [5,7]. Most prior works rely on a Historical Query Expansion +step [34], i.e. a query expansion mechanism that takes into account all past +queries and their associated answers. Such query expansion model is learned on +the CANARD dataset [8], which is composed of a series of questions and their +associated answers, together with a disambiguated query, referred to as gold +query in this paper. However, relying on a reformulation step is computationally + +2 +Le Hai et al. +costly and might be sub-optimal as underlined in [13,16]. Krasakis et al. [13] +proposed to use ColBERT [12] in a zero-shot manner, replacing the query by the +sequence of queries, without any training of the model. Lin et al. [16] proposed +to learn a dense contextualized representation of the query history, optimizing +a learning-to-rank loss over a dataset composed of weak labels. This makes the +training process complex (labels are not reliable) and long. +In this work, we follow this direction of research but propose a much lighter +training process for the first-stage ranker, where we focus on queries and do not +make use of any passage – and thus of a learning-to-rank training. It moreover +sidesteps the problem of having to derive weak labels from the CANARD dataset. +Given this strong supervision, we can consider more context – i.e. we use the +answers provided by the system the user is interacting with, which allows to +better contextualize the query, as shown in our experiments. The training loss we +propose leverages the sparse representation of queries and documents provided +by the SPLADE model [9]. In a nutshell, we require that the representation of +the query matches that of the disambiguated query (i.e. the gold query). Our +first-stage ranker achieves high performances, especially on recall – the most +important measure in a multi-stage approach, comparable to the best systems +in TREC CAsT [7], but also on precision-oriented measures – which shows the +potential of our methodology. +Finally, to perform well, the second-stage ranker (i.e. re-ranker) needs to +consider the conversation as well, which might require a set of heuristics to select +some content and/or query reformulation such as those used in [18]. Leveraging +the fact that our first-stage ranker outputs weights over the (BERT) vocabulary, +we propose a simple mechanism that provides a conversational context to the +re-ranker in the form of keywords selected by SPLADE. +In summary, our contributions are the following: +1. We propose a new loss to optimize a first-stage ranker resulting in a lightweight +training strategy and state-of-the-art results in terms of recall; +2. We show that, when combined with a second-stage ranker based on a context +derived from the SPLADE query representation of the first stage, we obtain +results on par with the best approaches in TREC CAsT 2020 and 2021. +2 +Related Works +The first edition [5] of the TREC Conversational Assistance Track (CAsT) was +implemented in 2019, providing a new challenge on Conversational Search. The +principle is the following: a user queries the system with questions in natural +language, and each time gets a response from the system. The challenge differs +from classical search systems as involving previous utterances (either queries +or answers) is key to better comprehending the user intent. In conversational +IR, and in TREC CAsT [6,5,7] in particular, the sheer size of the document +collection implies to design an efficient (and effective) search system. +Conversational IR is closely related to conversational Question-Answering +[25,27,26] in the sense that they both include interaction turns in natural lan- +guage. However, the objective is intrinsically different. While the topic or the + +CoSPLADE: Contextualizing SPLADE for Conversational IR +3 +context (i.e., the passage containing answers) is known in conversational QA, +conversational IR aims to search among a huge collection of documents with po- +tentially more exploratory topics. With this in mind, in the following, we focus +on the literature review of conversational IR. +We can distinguish two lines of work in conversational search. The first one +[29,30,32,3] focuses on a Contextual Query Reformulation (CQR) to produce +a (plain or bag-of-words) query, representing ideally the information need free +of context, which is fed into a search model. One strategy of CQR consists in +selecting keywords from previous utterances by relying on a graph weighted by +either word2vec similarity [29], term-based importance using BM25 [19], or clas- +sification models [30]. Other approaches [14,19,18,33,28] leverage the potential +of generative language models (e.g., GPT2 or T5) to rewrite the query. Such +approaches are particularly effective, reaching top performances in the TREC +CAsT 2020 edition [5]. Query reformulation models also differ in the selected +evidence sources. Models either focus on the early stage of the conversation [1], +on a set of the queries filtered either heuristically [2] or by a classification model +[21], or on both previous queries and documents [31]. Finally, to avoid the prob- +lem of generating a single query, [14,20] have proposed to use different generated +queries and aggregate the returned documents. +The reformulation step is however a bottleneck since there is no guarantee +that the “gold query” is optimal and thus generalizes well [16,13]. Moreover, +generating text is time-consuming. To avoid these problems, the second line of +work aims to directly integrate the conversation history into the retrieval model, +bypassing the query reformulation step. As far as we know, only a few studies +followed this path in conversational search. Qu et al [24] compute a query rep- +resentation using the k last queries in the dialogue [15]. Similarly Lin et al. [16] +average contextualized tokens embeddings over the whole query history. The +representation is learned by optimizing a learning-to-rank loss over a collection +with weak labels, which requires much care to ensure good generalization. Fi- +nally, Krasakis et al. [13] use a more lexical neural model, i.e. ColBERT [12], +to encode the query with its context – but they do not finetune it at all. In +this work, we go further by using a sparse model SPLADE [9], using a novel +loss tailored to such sparse representations, and by using a lightweight train- +ing procedure that does not rely on passages, but only on a dataset containing +reformulated queries. +3 +Model +In TREC CAsT [5,7], each retrieval session contains around 10 turns of ex- +change. Each turn corresponds to a query and its associated canonical answer5 +is provided as context for future queries. Let us now introduce some notations +that we use to describe our model. For each turn n ≤ N, where N is the last +turn of the conversation, we denote by qn and an respectively the corresponding +query and its response. Finally, the context of a query qn at turn n corresponds +5 Selected by the organizer as the most relevant answer of a baseline system. + +4 +Le Hai et al. +to all the previous queries and answers, i.e. q1, a1, q2, a2, ..., qn−1, an−1. The +main objective of the TREC CAsT challenges is to retrieve, for each query qn +and its context, the relevant passages. +In the next sections, we present our first-stage ranker and second-stage re- +ranker, along with their training procedure, both based, directly or indirectly, +on the SPLADE (v2) model described in [9]. SPLADE has shown results on +par with dense approaches on in-domain collections while exhibiting stronger +abilities to generalize in a zero-shot setting [9]. It outputs a sparse representation +of a document or a query in the BERT vocabulary, which is key to our model +during training and inference. The SPLADE model we use includes a contextual +encoding function, followed by some aggregation steps: ReLU, log saturation, +and max pooling over each token in the text. The output of SPLADE is a sparse +vector with only positive or zero components in the BERT vocabulary space R|V |. +In this work, we use several sets of parameters for the same SPLADE architecture +and distinguish each version by its parameters θ, and the corresponding model +by SPLADE(. . . ; θ). +3.1 +First stage +The original SPLADE model [9] scores a document using the dot product be- +tween the sparse representation of a document ( ˆd) and of a query (ˆq): +s(ˆq, ˆd) = ˆq · ˆd +(1) +In our work, like in [16], we suppose that the document representation has +been sufficiently well-tuned on the standard ad-hoc IR task. The document +embedding ˆd is thus obtained using the pre-trained SPLADE model, i.e. ˆd = +SPLADE([CLS] d; θSP LADE) where θSP LADE are the original SPLADE param- +eters obtained from HuggingFace6. These parameters are not fine-tuned during +the training process. We can thus use standard indices built from the original +SPLADE document representations to retrieve efficiently the top-k documents. +In the following, we present how to contextualize the query representation using +the conversation history. Then, we detail the training loss aiming at reducing +the gap between the representation of the gold query and the contextualized +representation. +Query representation. Like state-of-the-art approaches for first-stage conversa- +tional ranking [16,13], we contextualize the query with the previous ones. Going +further, we propose to include the answers in the query representation process, +which is easier to do thanks to our lightweight training. +To leverage both contexts, we use a simple model where the contextual query +representation at turn n, denoted by ˆqn,k, is the combination of two representa- +tions, ˆqqueries +n +which encodes the current query in the context of all the previous +6 The weights can be found at https://huggingface.co/naver/splade-cocondenser-ensembledistil + +CoSPLADE: Contextualizing SPLADE for Conversational IR +5 +queries, and ˆqanswers +n,k +which encodes the current query in the context of k the +past answers7. Formally, the contextualized query representation ˆqn,k is: +ˆqn,k = ˆqqueries +n ++ ˆqanswers +n,k +(2) +where we use two versions of SPLADE parameterized by θqueries for the full +query history and θanswers,k for the answers. These parameters are learned by +optimizing the loss defined in Eq. (8). +Following [16], we define ˆqqueries +n +to be the query representation produced by +encoding the concatenation of the current query and all the previous ones: +ˆqqueries +n += SPLADE([CLS] qn [SEP] q1 [SEP] . . . [SEP] qn−1; θqueries) +(3) +using a set of specific parameters θqueries. +To take into account the answers that the user had access to, we need to +include them in the representation. Following prior work [2], we can consider a +various number of answers k, and in particular, we can either choose k = 1 (the +last answer) or k = n−1 (all the previous answers). Formally, the representation +ˆqanswers +n,k +is computed as: +ˆqanswers +n,k += 1 +k +n−1 +� +i=n−k +SPLADE(qn [SEP] ai; θanswers,k) +(4) +Training Based on the above, training aims at obtaining a good representation +ˆqn for the last issued query qn, i.e. to contextualize qn using the previous queries +and answers. To do so, we can leverage the gold query q∗ +n, that is, a (hopefully) +contextualized and unambiguous query. We can compute the representation ˆq∗ +n +of this query by using the original SPLADE model, i.e. +ˆq∗ +n = SPLADE(q∗ +n; θSP LADE) +(5) +For example, for a query "How old is he?" the matching gold query could be +"How old is Obama?". The representation of the latter given by SPLADE would +be as follows: +[(”Obama”, 1.5), (”Barack”, 1.2), (”age”, 1.2), (”old”, 1.0), (”president”, 0.8), ...] +where the terms “Obama” and “Barack” clearly appear alongside other words +related to the current query (“old” and the semantically related “age”). +We can now define the goal of the training, which is to reduce the difference +between the gold query representation ˆq∗ +n and the representation ˆqn,k computed +by our model. An obvious choice of a loss function is to match the predicted +and gold representations using cosine loss (since the ranking is invariant when +scaling the query). However, as shown in the result section, we experimentally +7 In the experiments, we also explore an alternative model where answers and queries +are considered at once. + +6 +Le Hai et al. +found better results with a modified MSE loss, whose first component is the +standard MSE loss: +LossMSE(ˆqn,k, ˆq∗ +n) = MSE(ˆqn,k, ˆq∗ +n) +(6) +In our experiments, we observed that models trained with the direct MSE do +not capture well words from the context, especially for words from the answers. +The reason is that the manually reformulated gold query usually only contains a +few additional words from the previous turns that are directly implied by the last +query. Other potentially useful words from the answers may not be included. This +is a conservative expansion strategy which may not be the best example to follow +by an automatic query rewriting process. We thus added an asymmetric MSE, +designed to encourage term expansion from past answers, but avoid introducing +noise by restricting the terms to those present in the gold query q∗ +n. Formally, +our asymmetric loss is: +Lossasym(ˆqanswers +n,k +, ˆq∗ +n) = +� +max(ˆq∗ +n − ˆqanswers +n,k +, 0) +�2 +(7) +where the maximum is component-wise. This loss thus pushes the answer-biased +representation ˆqanswers +n,k +to include tokens from the gold answer. Contrarily to +MSE, it does not impose (directly) an upper bound on the components of the +ˆqanswers +n,k +representation – this is done indirectly through the final loss function +described below. +The final loss we optimize is a simple linear combination of the losses defined +above, and only relies on computing two query representations: +Loss(ˆqn,k, ˆq∗ +n) = LossMSE(ˆqn,k, ˆq∗ +n) + Lossasym(ˆqanswers +n,k +, ˆq∗ +n) +(8) +There is an interplay between the two components of the global loss. More pre- +cisely, Lossasym pushes the ˆqanswers +n,k +representation to match the golden query +representation ˆq∗ +n if it can, and LossMSE pushes the queries-biased representa- +tion ˆqn,k to compensate if not. It thus puts a strong focus on extracting infor- +mation from past answers, which is shown to be beneficial in our experiments. +Implementation details. For the first-stage, we initialize both encoders (one en- +coding the queries, and the other encoding the previous answer) with pre-trained +weights from SPLADE model for adhoc retrieval. We use the ADAM optimizer +with train batch size 16, learning rate 2e-5 for the first encoder and 3e-5 for the +second. We fine-tune for only 1 epoch over the CANARD dataset. +3.2 +Reranking +We perform reranking using a T5Mono [22] approach, where we enrich the raw +query qn with keywords identified by the first-stage ranker. Our motivation is +that these words should capture the information needed to contextualize the raw +query. The enriched query q+ +n for conversational turn n is as follows: +q+ +n = qn. Context : q1 q2 . . . qn−1. Keywords : w1, w2, ..., wK +(9) + +CoSPLADE: Contextualizing SPLADE for Conversational IR +7 +where the wi are the top-K most important words that we select by leveraging +the first-stage ranker as follows. First, to reduce noise, we only consider words +that appear either in any query qi or in the associated answers ai (for i ≤ n−1). +Second, we order words by using the maximum SPLADE weight over tokens +that compose the word.8 +We denote the T5 model fine-tuned for this input as T 5+. As in the original +paper [22], the relevance score of a document d for the query qn is the proba- +bility of generating the token “true” given a prompt pt(q+ +n , d) = “Query: q+ +n . +Document: d. Relevant:”: +score(q+ +n , d; θ) = +pT 5(true|pt(q+ +n , d); θ) +pT 5(true|pt(q+ +n , d); θ) + pT 5(false|pt(q+ +n , d); θ) +(10) +where θ are the parameters of the T5Mono model. +Differently to the first stage training, we fine-tune the ranker by aligning the +scores of the documents, and not the weight of a query (which is obviously not +possible with the T5 model). Here the “gold” score of a document is computed us- +ing the original T5Mono with the gold query q∗ +n. The T5 model is initialized with +weights made public by the original authors9, denoted as θT 5. More precisely, +we finetune the pre-trained T5Mono model using the MSE-Margin loss [11]. The +loss function for the re-ranker (at conversation turn n, given documents d1 and +d2) is calculated as follows: +LR = +�� +s(q+ +n , d1; θT 5+) − s(q+ +n , d2; θT 5+) +� +− (s(q∗ +n, d1; θT 5) − s(q∗ +n, d2; θT 5)) +�2 +We optimize the θT 5+ parameters by keeping the original θT 5 to evaluate the +score of gold queries. +Implementation details. We initialize θT 5+ as θT 5, and fine-tune for 3 epochs, +with a batch size of 8 and a learning rate 1e-4. We sample pairs (d1, d2) using the +first-stage top-1000 documents: d1 is sampled among the top-3, and d2 among +the remaining 997 to push the model to focus on important differences in scores. +4 +Experimental Protocol +We designed the evaluation protocol to satisfy two main evaluation objectives: +(i) Evaluating separately the effectiveness of the first-stage and the second-stage +ranking components of our CoSPLADE model; (ii) Comparing the effectiveness +of our CoSPLADE model with TREC CAsT 2020 and 2021 participants. +8 To improve coherence, we chose to make keywords follow their order of appearance +in the context, but did not vary this experimental setting. +9 We used the Huggingface checkpoint https://huggingface.co/castorini/monot5-base-msmarco + +8 +Le Hai et al. +4.1 +Datasets +To train our model, we used the CANARD corpus10, a conversational dataset fo- +cusing on context-based query rewriting. More specifically, the CANARD dataset +is a list of conversation histories, each being composed of a series of queries, short +answers (human written) and reformulated queries (contextualised). The train- +ing, development, and test sets include respectively 31.538, 3.418, and 5.571 +contextual and reformulated queries. +To evaluate our model, we used the TREC CAsT 2020 and 2021 datasets +which include respectively 25 and 26 information needs (topics) and a document +collection composed of the MS MARCO dataset, an updated dump of Wikipedia +from the KILT benchmark, and the Washington Post V4 collection. For each +topic, a conversation is available, alternating questions and responses (manually +selected passages from the collection, aka canonical answers). For each question +(216 and 239 in total), the dataset provides its manually rewritten form as well +as a set of about 20 relevant documents. We use the former to define an upper- +bound baseline (Splade_GoldQuery). +4.2 +Metrics and baselines +We used the official evaluation metrics considered in the TREC CAsT 2020 and +2021, namely nDCG@3, MRR, Recall@X, MAP@X, nDCG@X, where the cut-off +is set to 1000 for the CAsT 2020 and 500 for the CAsT 2021. For each metric, +we calculate the mean and variance of performance across the different queries +in the dataset. With this in mind, we present below the different baselines and +scenarios used to evaluate each component of our model. +First-stage ranking scenarios. To evaluate the effectiveness of our first-stage +ranking model (Section 3.1), we compare our approach CoSPLADE, based on +the query representation of Eq. (2) with different variants (the document en- +coder is set to the original SPLADE encoder throughout our experiments): +SPLADE_rawQuery (lower bound): SPLADE [10] using only the original +and ambiguous user queries qn; SPLADE_goldQuery (kind of upper bound): +SPLADE using the manually rewritten query q∗ +n; CQE [16], a state-of-the-art +dense contextualized query representation learned using learning-to-rank on a +dataset with pseudo-labels. +To model answers when representing the query using ˆqanswers +n,k +, we used two +historical ranges (“All” with k = n−1 answers and “Last” where we use only the +last one, i.e. k = 1) and three types of answer inputs: Answer in which answers +are the canonical answers; Answer-Short in which sentences are filtered as +in the best performing TREC CAsT approach [18]. This allows for consistent +input length, at the expense of losing information; Answer-Long As answers +from CANARD are short (a few sentences extracted from Wikipedia – contrarily +to CAsT ones), we expand them to reduce the discrepancy between training and +10 https://sites.google.com/view/qanta/projects/canard + +CoSPLADE: Contextualizing SPLADE for Conversational IR +9 +inference. For each sentence, we find the Wikipedia passage it appears in (if it +exists in ORConvQA [23]), and sample a short snippet of 3 adjacent sentences +from it. +Finally, we also conducted ablation studies (on the best of the above vari- +ants) by modifying either the way to use the historical context or the training +loss: flatContext a one-encoder version of our SPLADE approach in which we +concatenate all information of the context to apply SPLADE to obtain a single +representation of the query (instead of two representations ˆqqueries +n +and ˆqanswers +n,k +as in Equations 2 and 3) trained using a MSE loss function (Eq. 6) since there is +no more two representations. MSE the version of our SPLADE approach trained +with the MSE loss (Eq. 6) instead of the proposed one (Eq. 8); cosine the ver- +sion of our SPLADE approach trained with a cosine loss instead of the proposed +loss (Eq. 8). The cosine loss is interesting because it is invariant to the scaling +factor that preserves the document ordering (Eq.1). +Second-stage ranking scenarios. We consider different scenario for our second- +stage ranking model: T5Mono_RawQuery the T5Mono ranking model [22] +applied on raw queries; T5Mono_GoldQuery the T5Mono ranking model +applied on gold queries; T5Mono_CQR the T5Mono ranking model applied +on query reformulation generated with a pre-trained T5 (using the CANARD +dataset); CoSPLADE_[context]_[number] : different versions of our second- +stage ranking model input (Eq. 9), varying 1) the number K of keywords identi- +fied as relevant by the first-stage ranker: 5, 10, 20, and 2) the presence or absence +of the past queries within the reformulation. +TREC participant baselines. For each evaluation campaign (2020 and 2021), +we also compare our model with the best, the median and the lowest TREC +CAsT participants presented in the two overviews [5,7], where participant are +ranked according to the nDCG@3 metric. +5 +Results +5.1 +First-stage ranking effectiveness +In this section, we focus on the first-stage ranking component of our CoSPLADE +model. To do so, we experiment different scenarios aiming at evaluating the +impact of the designed loss (Eq. 8) and the modeling/utility of evidence sources +(Equations 3 and 4). Results of these different baselines and scenarios on the +TREC CAsT 2021 dataset are provided in Table 1. Similar trends are observed +on CAsT 2020, but are not reported due to space limit. +In general, one can see that all variants of our approach (CoSPLADE_* +models) outperform the scenario applying the initial version of SPLADE on raw +and, more importantly, gold queries. This is very encouraging since this latter +scenario might be considered an oracle, i.e. the query is manually disambiguated. +Finally, we improve the results over CQE [16] for all the metrics – showing that + +10 +Le Hai et al. +Recall@500 MAP@500 +MRR +nDCG@500 nDCG@3 +Baselines +SPLADE_rawQuery +30.8±2.7 +5.5±0.9 +21.3±2.9 +17.8±1.8 +13.1±2.1 +SPLADE_goldQuery +68.8±2.0 +16.1±1.2 +55.5±3.3 +42.8±1.7 +38.3±2.8 +CQE [17] from [7] +79.1 +28.9 +60.3 +55.7 +43.8 +Effect of answer processing: CoSPLADE_. . . +AllAnswers +79.5±2.2 +28.8±1.7 +61.7±3.1 +55.3±2.0 +46.5±2.9 +AllAnswers-short +72.8±2.6 +25.7±1.9 +54.4±3.3 +49.5±2.3 +40.1±3.0 +AllAnswers-long +80.4±2.1 +29.3±1.8 +62.0±3.2 +55.6±2.1 +48.9±3.0 +LastAnswer +83.4±2.0 +31.2±1.8 +61.8±3.1 +58.1±2.0 +47.4±3.0 +LastAnswer-short +79.2±2.2 +28.1±1.8 +61.4±3.3 +54.3±2.1 +46.4±3.0 +LastAnswer-long +85.2±1.8 +32.0±1.7 64.3±03.0 59.4±1.9 +48.6±3.0 +CoSPLADE_LastAnswer-long variants +flatContext +77.0±2.0 +26.0±2.0 +55.0±3.0 +52.0±2.0 +42.0±3.0 +MSE loss +70.9±2.4 +21.6±1.7 +48.7±3.4 +45.2±2.3 +39.6±3.1 +cosine loss +70.4±2.5 +22.6±1.7 +52.5±3.3 +46.9±2.2 +39.0±3.0 +Table 1. Effectiveness of different scenarios of our first-stage ranking model on the +TREC CAsT 2021. +our simple learning mechanism, combined with SPLADE, allows for achieving +SOTA performance. +Leveraging queries and answers history better contextualizes the current query. +The results of the flatContext scenario w.r.t. to the SPLADE_goldQuery allows +comparing the impact of evidence sources related to the conversation since they +both use the same architecture (SPLADE). We can observe that it obtains better +results than SPLADE_goldQuery (e.g., 77 vs. 68.8 for the Recall@500 metric), +highlighting the usefulness of context to better understand the information need. +More detailed answers perform better. Since answers are more verbose than ques- +tions, including them is more complex, and we need to study the different pos- +sibilities (CoSPLADE_AllAnswers* and CoSPLADE_LastAnswer*). One can +see that: 1) trimming answers (*-short) into a few keywords is less effective than +considering canonical answers, but 2) it might be somehow effective when com- +bined with the associated Wikipedia passage (*-long). Moreover, it seems more +effective to consider only the last answer rather than the whole set of answers +in the conversation history11. Taking all together, these observations highlight +the importance of the way to incorporate information from answers into the +reformulation process. +Dual query representation with asymmetric loss leverages sparse query represen- +tations. The results of the flatContext scenario show that considering at once +past queries and answers perform better (compared to the MSE loss scenario +which is directly comparable). However, if we separate the representations and +11 This might be due to the simple way to use past answers, i.e. Eq. 4, but all the other +variations we tried did not perform better + +CoSPLADE: Contextualizing SPLADE for Conversational IR +11 +Recall@500 MAP@500 +MRR +nDCG@500 nDCG@3 +Baselines +T5Mono_RawQuery +78.4±2.3 +21.0±1.8 +39.6±3.2 +45.9±2.1 +28.4±3.0 +T5Mono_GoldQuery +86.1±1.7 +44.1±1.9 +78.7±2.7 +68.5±1.8 +64.6±2.8 +T5Mono_CQR +80.4±2.2 +30.0±1.9 +58.2±3.4 +55.3±2.1 +44.6±3.2 +coSPLADE-based second stage variants +CoSPLADE_NoContext_5 +84.3±1.8 +31.7±2.0 +61.6±3.3 +58.1±2.0 +45.9±3.1 +CoSPLADE_NoContext_10 +83.1±1.9 +32.0±1.7 +66.0±3.1 +59.1±1.9 +49.8±2.9 +CoSPLADE_NoContext_20 +84.8±1.7 +33.4±1.8 +66.0±3.0 +60.4±1.8 +49.6±2.9 +CoSPLADE_Context_5 +85.0±1.7 +35.0±1.8 +68.4±3.0 +61.7±1.9 +51.5±02.9 +CoSPLADE_Context_10 +84.8±1.7 +36.5±1.9 67.8±3.1 +63.0±1.9 +53.3±3.1 +CoSPLADE_Context_20 +84.9±1.7 +35.5±1.8 69.8±3.0 +62.2±1.9 +54.4±2.9 +Table 2. Effectiveness of different scenarios of our second-stage ranking model on +TREC CAsT 2021. +use an asymmetric loss function, the conclusion changes. Moreover, the compar- +ison of our best scenario CoSPLADE_LastAnswer-long with a similar scenario +trained by simply using a MSE or a cosine losses reveals the effectiveness of our +asymmetric MSE (Equation 7). Remember that this asymmetric loss encourages +the consideration of previous answers in the query encoding. This reinforces our +intuition that the conversation context, and particularly verbose answers, is im- +portant for the conversational search task. It also reveals that the context should +be included at different levels in the architecture (input and loss). +5.2 +Second-stage ranking effectiveness +In this section, we rely on the CoSPLADE_LastAnswer-long model as a first +stage ranker, and evaluate different variants of the second-stage ranking method +relying on the T5Mono model. For fair comparison, we also mention results ob- +tained by a T5Mono ranking model applied on raw and gold queries, as well as +query reformulated using a T5 generative model. Results on the TREC CAsT +2021 dataset are presented in Table 2. +The analysis of the CoSPLADE model variants allows to highlight different +observations regarding the usability of the context and the number of keywords +added to the query. First, adding the previous questions to the current query +in the prompt (i.e., “Context”) seems to improve the query understanding and, +therefore, positively impacts the retrieval effectiveness. For instance, when 5 +keywords are added, the context allows reaching 51.5% for the nDCG@3 against +45.9% without context. Second, the effectiveness metrics tend to increase with +the number of additional keywords, particularly for scenarios without context, +which is sensible. This trend is less noticeable for the scenarios with context since +the best metrics are alternatively obtained by the scenario adding either 10 or +20 keywords. It is worth noting however that adding 10 or 20 keywords is more +valuable than adding only 5 (e.g. 54.4% vs. 51.5% for the nDCG@3 metric). It +thus seems that 1) keywords help to reformulate the initial information need, + +12 +Le Hai et al. +2) but they can lead to saturation when they are both numerous and combined +with other information. +By comparing the best model scenarios with the more basic scenarios apply- +ing the T5Mono second-stage ranker on raw and gold queries, we can observe that +our method allows improving the retrieval effectiveness regarding initial queries +but is not sufficient for reaching the performance of T5Mono_GoldQuery. How- +ever, results obtained when applying T5Mono on queries reformulated by T5 +highlight that the contextualization of an initial query is a difficult task. Indeed, +the T5Mono_CQR scenario is less effective than the T5Mono_GoldQuery one +with between 6 and 20 points of difference depending on the metrics. +Moreover, it is interesting to note that the SPLADE model applied on raw +and gold queries (first-stage ranking in Table 1) obtains lower results than the +T5Mono model on the same data (second-stage ranking in Table 2). It can be ex- +plained by the purpose of those two architectures which are different: SPLADE is +a sparse model focusing on query/document expansion while T5Mono is partic- +ularly devoted to increase precision. However, it is worth noting that combining +SPLADE and T5Mono as first and second-stage rankers reaches the highest ef- +fectiveness results in our experimental evaluation. This shows the effectiveness +of CoSPLADE to both contextualize queries and effectively rank documents. +5.3 +Effectiveness compared to TREC CAsT participants +We finally compare our approach with TREC CAsT participants for the 2020 +and 2021 evaluation campaigns. For both years, we can see that we obtain effec- +tiveness metrics that are very close or higher than the ones reached by the best +participants. Indeed, CoSPLADE surpasses the best TREC participant for the +2020 evaluation campaign regarding Recall@1000 and nDCG@1000. For 2021, +our model obtains better results than the best one for the MRR and nDCG@3 +metrics. For both years, the best participant is the h2oloo team [18,7] where +they use query reformulation techniques, either using AllenAI or T5. Our re- +sults suggest that our approach focusing on a sparse first-stage ranking model +allows combining the benefit of query expansion and document ranking in a sin- +gle model that eventually helps the final reranking step. In other words, simply +rewriting the query without performing a joint learning document ranking can +hinder the overall performance of the search task. +6 +Conclusion +In this paper, we have shown how a sparse retrieval neural IR model, namely +SPLADE [9], could be leveraged together with a lightweight learning process +to obtain a state-of-the-art first-stage ranker. We further showed that this first- +stage ranker could be used to provide context to the second-stage ranker, leading +to results comparable with the best-performing systems. Future work may ex- +plore strategies to better capture the information from the context or to explicitly +treat user feedback present in the evaluation dataset. + +CoSPLADE: Contextualizing SPLADE for Conversational IR +13 +TREC CAsT 2020 +Recall@1000 MAP@1000 +MRR +nDCG@1000 nDCG@3 +TREC Participant (best) +63.3 +30.2 +59.3 +52.6 +45.8 +TREC Participant (median) +52.1 +15.1 +42.2 +36.4 +30.4 +TREC Participant (low) +27.9 +1.0 +5.9 +11.1 +2.2 +CoSPLADE +82.4±2.0 +26.9±1.5 +58.1±2.9 +54.2±1.8 +44.0±2.7 +TREC CAsT 2021 +Recall@500 +MAP@500 +MRR +nDCG@500 nDCG@3 +TREC Participants 1 (best) +85.0 +37.6 +67.9 +63.6 +52.6 +TREC Participants 2 (median) +36.4 +17.6 +53.4 +33.6 +37.7 +TREC Participants 3 (low) +58.9 +7.6 +27.0 +31.4 +15.4 +CoSPLADE +84.9±1.7 +35.5±1.8 +69.8±3 +62.2±1.9 +54.4±2.9 +Table 3. TREC CAsT 2020 and 2021 performances regarding participants +References +1. 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Zamani, +H., +Trippas, +J.R., +Dalton, +J., +Radlinski, +F.: +Conversational +In- +formation Seeking (Jan 2022). https://doi.org/10.48550/arXiv.2201.08808, +http://arxiv.org/abs/2201.08808, arXiv:2201.08808 [cs] + diff --git a/5NE3T4oBgHgl3EQfQgli/content/tmp_files/load_file.txt b/5NE3T4oBgHgl3EQfQgli/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d3500e436c63b177e50019973ddb1ae48bd22c8 --- /dev/null +++ b/5NE3T4oBgHgl3EQfQgli/content/tmp_files/load_file.txt @@ -0,0 +1,985 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf,len=984 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='04413v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='IR] 11 Jan 2023 CoSPLADE: Contextualizing SPLADE for Conversational Information Retrieval Nam Le Hai1[0000−0002−9020−8790], Thomas Gerald2, Thibault Formal1,3, Jian-Yun Nie4, Benjamin Piwowarski1[0000−0001−6792−3262], and Laure Soulier1,2[0000−0001−9827−7400] 1 Sorbonne Université, CNRS, ISIR, F-75005 Paris, France first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='last @sorbonne-universite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='fr 2 Université Paris-Saclay, CNRS, LISN, 91405 Orsay France first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='last @lisn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='fr 3 Naver Labs Europe, Meylan, France first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='last @naverlabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='com 4 University of Montreal, Montreal, Canada nie@iro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='umontreal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='ca Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Conversational search is a difficult task as it aims at retriev- ing documents based not only on the current user query but also on the full conversation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Most of the previous methods have focused on a multi-stage ranking approach relying on query reformulation, a criti- cal intermediate step that might lead to a sub-optimal retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Other approaches have tried to use a fully neural IR first-stage, but are ei- ther zero-shot or rely on full learning-to-rank based on a dataset with pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' In this work, leveraging the CANARD dataset, we propose an innovative lightweight learning technique to train a first-stage ranker based on SPLADE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' By relying on SPLADE sparse representations, we show that, when combined with a second-stage ranker based on T5Mono, the results are competitive on the TREC CAsT 2020 and 2021 tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Keywords: information retrieval · conversational search · first-stage ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 1 Introduction With the introduction of conversational assistants like Siri, Alexa or Cortana, conversational Information Retrieval, a variant of adhoc IR, has emerged as an important research domain [4,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' In conversational IR, a search is conducted within a session, and the user’s information need is expressed through a sequence of queries, similarly to natural conversations – thus introducing complex inter- dependencies between queries and responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Not surprisingly, neural IR models have been shown to perform the best on conversational IR [5,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Most prior works rely on a Historical Query Expansion step [34], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' a query expansion mechanism that takes into account all past queries and their associated answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Such query expansion model is learned on the CANARD dataset [8], which is composed of a series of questions and their associated answers, together with a disambiguated query, referred to as gold query in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' However, relying on a reformulation step is computationally 2 Le Hai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' costly and might be sub-optimal as underlined in [13,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Krasakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' [13] proposed to use ColBERT [12] in a zero-shot manner, replacing the query by the sequence of queries, without any training of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' [16] proposed to learn a dense contextualized representation of the query history, optimizing a learning-to-rank loss over a dataset composed of weak labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' This makes the training process complex (labels are not reliable) and long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' In this work, we follow this direction of research but propose a much lighter training process for the first-stage ranker, where we focus on queries and do not make use of any passage – and thus of a learning-to-rank training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' It moreover sidesteps the problem of having to derive weak labels from the CANARD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Given this strong supervision, we can consider more context – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' we use the answers provided by the system the user is interacting with, which allows to better contextualize the query, as shown in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The training loss we propose leverages the sparse representation of queries and documents provided by the SPLADE model [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' In a nutshell, we require that the representation of the query matches that of the disambiguated query (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' the gold query).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Our first-stage ranker achieves high performances, especially on recall – the most important measure in a multi-stage approach, comparable to the best systems in TREC CAsT [7], but also on precision-oriented measures – which shows the potential of our methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Finally, to perform well, the second-stage ranker (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' re-ranker) needs to consider the conversation as well, which might require a set of heuristics to select some content and/or query reformulation such as those used in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Leveraging the fact that our first-stage ranker outputs weights over the (BERT) vocabulary, we propose a simple mechanism that provides a conversational context to the re-ranker in the form of keywords selected by SPLADE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' In summary, our contributions are the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We propose a new loss to optimize a first-stage ranker resulting in a lightweight training strategy and state-of-the-art results in terms of recall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We show that, when combined with a second-stage ranker based on a context derived from the SPLADE query representation of the first stage, we obtain results on par with the best approaches in TREC CAsT 2020 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 2 Related Works The first edition [5] of the TREC Conversational Assistance Track (CAsT) was implemented in 2019, providing a new challenge on Conversational Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The principle is the following: a user queries the system with questions in natural language, and each time gets a response from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The challenge differs from classical search systems as involving previous utterances (either queries or answers) is key to better comprehending the user intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' In conversational IR, and in TREC CAsT [6,5,7] in particular, the sheer size of the document collection implies to design an efficient (and effective) search system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Conversational IR is closely related to conversational Question-Answering [25,27,26] in the sense that they both include interaction turns in natural lan- guage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' However, the objective is intrinsically different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' While the topic or the CoSPLADE: Contextualizing SPLADE for Conversational IR 3 context (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=', the passage containing answers) is known in conversational QA, conversational IR aims to search among a huge collection of documents with po- tentially more exploratory topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' With this in mind, in the following, we focus on the literature review of conversational IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We can distinguish two lines of work in conversational search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The first one [29,30,32,3] focuses on a Contextual Query Reformulation (CQR) to produce a (plain or bag-of-words) query, representing ideally the information need free of context, which is fed into a search model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' One strategy of CQR consists in selecting keywords from previous utterances by relying on a graph weighted by either word2vec similarity [29], term-based importance using BM25 [19], or clas- sification models [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Other approaches [14,19,18,33,28] leverage the potential of generative language models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=', GPT2 or T5) to rewrite the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Such approaches are particularly effective, reaching top performances in the TREC CAsT 2020 edition [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Query reformulation models also differ in the selected evidence sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Models either focus on the early stage of the conversation [1], on a set of the queries filtered either heuristically [2] or by a classification model [21], or on both previous queries and documents [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Finally, to avoid the prob- lem of generating a single query, [14,20] have proposed to use different generated queries and aggregate the returned documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The reformulation step is however a bottleneck since there is no guarantee that the “gold query” is optimal and thus generalizes well [16,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Moreover, generating text is time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' To avoid these problems, the second line of work aims to directly integrate the conversation history into the retrieval model, bypassing the query reformulation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' As far as we know, only a few studies followed this path in conversational search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Qu et al [24] compute a query rep- resentation using the k last queries in the dialogue [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Similarly Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' [16] average contextualized tokens embeddings over the whole query history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The representation is learned by optimizing a learning-to-rank loss over a collection with weak labels, which requires much care to ensure good generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Fi- nally, Krasakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' [13] use a more lexical neural model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' ColBERT [12], to encode the query with its context – but they do not finetune it at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' In this work, we go further by using a sparse model SPLADE [9], using a novel loss tailored to such sparse representations, and by using a lightweight train- ing procedure that does not rely on passages, but only on a dataset containing reformulated queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 3 Model In TREC CAsT [5,7], each retrieval session contains around 10 turns of ex- change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Each turn corresponds to a query and its associated canonical answer5 is provided as context for future queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Let us now introduce some notations that we use to describe our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' For each turn n ≤ N, where N is the last turn of the conversation, we denote by qn and an respectively the corresponding query and its response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Finally, the context of a query qn at turn n corresponds 5 Selected by the organizer as the most relevant answer of a baseline system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 4 Le Hai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' to all the previous queries and answers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' q1, a1, q2, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=', qn−1, an−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The main objective of the TREC CAsT challenges is to retrieve, for each query qn and its context, the relevant passages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' In the next sections, we present our first-stage ranker and second-stage re- ranker, along with their training procedure, both based, directly or indirectly, on the SPLADE (v2) model described in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' SPLADE has shown results on par with dense approaches on in-domain collections while exhibiting stronger abilities to generalize in a zero-shot setting [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' It outputs a sparse representation of a document or a query in the BERT vocabulary, which is key to our model during training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The SPLADE model we use includes a contextual encoding function, followed by some aggregation steps: ReLU, log saturation, and max pooling over each token in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The output of SPLADE is a sparse vector with only positive or zero components in the BERT vocabulary space R|V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' In this work, we use several sets of parameters for the same SPLADE architecture and distinguish each version by its parameters θ, and the corresponding model by SPLADE(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 First stage The original SPLADE model [9] scores a document using the dot product be- tween the sparse representation of a document ( ˆd) and of a query (ˆq): s(ˆq, ˆd) = ˆq · ˆd (1) In our work, like in [16], we suppose that the document representation has been sufficiently well-tuned on the standard ad-hoc IR task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The document embedding ˆd is thus obtained using the pre-trained SPLADE model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' ˆd = SPLADE([CLS] d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' θSP LADE) where θSP LADE are the original SPLADE param- eters obtained from HuggingFace6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' These parameters are not fine-tuned during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We can thus use standard indices built from the original SPLADE document representations to retrieve efficiently the top-k documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' In the following, we present how to contextualize the query representation using the conversation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Then, we detail the training loss aiming at reducing the gap between the representation of the gold query and the contextualized representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Query representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Like state-of-the-art approaches for first-stage conversa- tional ranking [16,13], we contextualize the query with the previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Going further, we propose to include the answers in the query representation process, which is easier to do thanks to our lightweight training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' To leverage both contexts, we use a simple model where the contextual query representation at turn n, denoted by ˆqn,k, is the combination of two representa- tions, ˆqqueries n which encodes the current query in the context of all the previous 6 The weights can be found at https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='co/naver/splade-cocondenser-ensembledistil CoSPLADE: Contextualizing SPLADE for Conversational IR 5 queries, and ˆqanswers n,k which encodes the current query in the context of k the past answers7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Formally, the contextualized query representation ˆqn,k is: ˆqn,k = ˆqqueries n + ˆqanswers n,k (2) where we use two versions of SPLADE parameterized by θqueries for the full query history and θanswers,k for the answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' These parameters are learned by optimizing the loss defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Following [16], we define ˆqqueries n to be the query representation produced by encoding the concatenation of the current query and all the previous ones: ˆqqueries n = SPLADE([CLS] qn [SEP] q1 [SEP] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' [SEP] qn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' θqueries) (3) using a set of specific parameters θqueries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' To take into account the answers that the user had access to, we need to include them in the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Following prior work [2], we can consider a various number of answers k, and in particular, we can either choose k = 1 (the last answer) or k = n−1 (all the previous answers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Formally, the representation ˆqanswers n,k is computed as: ˆqanswers n,k = 1 k n−1 � i=n−k SPLADE(qn [SEP] ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' θanswers,k) (4) Training Based on the above, training aims at obtaining a good representation ˆqn for the last issued query qn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' to contextualize qn using the previous queries and answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' To do so, we can leverage the gold query q∗ n, that is, a (hopefully) contextualized and unambiguous query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We can compute the representation ˆq∗ n of this query by using the original SPLADE model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' ˆq∗ n = SPLADE(q∗ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' θSP LADE) (5) For example, for a query "How old is he?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='" the matching gold query could be "How old is Obama?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The representation of the latter given by SPLADE would be as follows: [(”Obama”, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5), (”Barack”, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2), (”age”, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2), (”old”, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0), (”president”, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='] where the terms “Obama” and “Barack” clearly appear alongside other words related to the current query (“old” and the semantically related “age”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We can now define the goal of the training, which is to reduce the difference between the gold query representation ˆq∗ n and the representation ˆqn,k computed by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' An obvious choice of a loss function is to match the predicted and gold representations using cosine loss (since the ranking is invariant when scaling the query).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' However, as shown in the result section, we experimentally 7 In the experiments, we also explore an alternative model where answers and queries are considered at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 6 Le Hai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' found better results with a modified MSE loss, whose first component is the standard MSE loss: LossMSE(ˆqn,k, ˆq∗ n) = MSE(ˆqn,k, ˆq∗ n) (6) In our experiments, we observed that models trained with the direct MSE do not capture well words from the context, especially for words from the answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The reason is that the manually reformulated gold query usually only contains a few additional words from the previous turns that are directly implied by the last query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Other potentially useful words from the answers may not be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' This is a conservative expansion strategy which may not be the best example to follow by an automatic query rewriting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We thus added an asymmetric MSE, designed to encourage term expansion from past answers, but avoid introducing noise by restricting the terms to those present in the gold query q∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Formally, our asymmetric loss is: Lossasym(ˆqanswers n,k , ˆq∗ n) = � max(ˆq∗ n − ˆqanswers n,k , 0) �2 (7) where the maximum is component-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' This loss thus pushes the answer-biased representation ˆqanswers n,k to include tokens from the gold answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Contrarily to MSE, it does not impose (directly) an upper bound on the components of the ˆqanswers n,k representation – this is done indirectly through the final loss function described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The final loss we optimize is a simple linear combination of the losses defined above, and only relies on computing two query representations: Loss(ˆqn,k, ˆq∗ n) = LossMSE(ˆqn,k, ˆq∗ n) + Lossasym(ˆqanswers n,k , ˆq∗ n) (8) There is an interplay between the two components of the global loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' More pre- cisely, Lossasym pushes the ˆqanswers n,k representation to match the golden query representation ˆq∗ n if it can, and LossMSE pushes the queries-biased representa- tion ˆqn,k to compensate if not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' It thus puts a strong focus on extracting infor- mation from past answers, which is shown to be beneficial in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' For the first-stage, we initialize both encoders (one en- coding the queries, and the other encoding the previous answer) with pre-trained weights from SPLADE model for adhoc retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We use the ADAM optimizer with train batch size 16, learning rate 2e-5 for the first encoder and 3e-5 for the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We fine-tune for only 1 epoch over the CANARD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2 Reranking We perform reranking using a T5Mono [22] approach, where we enrich the raw query qn with keywords identified by the first-stage ranker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Our motivation is that these words should capture the information needed to contextualize the raw query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The enriched query q+ n for conversational turn n is as follows: q+ n = qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Context : q1 q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' qn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Keywords : w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=', wK (9) CoSPLADE: Contextualizing SPLADE for Conversational IR 7 where the wi are the top-K most important words that we select by leveraging the first-stage ranker as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' First, to reduce noise, we only consider words that appear either in any query qi or in the associated answers ai (for i ≤ n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Second, we order words by using the maximum SPLADE weight over tokens that compose the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 We denote the T5 model fine-tuned for this input as T 5+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' As in the original paper [22], the relevance score of a document d for the query qn is the proba- bility of generating the token “true” given a prompt pt(q+ n , d) = “Query: q+ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Document: d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Relevant:”: score(q+ n , d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' θ) = pT 5(true|pt(q+ n , d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' θ) pT 5(true|pt(q+ n , d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' θ) + pT 5(false|pt(q+ n , d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' θ) (10) where θ are the parameters of the T5Mono model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Differently to the first stage training, we fine-tune the ranker by aligning the scores of the documents, and not the weight of a query (which is obviously not possible with the T5 model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Here the “gold” score of a document is computed us- ing the original T5Mono with the gold query q∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The T5 model is initialized with weights made public by the original authors9, denoted as θT 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' More precisely, we finetune the pre-trained T5Mono model using the MSE-Margin loss [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The loss function for the re-ranker (at conversation turn n, given documents d1 and d2) is calculated as follows: LR = �� s(q+ n , d1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' θT 5+) − s(q+ n , d2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' θT 5+) � − (s(q∗ n, d1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' θT 5) − s(q∗ n, d2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' θT 5)) �2 We optimize the θT 5+ parameters by keeping the original θT 5 to evaluate the score of gold queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We initialize θT 5+ as θT 5, and fine-tune for 3 epochs, with a batch size of 8 and a learning rate 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We sample pairs (d1, d2) using the first-stage top-1000 documents: d1 is sampled among the top-3, and d2 among the remaining 997 to push the model to focus on important differences in scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 4 Experimental Protocol We designed the evaluation protocol to satisfy two main evaluation objectives: (i) Evaluating separately the effectiveness of the first-stage and the second-stage ranking components of our CoSPLADE model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' (ii) Comparing the effectiveness of our CoSPLADE model with TREC CAsT 2020 and 2021 participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 8 To improve coherence, we chose to make keywords follow their order of appearance in the context, but did not vary this experimental setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 9 We used the Huggingface checkpoint https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='co/castorini/monot5-base-msmarco 8 Le Hai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 Datasets To train our model, we used the CANARD corpus10, a conversational dataset fo- cusing on context-based query rewriting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' More specifically, the CANARD dataset is a list of conversation histories, each being composed of a series of queries, short answers (human written) and reformulated queries (contextualised).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The train- ing, development, and test sets include respectively 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='538, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='418, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='571 contextual and reformulated queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' To evaluate our model, we used the TREC CAsT 2020 and 2021 datasets which include respectively 25 and 26 information needs (topics) and a document collection composed of the MS MARCO dataset, an updated dump of Wikipedia from the KILT benchmark, and the Washington Post V4 collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' For each topic, a conversation is available, alternating questions and responses (manually selected passages from the collection, aka canonical answers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' For each question (216 and 239 in total), the dataset provides its manually rewritten form as well as a set of about 20 relevant documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We use the former to define an upper- bound baseline (Splade_GoldQuery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2 Metrics and baselines We used the official evaluation metrics considered in the TREC CAsT 2020 and 2021, namely nDCG@3, MRR, Recall@X, MAP@X, nDCG@X, where the cut-off is set to 1000 for the CAsT 2020 and 500 for the CAsT 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' For each metric, we calculate the mean and variance of performance across the different queries in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' With this in mind, we present below the different baselines and scenarios used to evaluate each component of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' First-stage ranking scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' To evaluate the effectiveness of our first-stage ranking model (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1), we compare our approach CoSPLADE, based on the query representation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' (2) with different variants (the document en- coder is set to the original SPLADE encoder throughout our experiments): SPLADE_rawQuery (lower bound): SPLADE [10] using only the original and ambiguous user queries qn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' SPLADE_goldQuery (kind of upper bound): SPLADE using the manually rewritten query q∗ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' CQE [16], a state-of-the-art dense contextualized query representation learned using learning-to-rank on a dataset with pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' To model answers when representing the query using ˆqanswers n,k , we used two historical ranges (“All” with k = n−1 answers and “Last” where we use only the last one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' k = 1) and three types of answer inputs: Answer in which answers are the canonical answers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Answer-Short in which sentences are filtered as in the best performing TREC CAsT approach [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' This allows for consistent input length, at the expense of losing information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Answer-Long As answers from CANARD are short (a few sentences extracted from Wikipedia – contrarily to CAsT ones), we expand them to reduce the discrepancy between training and 10 https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='com/view/qanta/projects/canard CoSPLADE: Contextualizing SPLADE for Conversational IR 9 inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' For each sentence, we find the Wikipedia passage it appears in (if it exists in ORConvQA [23]), and sample a short snippet of 3 adjacent sentences from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Finally, we also conducted ablation studies (on the best of the above vari- ants) by modifying either the way to use the historical context or the training loss: flatContext a one-encoder version of our SPLADE approach in which we concatenate all information of the context to apply SPLADE to obtain a single representation of the query (instead of two representations ˆqqueries n and ˆqanswers n,k as in Equations 2 and 3) trained using a MSE loss function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 6) since there is no more two representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' MSE the version of our SPLADE approach trained with the MSE loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 6) instead of the proposed one (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' cosine the ver- sion of our SPLADE approach trained with a cosine loss instead of the proposed loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The cosine loss is interesting because it is invariant to the scaling factor that preserves the document ordering (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Second-stage ranking scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We consider different scenario for our second- stage ranking model: T5Mono_RawQuery the T5Mono ranking model [22] applied on raw queries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' T5Mono_GoldQuery the T5Mono ranking model applied on gold queries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' T5Mono_CQR the T5Mono ranking model applied on query reformulation generated with a pre-trained T5 (using the CANARD dataset);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' CoSPLADE_[context]_[number] : different versions of our second- stage ranking model input (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 9), varying 1) the number K of keywords identi- fied as relevant by the first-stage ranker: 5, 10, 20, and 2) the presence or absence of the past queries within the reformulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' TREC participant baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' For each evaluation campaign (2020 and 2021), we also compare our model with the best, the median and the lowest TREC CAsT participants presented in the two overviews [5,7], where participant are ranked according to the nDCG@3 metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 5 Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 First-stage ranking effectiveness In this section, we focus on the first-stage ranking component of our CoSPLADE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' To do so, we experiment different scenarios aiming at evaluating the impact of the designed loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 8) and the modeling/utility of evidence sources (Equations 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Results of these different baselines and scenarios on the TREC CAsT 2021 dataset are provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Similar trends are observed on CAsT 2020, but are not reported due to space limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' In general, one can see that all variants of our approach (CoSPLADE_* models) outperform the scenario applying the initial version of SPLADE on raw and, more importantly, gold queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' This is very encouraging since this latter scenario might be considered an oracle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' the query is manually disambiguated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Finally, we improve the results over CQE [16] for all the metrics – showing that 10 Le Hai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Recall@500 MAP@500 MRR nDCG@500 nDCG@3 Baselines SPLADE_rawQuery 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 SPLADE_goldQuery 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 CQE [17] from [7] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 Effect of answer processing: CoSPLADE_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' AllAnswers 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 AllAnswers-short 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 AllAnswers-long 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 LastAnswer 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 LastAnswer-short 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 LastAnswer-long 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3±03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 CoSPLADE_LastAnswer-long variants flatContext 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 MSE loss 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 cosine loss 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Effectiveness of different scenarios of our first-stage ranking model on the TREC CAsT 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' our simple learning mechanism, combined with SPLADE, allows for achieving SOTA performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Leveraging queries and answers history better contextualizes the current query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The results of the flatContext scenario w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' to the SPLADE_goldQuery allows comparing the impact of evidence sources related to the conversation since they both use the same architecture (SPLADE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We can observe that it obtains better results than SPLADE_goldQuery (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=', 77 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 for the Recall@500 metric), highlighting the usefulness of context to better understand the information need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' More detailed answers perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Since answers are more verbose than ques- tions, including them is more complex, and we need to study the different pos- sibilities (CoSPLADE_AllAnswers* and CoSPLADE_LastAnswer*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' One can see that: 1) trimming answers (*-short) into a few keywords is less effective than considering canonical answers, but 2) it might be somehow effective when com- bined with the associated Wikipedia passage (*-long).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Moreover, it seems more effective to consider only the last answer rather than the whole set of answers in the conversation history11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Taking all together, these observations highlight the importance of the way to incorporate information from answers into the reformulation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Dual query representation with asymmetric loss leverages sparse query represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The results of the flatContext scenario show that considering at once past queries and answers perform better (compared to the MSE loss scenario which is directly comparable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' However, if we separate the representations and 11 This might be due to the simple way to use past answers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 4, but all the other variations we tried did not perform better CoSPLADE: Contextualizing SPLADE for Conversational IR 11 Recall@500 MAP@500 MRR nDCG@500 nDCG@3 Baselines T5Mono_RawQuery 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 T5Mono_GoldQuery 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 T5Mono_CQR 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2 coSPLADE-based second stage variants CoSPLADE_NoContext_5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 CoSPLADE_NoContext_10 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 CoSPLADE_NoContext_20 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 CoSPLADE_Context_5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5±02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 CoSPLADE_Context_10 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 CoSPLADE_Context_20 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Effectiveness of different scenarios of our second-stage ranking model on TREC CAsT 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' use an asymmetric loss function, the conclusion changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Moreover, the compar- ison of our best scenario CoSPLADE_LastAnswer-long with a similar scenario trained by simply using a MSE or a cosine losses reveals the effectiveness of our asymmetric MSE (Equation 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Remember that this asymmetric loss encourages the consideration of previous answers in the query encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' This reinforces our intuition that the conversation context, and particularly verbose answers, is im- portant for the conversational search task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' It also reveals that the context should be included at different levels in the architecture (input and loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2 Second-stage ranking effectiveness In this section, we rely on the CoSPLADE_LastAnswer-long model as a first stage ranker, and evaluate different variants of the second-stage ranking method relying on the T5Mono model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' For fair comparison, we also mention results ob- tained by a T5Mono ranking model applied on raw and gold queries, as well as query reformulated using a T5 generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Results on the TREC CAsT 2021 dataset are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' The analysis of the CoSPLADE model variants allows to highlight different observations regarding the usability of the context and the number of keywords added to the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' First, adding the previous questions to the current query in the prompt (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=', “Context”) seems to improve the query understanding and, therefore, positively impacts the retrieval effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' For instance, when 5 keywords are added, the context allows reaching 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5% for the nDCG@3 against 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9% without context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Second, the effectiveness metrics tend to increase with the number of additional keywords, particularly for scenarios without context, which is sensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' This trend is less noticeable for the scenarios with context since the best metrics are alternatively obtained by the scenario adding either 10 or 20 keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' It is worth noting however that adding 10 or 20 keywords is more valuable than adding only 5 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5% for the nDCG@3 metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' It thus seems that 1) keywords help to reformulate the initial information need, 12 Le Hai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 2) but they can lead to saturation when they are both numerous and combined with other information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' By comparing the best model scenarios with the more basic scenarios apply- ing the T5Mono second-stage ranker on raw and gold queries, we can observe that our method allows improving the retrieval effectiveness regarding initial queries but is not sufficient for reaching the performance of T5Mono_GoldQuery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' How- ever, results obtained when applying T5Mono on queries reformulated by T5 highlight that the contextualization of an initial query is a difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Indeed, the T5Mono_CQR scenario is less effective than the T5Mono_GoldQuery one with between 6 and 20 points of difference depending on the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Moreover, it is interesting to note that the SPLADE model applied on raw and gold queries (first-stage ranking in Table 1) obtains lower results than the T5Mono model on the same data (second-stage ranking in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' It can be ex- plained by the purpose of those two architectures which are different: SPLADE is a sparse model focusing on query/document expansion while T5Mono is partic- ularly devoted to increase precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' However, it is worth noting that combining SPLADE and T5Mono as first and second-stage rankers reaches the highest ef- fectiveness results in our experimental evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' This shows the effectiveness of CoSPLADE to both contextualize queries and effectively rank documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3 Effectiveness compared to TREC CAsT participants We finally compare our approach with TREC CAsT participants for the 2020 and 2021 evaluation campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' For both years, we can see that we obtain effec- tiveness metrics that are very close or higher than the ones reached by the best participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Indeed, CoSPLADE surpasses the best TREC participant for the 2020 evaluation campaign regarding Recall@1000 and nDCG@1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' For 2021, our model obtains better results than the best one for the MRR and nDCG@3 metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' For both years, the best participant is the h2oloo team [18,7] where they use query reformulation techniques, either using AllenAI or T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Our re- sults suggest that our approach focusing on a sparse first-stage ranking model allows combining the benefit of query expansion and document ranking in a sin- gle model that eventually helps the final reranking step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' In other words, simply rewriting the query without performing a joint learning document ranking can hinder the overall performance of the search task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' 6 Conclusion In this paper, we have shown how a sparse retrieval neural IR model, namely SPLADE [9], could be leveraged together with a lightweight learning process to obtain a state-of-the-art first-stage ranker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' We further showed that this first- stage ranker could be used to provide context to the second-stage ranker, leading to results comparable with the best-performing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Future work may ex- plore strategies to better capture the information from the context or to explicitly treat user feedback present in the evaluation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' CoSPLADE: Contextualizing SPLADE for Conversational IR 13 TREC CAsT 2020 Recall@1000 MAP@1000 MRR nDCG@1000 nDCG@3 TREC Participant (best) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='3 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 TREC Participant (median) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4 TREC Participant (low) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2 CoSPLADE 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 TREC CAsT 2021 Recall@500 MAP@500 MRR nDCG@500 nDCG@3 TREC Participants 1 (best) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6 TREC Participants 2 (median) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 TREC Participants 3 (low) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4 CoSPLADE 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='8±3 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='9 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' TREC CAsT 2020 and 2021 performances regarding participants References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' Aliannejadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=', Chakraborty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=', Ríssola, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=': Conversational In- formation Seeking (Jan 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='08808, http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='org/abs/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='08808, arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} +page_content='08808 [cs]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE3T4oBgHgl3EQfQgli/content/2301.04413v1.pdf'} diff --git a/5tE2T4oBgHgl3EQfkQe0/content/tmp_files/2301.03977v1.pdf.txt b/5tE2T4oBgHgl3EQfkQe0/content/tmp_files/2301.03977v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..844ce757620df59b03178c2148f1ce245a03dc43 --- /dev/null +++ b/5tE2T4oBgHgl3EQfkQe0/content/tmp_files/2301.03977v1.pdf.txt @@ -0,0 +1,1182 @@ +Service Differentiation and Fair Sharing +in Distributed Quantum Computing +Claudio Cicconettia,∗, Marco Contia, Andrea Passarellaa +aIIT, National Research Council, Pisa, Italy +Abstract +In the future, quantum computers will become widespread and a network of +quantum repeaters will provide them with end-to-end entanglement of remote +quantum bits. As a result, a pervasive quantum computation infrastructure will +emerge, which will unlock several novel applications, including distributed quan- +tum computing, that is the pooling of resources on multiple computation nodes +to address problem instances that are unattainable by any individual quantum +computer. In this paper, we first investigate the issue of service differentiation +in this new environment. Then, we define the problem of how to select which +computation nodes should participate in each pool, so as to achieve a fair share +of the quantum network resources available. The analysis is performed via an +open source simulator and the results are fully and readily available. +Keywords: +Distributed Quantum Computing, Quantum Internet, Quantum +Routing +1. Introduction +Quantum Computing (QC) exploits the properties of matter at very small +scale to solve some problems much faster than a classical counterpart. Even +though QC has been theorized 40 years ago [1], only recently the technology +evolution and a spur of investments have made it possible to obtain practical +∗Corresponding author +Email addresses: c.cicconetti@iit.cnr.it (Claudio Cicconetti), m.conti@iit.cnr.it +(Marco Conti), a.passarella@iit.cnr.it (Andrea Passarella) +Preprint submitted to Elsevier +January 11, 2023 +arXiv:2301.03977v1 [quant-ph] 10 Jan 2023 + +results and speculate about approaching mass deployments [2]. QC is being al- +ready used in the chemical and pharmaceutical industry, while new applications +are being progressively unlocked in material science, Machine Learning (ML) +and engineering optimization, production and logistics, post-quantum security +[3]. Essentially, the computational advantage of QC stems from the proper- +ties of superposition and entanglement of the qubits (i.e., the “quantum bits”): +(i) superposition, which means that a qubit can be in a combination of multiple +states at the same time; and (ii) entanglement, which is a property exhibited by +a set of qubits that maintain their correlation even separated in space or time. +We can expect that the computational power of a single QC will remain +relatively limited in the near future, due to scalability issues in maintaining +a very stable and controlled environment to cope with the flimsy nature of +qubits. On the other hand, the realization of the Quantum Internet is progress- +ing steadily [4], with the long-term goal to enable the entanglement of qubits that +reside in QCs across geographical distances. With the diffusion of QCs and their +gradual interconnection via quantum networks, a pervasive infrastructure will +therefore materialize, with the potential to combine opportunistically resources +from multiple QCs for the execution of specialized algorithms in a distributed +fashion. A general framework for such distributed quantum computing has been +proposed in [5], where the authors propose practical examples, e.g., a quantum +version of the k-means clustering, which is used in unsupervised ML. +A preliminary analysis of the allocation of resources among multiple quantum +computers based on the characteristics of the underlying quantum network has +been presented in [6]. In the same work, we have also proposed a practical solu- +tion inspired by a well-known algorithm in classical data networks, i.e., Deficit +Round Robin (DRR) [7], which we have evaluated through simulations. We +have found that some fundamental properties of quantum networks immensely +impact on the provisioning of resources, which calls for new research in this +area. This is especially manifest when considering networks of first-generation +(1G) quantum repeaters [8], which do not have error-correction capabilities and +are expected to be next in line for the industrialization and mass deployment +2 + +in the following years [9]. +The contribution of this paper is twofold. +1. We evaluate the performance of the resource allocation algorithm proposed +in [6] with differentiated services coexisting within the same quantum net- +work. Furthermore, we do so by comparing the performance with two alter- +native algorithms, inspired by equivalents in classical problems with similar +settings. This extends and completes the preliminary analysis in our previous +work. +2. We define a new problem related to fair share of resources in a quantum +network among multiple applications wishing to perform distributed QC: +how to best choose the peers among those available? After introducing a +mathematical formulation of the problem, we propose a greedy approxima- +tion algorithm, which is then evaluated thoroughly and compared to two +alternatives. +All the experiments in the paper are carried out via simulations, which are +fully reproducible and publicly available on GitHub, including the simulation +software source code, the scripts to run the analysis, and the artifacts and plots. +The rest of this paper is structured as follows. We summarize the system +model assumptions and findings in [6] in Sec. 2. We then review the related work +on routing in quantum network in Sec. 3. The main contributions are reported +in Sec. 4, where we study the service differentiation, and in Sec. 5, where we +tackle the problem of fair sharing of resources. Sec. 6 concludes the paper and +identifies the most important open research directions in this context. +2. System Model +In this section, we describe in short the quantum network abstract model +adopted in the paper (Sec. 2.1), the resource allocation algorithm proposed in [6] +(Sec. 2.2), and the simulation methodology and tool (Sec. 2.3). For more details, +we refer the reader to [6], in particular sections II and IV, and references within. +3 + +end-to-end +entanglement +path +Figure 1: Quantum network model. End-to-end entanglement can be established between two +nodes s and d for which there is a path in G(V, E), where the intermediate nodes perform +entanglement swapping. F is the fidelity with which the local link EPR pairs are generated; q +is the measurement success probability, which affects the entanglement swapping procedure; +Cij is the capacity of edge eij, in EPR-pairs/s. +2.1. Quantum network model +The quantum network model is illustrated in Fig. 1 as a graph G(V, E), +where nodes represent quantum devices (repeaters or computers), and edges +represent direct quantum communication links between them [10]. We assume +that maximally entangled EPR (Einstein–Podolsky–Rosen) pairs, e.g., |Φ+⟩, +are generated periodically at each link, and they have initial fidelity equal to +¯F ∈ [0.5, 1]. The fidelity is a measure of how close a given quantum state is to +a reference state, with 1 meaning that they are identical. Every edge eij ∈ E +has a given capacity Cij, which is the rate of generation of EPR-pairs, which +in turn depends on the physical characteristics of the quantum network devices +and links. +Quantum networks will offer the capability to produce entangled +EPR-pairs between remote nodes, i.e., nodes that are interconnected through +intermediate hops, which will perform the procedure of entanglement swapping +to this purpose. +Such a procedure is stochastic in nature, in 1G quantum +repeaters, and we assume that it succeeds with probability q [11]. An end-to- +end EPR-pair can only be used in a meaningful manner if all the entanglement +4 + +swaps along the path have succeeded, which leads to the following formula to +compute the maximum net rate that can be used by a quantum application +consuming resources along a path p = {(s, v1), . . . , (vN, d)}: +r(p) ≤ mineij∈p{Cij} +q|p|−1 +, +(1) +where |p| is the length of the path p, in number of edges. Furthermore, en- +tanglement swapping reduces the fidelity of the end-to-end entangled EPR-pair +according to the following formula [12]: +F(p) = 1 +4 + 3 +4 +�4 ¯F − 1 +3 +�|p| +. +(2) +We can say that r(p) and F(p) define the effective rate at which two end-points +can transfer EPR-pairs, which is the logical equivalent of throughput in classical +networks. +In our previous work [6] we have classified the quantum applications in two +categories: +– Flows. They are characterized by the need for two specific nodes to exchange +a constant flow of EPR pairs in a point-to-point manner for the whole dura- +tion of a session. If the rate of EPR pairs falls below the requested amount, +then the application Quality of Service (QoS) degrades. Examples of such ap- +plications are: clock synchronization and Quantum Key Distribution (QKD). +– Apps. In this category we find distributed quantum computing applications, +each characterized by a given quantum computer (host) running an algorithm +that pools the resources of a number of other quantum computers (peers or +workers). There is no required EPR-pair rate, but the application wishes +to consume as many EPR-pairs as possible to complete the execution faster. +Such kind of service is equivalent to best-effort or elastic traffic in a classical +data network. +Since the network operator might provide a differentiated +service, we foresee that each app is also assigned a weight (ρ), which is a +relative indication of how much throughput (in EPR-pair/s) it should be +given in the long-term compared to another app with a different weight. +5 + +For both categories, we foresee the minimum fidelity (F min) can be also a +user requirement. Since in this work we focus on distributed quantum computing +applications, whose traffic is better modeled by apps than flows, in the rest of +the paper we do not consider further the latter. +2.2. QDRR resource allocation algorithm +We report below a recap of the apps’ resource allocation algorithm in [6], +which in the following will be called Quantum DRR (QDRR) as it was inspired +by the well-known DRR algorithm [7]. The basic idea of QDRR is to provide all +applications with a fair chance to be allocated a fraction of the quantum network +resources. This is enforced by visiting the applications in a round robin: at each +visit, the application can be allocated capacity across multiple paths towards +its peers, up until a given amount that is proportional to the application’s +weight. Shortest paths are always preferred to longer ones, because they are +more efficient. A more detailed explanation follows. +The algorithm has two system parameters, which are set based on our pre- +vious results: k = 4 and the round size φ = 10 EPR-pairs/s. For a set of apps +i ∈ A, each defined by a host node {hi} and a set of candidate peers Wi, QDRR +consists of the following steps: +0. ∀i ∈ A, ∀j ∈ Wi: find k shortest paths from hi to wij and add them to +Pi; at the end, each Pi contains up to k paths for each possible peer of i. +Initialize the active list of apps L with the identifiers of all the apps A. Copy +the graph G(V, E) into a temporary copy G′. +1. If L = ∅ terminate. Otherwise, let a be the next application to be visited in +L in round robin order. +2. Set the residual capacity that can be used by the current app a in this round +to δa = φ +ρa +� +i∈A ρi , i.e., a fraction of the round size φ proportional to its +priority. +3. Select the shortest path p ∈ Pa. The shortest path is the one that requires +the least amount of resources among those available for the current app, +6 + +according to Eq. (1), and gives the maximum fidelity, according to Eq. (2). +If p is not feasible anymore because it contains edges that have been removed +from G′ discard it and move to the next shortest path. If Pa = ∅ remove a +from the active list L and continue from Step 1. +4. Determine the gross rate to be assigned to the current application a along +path p at this round as R = min {δa, mine∈p Ce} ≥ 0. The corresponding net +rate will be r = R · q|p|−1, as per Eq. (1). +5. Remove R from the capacity of all the edges along the path p. Remove from +G′ all the vanishing edges. +6. Update δa ← δa − R. If δa = 0 restart from Step 1, otherwise continue from +Step 3. +2.3. Simulation methodology and tool +We conclude the section by describing the methodology and tool adopted +for the performance evaluation in Sec. 4 and Sec. 5. +Like in [13], we use a Poisson Point Process (PPP) to generate the position +of an average of µ nodes in a flat square grid with edge size 60 km; a link is +added between two nodes with probability plink = 0.5 if their Euclidean distance +is smaller than a threshold τ. The capacity of each link is drawn from a r.v. +uniformly distributed between 1 Bell pair/s and 400 Bell pairs/s, as in [10]. The +initial fidelity of Bell pairs is ¯F = 0.95, which is widely used in the literature, +and the entanglement swapping success probability is q = 0.5, which is the +best value that can be obtained with linear optics components [11]. Based on +previous results in [6] we have selected two representative topologies: +– dense: µ = 100, τ = 20 km; +– sparse: µ = 50, τ = 15 km. +The following metrics are used to evaluate the performance: +– The net rate of the apps, i.e., the number of EPR-pairs that the end-points +can consume in the unit of time, which is a direct operational measure from +the point of view of the end users; +7 + +– The max-min fairness, which is the difference between the top and lowest net +rates assigned to the apps. +– The fidelity, weighted for each app on the net rate assigned to the correspond- +ing peer, which impacts on the accuracy and convergence of the distributed +QC applications. +– The inter-class unfairness index, for a class of apps with R priority weights +ρj, provided that each app is allocated net rate ri, defined as: +� +� +� +� +R +� +i=2 +� ri +r1 +− ρi +ρ1 +�2 +(3) +This measures the distance of the net allocations with respect to an ideal case +where the proportions between rates is exactly the same as the proportions +between priorities. +We used a Monte Carlo approach: for any combination of the parameters +under study, we simulated 6,000 drops with randomly generated networks and +random workload. Statistical significance has been verified for all the metrics +in the experiments performed, but we seldom include error bars in plots for +better readability. The simulation tool used is a custom simulator, developed +in C++ and using the Boost Graph Library, available as open source under a +MIT license on GitHub: +https://github.com/ccicconetti/quantum-routing/ +For full reproducibility, the repository also includes the scripts to run the +experiments, as well as the artifacts obtained and the Gnuplot files to produce +the plots: see tag v1.5, experiments labeled 004 and 005. +3. Related Work +The literature on quantum networking and distributed QC is not vast: even +though the basic ingredients have been known since a long time ago —consider +8 + +for instance the seminal paper by Bouwmeester et al. +on quantum telepor- +tation [14] published on Nature in 1997— only recently there have been in- +vestments in an order of magnitude sufficient for technology to take off. This +revamped interest has triggered new research activities in this area, briefly re- +viewed below. +In general terms, the problem of quantum routing is formulated as follows: +given a network of quantum nodes (repeaters or computers) and a set of traffic +flows identified by their sources, destinations, and application requirements (e.g., +the minimum fidelity), find the “best” paths that fulfill the constraints. Some +works have studied the problem by reusing the findings in the area of routing +in classical networks. +Van Meter et al. +proposed a quantum version of the +famous Dijkstra’s shortest path algorithm, which was shown to give very good +performance with an appropriate selection of the routing metric that considers +the specific properties of quantum networks [15]. More recently, Caleffi et al. +have proposed a slightly less efficient variation of Dijkstra’s algorithm that can +work with non-isotonic routing metrics, which they have advocated to provide +superior performance in selected use cases [16]. +Dijkstra’s algorithm is also +the subject of [17], where the authors lay some mathematical foundations that +allow them to derive upper bounds of performance in specific network topologies, +including grid and ring. +A different direction is explored by Pant et al., who studied the distribution +of routing information to the nodes [18]; for this they propose a time-slotted +approach: in the first part of the slot every repeater tries to create a local entan- +glement with all its neighbors, then in the second part the paths are established +as instructed by a centralized authority. One interesting aspect of the paper +is that multiple paths are selected for the same (source, destination) to maxi- +mize the rate of end-to-end EPR-pairs. We have also adopted this time-slotted +model in [19], where we have investigated the issue of “scheduling” of traffic +flows, i.e., determining the order in which to assign paths to pending requests, +in case the network resources are not sufficient to serve them all. This prob- +lem is called “distribution” in [13], where the authors formulate it as an Integer +9 + +Linear Programming (ILP), for which they derive closed formula performance +bounds in the case of a homogeneous chain of quantum repeaters. The issue +is also addressed in [10], where the authors have proposed to split the overall +quantum routing problem in two to reduce the computational complexity: first, +they determine the rates achievable by the traffic flows under the given network +constraints using an approach based on multi-commodity flow optimization, +then they map these rates to paths. The paper adopts a network model using +probabilistic entanglement swapping, which we reuse in this work (described in +Sec. 2). +An important reference for our study is [20], where the authors study the +allocation strategy of traffic flows for which the paths have been pre-determined: +they do so by borrowing the fairness concept from data networks and re-using +traditional algorithms from the relevant literature. In our paper, we also bor- +row from the same literature, though we apply the concepts to a different +class of applications, as it will be clear in the next section. +As a matter of +fact, all the scientific works cited above have focused on point-to-point traffic +flows, while in this paper we focus on a different type of traffic that is more +suitable to model distributed QC, with distinguishing features that do not al- +low the reuse of state-of-the-art solutions. Rather, we claim that any existing +routing/allocation/scheduling solutions should work in parallel to our proposed +scheme to provide an effective resource allocation to each of the two traffic +classes. +In addition to mere routing aspects, system-wide studies have also been +published. We mention [21], which is a compendium of several previous studies +from the same authors that illustrates an overall architecture of the Quantum +Internet, also including application, protocol, and deployment aspects at a high +level. On the other hand, other works have focused on specific components, +which are complementary to the research activity presented, e.g., [22] on con- +gestion control in transport protocols and [23] on the link layer, with a focus on +hardware and physical-layer considerations. +Furthermore, some research groups have been working to define the basic +10 + +principles of distributed QC. Parekh et al. have defined an elegant frame- +work for the parallel execution of a broad class of quantum algorithms on mul- +tiple nodes [5], both using remote entanglement and with Local Operations and +Classical Communication (LOCC) only, also studying in depth three classes +of algorithms: variational quantum eigensolver, low-depth quantum amplitude +estimation, and quantum k-means clustering. In [24] the authors address the +problem of the efficient compilation of circuits for distributed QC by considering +that some gate operations will be executed remotely, hence with much different +latency and reliability than on-chip operations. The research of Dahlber et al. +moved in the same direction and went as far as defining a set of low-level in- +structions (called NetQASM) for distributed QC systems seamlessly supporting +local and remote gates [25]. These works confirm that there is a growing interest +in distributed QC, which is a motivation for our work. +On another line of research, solutions have been proposed to trade capacity +for fidelity, by using purification (or distillation) techniques [26]. +In brief, +they consist in entangling multiple pairs of qubits with low fidelity and then +collapsing them into a single one with high fidelity. We do not consider network- +level purification in this work to remain consistent with the positioning of our +contribution within the realm of 1G-repeater quantum networks. End-to-end +purification is also possible, that is the operation is performed by quantum +computers after the qubits have been entangled all along the path(s). This is +studied, e.g., in [27], where the authors propose a quantum routing algorithm +that maximizes the rate of EPR-pairs, while deciding not only the paths but +also the purification patterns. These works complement our contributions since +they operate on constant-rate point-to-point flows only and they do not take into +account network provisioning issues. +Finally, in line with the vast majority of prior works, we only consider bi- +partite entanglement, i.e., made of two qubits, each situated in a quantum +computer. +While there are some promising theoretical studies on repeater- +assisted multi-partite entanglement, i.e., involving more than two qubits (e.g., +[28]), the research in that area is in still its infancy. One noteworthy contribution +11 + +is [29], where the authors propose to adopt n-fusion of bi-partite entanglements +to create higher level entanglements between n > 2 quantum computers. An +appealing property that they demonstrate, under some assumptions, is that the +entanglement rate between nodes remains constant with increasing distance, +in number of hops. Ways to exploit this phenomenal quality are still under +study. Multi-partite entanglement in a quantum network is generated starting +from elementary bi-partite entanglements, which is the subject of this work. +4. Service Differentiation +In this section we extend the study in [6] by analyzing the QDRR algorithm +along two directions which have remained so far uninvestigated: service differ- +entiation by assigning apps different ρ values (Sec. 4.1) and apps with different +fidelity thresholds F min (Sec. 4.2). In all the simulations in this section the +peers are selected as follows: for each app i on node v we draw at random be- +tween 2 and 4 candidate nodes by sampling in a uniform manner from the set +of all nodes that are reachable from v in 2–7 hops. For benchmarking purposes, +QDRR is compared to two baseline algorithms: random and best-fit. The Step 0 +in Sec. 2.2 is the same for all the algorithms, that is for each app we find k = 4 +shortest paths to reach any peer. All the algorithms then loop through all the +possible paths for all candidates for each app until there are no more feasible +paths to be assigned, but they differ in how they do so: QDRR is descriped by +Steps 1–6 in Sec. 2.2 (and in far more details in Sec. IV-C in [6]), while: +– Random: at each iteration one app with remaining paths is chosen at random +and assigned its shortest path among any of its peers, which is allocated +the maximum rate along the path. Random is representative of allocation +algorithms that provide fair access to the quantum network resources in a +per-app manner. +– Best-fit: at each iteration, select the app with the shortest path to reach +one of its peers and allocate the maximum rate along that path. Best-fit is +representative of allocation algorithms that strive to maximize the efficiency, +12 + +that is the ratio between net entanglement rate and the quantum network +capacity allocated. + 0 + 10 + 20 + 30 + 40 + 50 + 60 + 70 + 0 + 100 + 200 + 300 + 400 + 500 + 600 + 700 + 800 + 900 + 1000 +Iterations (x1000) +Load (#apps) +Dense|Random +Dense|BestFit +Dense|QDRR +Sparse|Random +Sparse|BestFit +Sparse|QDRR +Figure 2: Number of iterations (expect Step 0 in Sec. 2.2) with random, best-fit, QDRR, in a +dense vs. sparse topology, when increasing the number of apps with ρ ∈ {1, 2, 4}. +Unlike QDRR, both random and best-fit can be considered greedy algo- +rithms, since they never backtrack to a previously selected combination of +(app, peer, path), which is always allocated as much throughput as possi- +ble. Therefore, their worst-case computational complexity (expect Step 0) is +O (k|A| log |A|E[Wi]): k is the number of shortest paths selected in Step 0; |A| +is the number of apps; log |A| takes into account the random selection or the ex- +traction from an sorted data structure, respectively for the random and best-fit +resource allocation algorithms; and, E[Wi] is the average number of peers per +app. The complexity of QDRR is discussed in [6] and it depends on the choice +of φ. To give an idea of the relative average complexity between QDRR and +random/best-fit, we report their number of iterations in the simulations dis- +cussed in Sec. 4.1 below in Fig. 2. As can be seen, in a sparse scenario the time +complexity of QDRR is only marginally higher than that of greedy algorithms +random and best-fit, but it becomes clearly higher in a dense scenario. If this is +an issue, the value of φ can always be tuned to reduce the number of iterations, +trading off fairness for speed. +4.1. Different traffic priorities +In a first batch of results we increase the load of the network from 10 to +1000 apps. For every app, its priority weight ρ is drawn randomly in {1, 2, 4}, +13 + + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 0 + 100 + 200 + 300 + 400 + 500 + 600 + 700 + 800 + 900 + 1000 +Residual/total capacity +Load (#apps) +Dense|Random +Dense|BestFit +Dense|QDRR +Sparse|Random +Sparse|BestFit +Sparse|QDRR +Figure 3: Ratio between the residual capacity and the total capacity with random, best-fit, +QDRR, in a dense vs. sparse topology, when increasing the number of apps with ρ ∈ {1, 2, 4}. +while F min is the same for all and equal to 0.7. As shown in Fig. 3, for all +the allocation algorithms and topologies the relative residual capacity decreases +with a sub-linear trend as the load increases, which is due to the exponential +relation between the net rate, in EPR-pairs/s, and the number of hops as per +Eq. (1). +The utilization is higher in a sparse topology, while the difference +between the allocation algorithms is negligible. In the following we report only +the results in a dense topology, due to limited space; the complete results can +be retrieved from the public GitHub repo above. + 120 + 140 + 160 + 180 + 200 + 220 + 240 + 260 + 280 + 300 + 320 + 0 + 100 + 200 + 300 + 400 + 500 + 600 + 700 + 800 + 900 + 1000 +Max-min fairness (EPR pairs/s) +Load (#apps) +Random (class avg) +Best-fit (class avg) +QDRR (ρ = 1) +QDRR (ρ = 2) +QDRR (ρ = 4) +Figure 4: Max-min fairness with random, best-fit, QDRR, in a dense topology, when increasing +the number of apps with ρ ∈ {1, 2, 4}. +We begin by showing the max-min fairness in Fig. 4. Since random and +best-fit do not differentiate based on the ρ values, for them we show an aggre- +gate average, while we keep separate curves for QDRR. With very low loads, the +max-min fairness is good, i.e., low, for all allocation algorithms, because there is +14 + +little contention on resources. However, as the load increases, the max-min fair- +ness increases steeply and the behavior is significantly affected by the allocation +algorithm: with random the curve reaches a peak, which then slowly decreases +towards high loads; best-fit performs worst, as expected, by continuing to in- +crease, even though only slightly after the initial spur; for all traffic categories, +QDRR provides increasingly better fairness as the load increases. The latter can +be explained as follows. When there are few apps, it is likely that there are not +many shared nodes/paths, hence QDRR does not really have a chance to dis- +tribute the resources proportional to the apps’ weights; on the other hand, with +more apps, it is increasingly easier for QDRR to enforce priorities by regulating +the resources in common. + 1 + 1.2 + 1.4 + 1.6 + 1.8 + 2 + 2.2 + 2.4 + 2.6 + 2.8 + 3 + 3.2 + 0 + 100 + 200 + 300 + 400 + 500 + 600 + 700 + 800 + 900 + 1000 +Inter-class unfairness +Load (#apps) +Random +Best-fit +QDRR +Figure 5: Inter-class unfairness index with random, best-fit, QDRR, in a dense topology, when +increasing the number of apps with ρ ∈ {1, 2, 4}. +To better show the service differentiating behavior of QDRR, we show the +inter-class unfairness index, as defined in Eq. (3), in Fig. 5. It is clear that +QDRR is the only allocation algorithm providing the apps with a clear service +differentiation, which improves as the load increases for the same reason above. +Finally, in Fig. 6 we show the net rate/app, in EPR-pairs/s: even though +it slowly decreases for all the resource allocation algorithms, we can see that +random and best-fit achieve better rates. Therefore, QDRR is effective in dif- +ferentiating service across apps with different priority weights, but this incurs a +cost, in terms of net rate. +15 + + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 0 + 100 + 200 + 300 + 400 + 500 + 600 + 700 + 800 + 900 + 1000 +Net rate/app (EPR pairs/s) +Load (#apps) +Random (class avg) +Best-fit (class avg) +QDRR (ρ = 1) +QDRR (ρ = 2) +QDRR (ρ = 4) +Figure 6: Net rate/app with random, best-fit, QDRR, in a dense topology, when increasing +the number of apps with ρ ∈ {1, 2, 4}. + 50 + 100 + 150 + 200 + 250 + 300 + 350 + 0 + 100 + 200 + 300 + 400 + 500 + 600 + 700 + 800 + 900 + 1000 +Max-min fairness (EPR pairs/s) +Load (#apps) +Dense|Random +Dense|Best-fit +Dense|QDRR +Sparse|Random +Sparse|Best-fit +Sparse|QDRR +Figure 7: Max-min fairness with random, best-fit, QDRR, in dense vs. sparse topologies, when +increasing the number of apps with F min ∈ {0.7, 0.8, 0.9}. +16 + +4.2. Different fidelity thresholds +We now report the results obtained in a new batch, which follows the same +direction as above, but we set ρ = 1 for all apps and draw randomly the mini- +mum fidelity F min from {0.7, 0.8, 0.9}, instead. We show the max-min fairness +in Fig. 7, for both topologies. Like in Sec. 4.1, QDRR achieves significantly +better performance than random and best-fit, the latter performing worst. + 0 + 20 + 40 + 60 + 80 + 100 + 120 + 0 + 100 + 200 + 300 + 400 + 500 + 600 + 700 + 800 + 900 + 1000 +Net rate/app (EPR pairs/s) +Load (#apps) +Dense|Random +Dense|Best-fit +Dense|QDRR +Sparse|Random +Sparse|Best-fit +Sparse|QDRR +Figure 8: Net rate/app with random, best-fit, QDRR, in dense vs. sparse topologies, when +increasing the number of apps with F min ∈ {0.7, 0.8, 0.9}. +However, as can be seen in Fig. 8, also with a mix of fidelity thresholds, the +net rate/app of QDRR is slightly less than that with both greedy allocation +strategies, which confirms the trade-off already identified in the previous batch +of experiments. It is worth noting that such a trade-off is well-known also in +different contexts: for instance, in cellular systems, greedy scheduling algorithms +that prioritize user terminals with good channel conditions (often known as “max +C/I”) are bound to provide a higher cell throughput at the cost of an inferior +fairness compared to milder strategies such as Proportional Fair [30]. +In conclusion, like for heterogeneous weights, QDRR can provide service dif- +ferentiation to apps with different fidelity thresholds, but the net rate achievable +is slightly reduced compared to alternatives that do not differentiate. +5. Fair Sharing +In this section we address a problem that is preliminary and complementary +to that defined in our previous work [6] and investigated in Sec. 4 with differen- +17 + +tiated service. So far we have assumed that all nodes are equal, and each host +is wishing to cooperate with a given set of workers, without elaborating further +on how such a set is selected; for performance evaluation purposes, such a set +was selected randomly from candidates depending only on the distance as per +the specific scenario simulated. In the following, instead, we address specifically +this issue: indeed, how does one decide which are the possible workers of a host +node? +end users +data centers +Figure 9: Example of quantum network with three end users, labeled from u1 to u3, wishing to +host distributed quantum computing algorithms with data center nodes d1 or d2, represented +with multiple co-located circles to indicate that they are expected to be more powerful than end +users and, possibly, they might have a more complex internal structure that is not elaborated +further in this paper; the other nodes in G(V, E) participate to the end-to-end entanglement of +qubits as intermediate hops. Like in Sec. 2.1/Fig. 1, the network is characterized by capacity +Cij of the link between nodes i and j, initial generation fidelity F, and entanglement swapping +success probability q. +We adapt our system model to the new landscape by specializing the role of +nodes. As illustrated in the example in Fig. 9, we assume that nodes can be of +three types: (i) end users (ui): they act as the “home QCs” of customers wish- +ing to run quantum algorithms on them, also exploiting quantum computation +resources offered by other nodes through distributed quantum computing; a cus- +tomer operates the end user via a classical computer for, e.g., input preparation, +18 + +circuit compilation, and quantum network resource reservations; (ii) data centers +(dj): these are QCs that can provide customers with extra quantum computa- +tion capacity to be added to their respective end users for the purpose of solving +bigger instances of their problems via distributed quantum computing, exploit- +ing end-to-end entanglement of qubits through an underlying quantum network; +(iii) intermediate nodes: quantum repeaters who do not consume or offer QC +capacity but contribute to the quantum network by performing entanglement +swapping between links as instructed by the resource allocation algorithm (e.g., +QDRR). A node can play any combination of the three roles above. We assume +that end user ui may perform distributed quantum computing with any combi- +nation of data centers {dj}, following commercial agreements that are out of the +scope of this work. All other quantum network assumptions in Sec. 2.1 remain +the same. In Sec. 5.1 we formulate the problem in mathematical terms and +propose a solution, which is then evaluated via simulation in Sec. 5.2, compared +to two alternatives. +5.1. Quantum Workers’ Assignment Problem +In its most general formulation, the problem of how to best assign each host +a set of workers depends on several factors that may be not known or under the +control of the quantum network operator. They include, for instance: the quan- +tum algorithms to be run, the different characteristics of the end user and data +center QCs, the schedule of the task execution, not to mention administrative +factors (contracts, partnerships, billing issues) and technical constraints (do the +QCs need to have the same hardware or software?). Since both quantum net- +working and distributed quantum computing are in their infancy, we consider +unrealistic to address the problem under such general settings. Rather, we focus +on aspects that are captured by our network model (in Sec. 2.1) with the goal +of providing an initial understanding of the problem, to be used as a stepping +stone by future studies as the technologies involved become more mature. Our +formulation, in natural language, is the following: +Quantum Workers’ Assignment Problem (QWAP): find the sets of workers, se- +19 + +lected from the data center nodes, to be assigned to each host, from the end +user nodes, so that the overall profit of the hosts is maximum while balancing +the load of data centers. +The problem can be formulated in a formal manner once we define the no- +tions of “profit” and “load balancing”. Based on the prior works in the literature +(see Sec. 3), we consider the net entanglement rate in Eq. (1) as the profit, i.e., +between two possible data centers d1 and d2 considered as candidate workers +for end user u, we prefer the one that potentially achieves the highest net en- +tanglement rate. On the other hand, we introduce load balancing as follows. +First, we define the system parameter W as the target number of workers per +end user. Then, we force each data center to be assigned as worker to at most +B end users, where B is determined as the minimum value that allows this con- +straint to be provided with given W, Nu end users, and Nd data centers. This +way, we force an even load across data centers. Note that this formulation can +be trivially extended to the case where data centers have different capabilities +by introducing appropriate weights, which we do not consider in this work to +keep the notation succinct. +Assuming without loss of generality, again to simplify notation, that all end +users have one and only one request to run distributed quantum computing, the +QWAP can be expressed as an optimization problem with objective function: +max +Nu +� +u=1 +Nd +� +d=1 +πudxud +(4) +such that: +20 + +Nu +� +u=1 +xud ≤ B +d = 1, . . . , Nd +(5) +Nd +� +d=1 +xud ≤ W +u = 1, . . . , Nu +(6) +xud ∈ {0, 1} +(7) +B = +�Nu · W +Nd +� +(8) +where the profit πud between end user u and data center d is defined by: +πud = +� +� +� +� +� +0 +if ∀p ∈ Pu,d : F(p) ≤ F min +u +rud¯p +where ¯p = arg maxp +� +rudp|F(p) ≥ F min +u +� , +(9) +where F(p) is the fidelity of the end-to-end entanglement along the path p, +according to Eq. (2), F min +u +is the minimum fidelity requested by end user u, +Pu,d is the set of all paths between u and d in G(V, E), and the net rate rudp +between end user u and data center d along path p is as follows: +rudp = max(i,j)∈p {Cij} +q|p|−1 +. +(10) +The output assignment integer variables, as per Eq. (8), are the xud, with +the constraints as follows: Eq. (5) ensures that no data center is assigned more +than its fair share of B users, where B is computed via Eq. (8); Eq. (6) ensures +that no end user is assigned more than W workers. The profit πud, defined +through Eqs. (9–10), corresponds to the maximum net rate of EPR-pairs/s that +can assigned to end user u along any path towards data center d that fulfils its +minimum fidelity requirement. Before proceeding, we state two key observations +about the QWAP. +Observation#1. +In a general graph G(V, E), the number of paths between +two nodes can be exponential with the size of the graph. For instance, in a +complete graph this number is ⌊(V − 2)!e⌋. Therefore, the preparation of the +problem input in Eq. (9) might be very computation-intensive in practice. In +21 + +the evaluation below, we adopt a reasonable approximation: rather than finding +all the paths Pud between nodes u and d, we use a reduced set P¯k +ud that consists +of the ¯k shortest paths (¯k = 10 in the simulations in Sec. 5.2). The rationale is +that long paths have a small chance of being selected as ¯p in the second branch +of Eq. (9), because the net rate decreases exponentially with the path length as +per Eq. (10). +Observation#2. When W = 1, then Eqs. (4–10) above can be trivially trans- +formed into an assignment problem, whose optimum solution can be found ef- +ficiently. +With W > 1, however, the QWAP becomes a “multiple knapsack +problem”, which is NP-hard, but for which several efficient heuristics are well- +known in the operations research literature (e.g., [31]). +Based on the two observations above, we propose our load balancing al- +gorithm to solve the QWAP, which we implemented in our simulator and eval- +uated in the next section. The idea of the load balancing algorithm is to find +the optimal allocation for the first worker of each end user using the Hungar- +ian algorithm [32], which finds the best (exact) solution in assignment problem +instances; then, it progresses by considering one more worker at a time, until +W, each time invoking the Hungarian algorithm again on the data centers with +residual slots. The algorithm is greedy because it never backtracks prior deci- +sions and it always terminates after a fixed number of iterations. More formally, +the load balancing algorithm consists of the following three steps: +1. Prepare the problem input, in particular the profits πud, by finding up to ¯k +shortest paths between u and d in G(E, V ) using Yen’s algorithm [33]. +2. Determine B based on Nu, Nd, and W using Eq. (8). +3. Arrange the output of Step 1 in a profit matrix where the rows are the end +users and there are B columns for each data center, i.e., the matrix size is +Nu × BNd. Then run W iterations of the Hungarian algorithm. After each +iteration, set πud ← 0 in all columns where a data center has been assigned +(avoids that the constraint Eq. (5) is violated) and in all cells that refer to the +same pair u, d that has been assigned (avoids that a user is assigned multiple +22 + +times the same data center). +5.2. Evaluation +Yen's algorithm using Dijkstra with Fibonacci heap +preparation +execution +for each end user +and data center +number of shortest +paths to search +Hungarian algorithm +number of iterations +Dijkstra with Fibonacci heap +for each end user +and worker +for each end user and worker +random selection +of data center +a) +b) +c) +Figure 10: Worst case time complexity of a) the load balancing algorithm to solve the QWAP +in Sec. 5.1 vs. the comparison algorithms b) random and c) shortest path. +In this section we evaluate the load balancing algorithm defined in Sec. 5.1 +using the simulator described in Sec. 2.3. End users and data centers are selected +randomly from the set of nodes V , with the following composition of the nodes: +10% end users, 10% data centers, 80% intermediate nodes. +As comparison +algorithms, we define: +– Random, which assigns each end user W data centers at random, thus it +maximizes fairness. +– Shortest path, which assigns each end user the W data centers that are closest +in G(E, V ), in number of hops, thus it maximizes the net rate. +After the assignment, the resources are allocated using QDRR. +We report in Fig. 10 the worst case time complexity of the three algorithms, +where we assume that the shortest path is computed using Dijkstra’s algorithm +with the help of a Fibonacci heap to keep edges sorted [34]. As can be seen, the +23 + + 50 + 100 + 150 + 200 + 250 + 300 + 350 + 400 + 450 + 500 + 1 + 2 + 3 + 4 + 5 +Net rate/app (EPR-pairs/s) +W +Random +Shortest path +Load balancing + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 + 4.5 + 1 + 2 + 3 + 4 + 5 +Max-min num users per data center +W +Random +Shortest path +Load balancing +Figure 11: Net rate/app (top) and max-min number of users per data center (bottom) with +random, shortest path, and load balancing, in a dense topology, with 10 apps, when increasing +W from 1 to 5. +load balancing algorithm is far more complex than random and shortest-path, +in both the preparation and the execution phases; random, in particular, does +not even depend on the graph size. However, in the following we will see that +the added complexity brings benefits that can be of potential interest to the +future quantum network operators. +In Fig. 11 (top) we show the net rate/app with Nu = 10 and W increasing +from 1 to 5, in a dense topology. As can be seen, the random algorithm per- +forms poorly, because it does not consider at all the network topology. On the +other hand, shortest path and load balancing perform similarly, with the latter +exhibiting a higher net rate with small values of W. In Fig. 11 (bottom) we +plot a measure of the spread of resources, as the max-min number of users per +data center. Load balancing performs consistently and significantly better than +both random and shortest path, with the latter exhibiting the highest unfair- +ness. We note that our conclusions are limited to the settings of the scenarios +24 + +simulated; in particular, different topologies or link capacity distributions can +lead to situations where the gap between load balancing and either random or +shortest path is reduced significantly. However, a crucial advantage of our pro- +posed solution, compared to its alternatives under test, is that it can adapt to +different settings, thus it can perform well even when the scenario is not known +or changes dynamically. + 0.82 + 0.84 + 0.86 + 0.88 + 0.9 + 0.92 + 0.94 + 0.96 +load balancing +random +shortest-path +load balancing +random +shortest-path +Fidelity +Dense|10 apps +Dense|20 apps +Sparse|10 apps +Sparse|20 apps +Figure 12: +Fidelity with random, shortest path, and load balancing, in dense vs. sparse +topologies, with 10 vs. 20 apps and W = 1. +The fidelity is shown in Fig. 12, with Nu = {10, 20} and in dense vs. sparse +topologies. The number of end users/apps, i.e., Nu, does not affect the perfor- +mance in a noticeable manner. On the other hand, the fidelity is generally lower +in sparse topologies, as expected. Random performs worse, while load balanc- +ing and shortest path give similar results, with the former performing slightly +worse only in the sparse case. This can be explained by following the same line +of reasoning for the net rate/app above. +To conclude the analysis, we modify the mix of nodes. +In one batch of +simulations we increase the ratio of end users from 10% (like in the results so +far) to 50%; in another one we do the same for data centers. Results are shown +only for load balancing in Fig. 13, in terms of the net rate, with Nu = 15 and +W = 3. As the ratio of data centers increases (green curves), the net rate/app +increases almost linearly, as well. On the other hand, increasing the number +of users (blue curves), only provides a sub-linear performance improvement. +This is because the former case corresponds to increasing the physical resources +25 + + 50 + 100 + 150 + 200 + 250 + 300 + 350 + 400 + 450 + 500 + 550 + 600 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 +Net rate per app (EPR-pairs/s) +Ratio of (end users | data centers) +[end users]|Dense +[end users]|Sparse +[data centers]|Dense +[data centers]|Sparse +Figure 13: Net rate/app with load balancing, in dense vs. sparse topologies, with 15 apps and +W = 3, when increasing the fraction of nodes as either end users or data centers. +provided to the users, while the latter only to a more uniform distribution of the +same resources. The conclusions are the same for dense and spare topologies, +though the net rate for the latter is significantly lower. +In conclusion, assigning data centers to end users through the load balancing +algorithm, which provides an approximate solution of the QWAP, achieves an +even distribution of resources, better than both random and shortest path as- +signment, without compromising on the net rate and fidelity of the end-to-end +entanglement paths. +6. Conclusions +In this paper we have studied two open issues in quantum networking for +distributed quantum computing. First, we have assessed through simulation +the effectiveness of the QDRR resource allocation algorithm [6] in handling sce- +narios where applications have different priority weights or minimum fidelity +requirements. Second, we have defined a novel problem involving the selection +of workers for a set of nodes hosting computation, called the Quantum Work- +ers’ Assignment Problem (QWAP), which we have modeled as an optimization +problem and for which we have proposed a heuristic called “load balancing”. +The latter has been evaluated in comparison to alternatives seeking to maxi- +mize only fairness and the net rate of end-to-end entanglement, respectively, and +the results have shown that load balancing achieves an excellent compromise in +26 + +terms of the two metrics. The source code and simulation scripts are publicly +available to the community. +Further open research areas are: the use of purification to increase fidelity +at the expense of capacity; modeling distributed QC applications to understand +their characteristic time scales and requirements; integration with link layer pro- +tocols; incorporation in the simulation of more realistic models for the quantum +channel and repeaters. +Acknowledgment +Work co-funded by EU, PON Ricerca e Innovazione 2014–2020 FESR/FSC +Project ARS01_00734 QUANCOM, and European High-Performance Comput- +ing Joint Undertaking (JU) under grant agreement No 101018180 HPCQS. The +paper reflects only the authors’ view and the funding agencies are not respon- +sible for any use that may be made of its content. +References +[1] J. Preskill, Quantum computing 40 years later, arXiv:2106.10522 [quant- +ph]ArXiv: 2106.10522. +[2] J. Sevilla, C. J. 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ACM 34, 3 (July 1987), 596–615. +30 + diff --git a/5tE2T4oBgHgl3EQfkQe0/content/tmp_files/load_file.txt b/5tE2T4oBgHgl3EQfkQe0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b4684ebd71e70cd18f8128ccf26c5b0bb9b8e19c --- /dev/null +++ b/5tE2T4oBgHgl3EQfkQe0/content/tmp_files/load_file.txt @@ -0,0 +1,773 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf,len=772 +page_content='Service Differentiation and Fair Sharing in Distributed Quantum Computing Claudio Cicconettia,∗, Marco Contia, Andrea Passarellaa aIIT, National Research Council, Pisa, Italy Abstract In the future, quantum computers will become widespread and a network of quantum repeaters will provide them with end-to-end entanglement of remote quantum bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' As a result, a pervasive quantum computation infrastructure will emerge, which will unlock several novel applications, including distributed quan- tum computing, that is the pooling of resources on multiple computation nodes to address problem instances that are unattainable by any individual quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In this paper, we first investigate the issue of service differentiation in this new environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Then, we define the problem of how to select which computation nodes should participate in each pool, so as to achieve a fair share of the quantum network resources available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The analysis is performed via an open source simulator and the results are fully and readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Keywords: Distributed Quantum Computing, Quantum Internet, Quantum Routing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Introduction Quantum Computing (QC) exploits the properties of matter at very small scale to solve some problems much faster than a classical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Even though QC has been theorized 40 years ago [1], only recently the technology evolution and a spur of investments have made it possible to obtain practical ∗Corresponding author Email addresses: c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='cicconetti@iit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='cnr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='it (Claudio Cicconetti), m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='conti@iit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='cnr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='it (Marco Conti), a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='passarella@iit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='cnr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='it (Andrea Passarella) Preprint submitted to Elsevier January 11, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='03977v1 [quant-ph] 10 Jan 2023 results and speculate about approaching mass deployments [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' QC is being al- ready used in the chemical and pharmaceutical industry, while new applications are being progressively unlocked in material science, Machine Learning (ML) and engineering optimization, production and logistics, post-quantum security [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Essentially, the computational advantage of QC stems from the proper- ties of superposition and entanglement of the qubits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', the “quantum bits”): (i) superposition, which means that a qubit can be in a combination of multiple states at the same time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' and (ii) entanglement, which is a property exhibited by a set of qubits that maintain their correlation even separated in space or time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We can expect that the computational power of a single QC will remain relatively limited in the near future, due to scalability issues in maintaining a very stable and controlled environment to cope with the flimsy nature of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' On the other hand, the realization of the Quantum Internet is progress- ing steadily [4], with the long-term goal to enable the entanglement of qubits that reside in QCs across geographical distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' With the diffusion of QCs and their gradual interconnection via quantum networks, a pervasive infrastructure will therefore materialize, with the potential to combine opportunistically resources from multiple QCs for the execution of specialized algorithms in a distributed fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' A general framework for such distributed quantum computing has been proposed in [5], where the authors propose practical examples, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', a quantum version of the k-means clustering, which is used in unsupervised ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' A preliminary analysis of the allocation of resources among multiple quantum computers based on the characteristics of the underlying quantum network has been presented in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In the same work, we have also proposed a practical solu- tion inspired by a well-known algorithm in classical data networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', Deficit Round Robin (DRR) [7], which we have evaluated through simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We have found that some fundamental properties of quantum networks immensely impact on the provisioning of resources, which calls for new research in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' This is especially manifest when considering networks of first-generation (1G) quantum repeaters [8], which do not have error-correction capabilities and are expected to be next in line for the industrialization and mass deployment 2 in the following years [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The contribution of this paper is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We evaluate the performance of the resource allocation algorithm proposed in [6] with differentiated services coexisting within the same quantum net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Furthermore, we do so by comparing the performance with two alter- native algorithms, inspired by equivalents in classical problems with similar settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' This extends and completes the preliminary analysis in our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We define a new problem related to fair share of resources in a quantum network among multiple applications wishing to perform distributed QC: how to best choose the peers among those available?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' After introducing a mathematical formulation of the problem, we propose a greedy approxima- tion algorithm, which is then evaluated thoroughly and compared to two alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' All the experiments in the paper are carried out via simulations, which are fully reproducible and publicly available on GitHub, including the simulation software source code, the scripts to run the analysis, and the artifacts and plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The rest of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We summarize the system model assumptions and findings in [6] in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We then review the related work on routing in quantum network in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The main contributions are reported in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 4, where we study the service differentiation, and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 5, where we tackle the problem of fair sharing of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 6 concludes the paper and identifies the most important open research directions in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' System Model In this section, we describe in short the quantum network abstract model adopted in the paper (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='1), the resource allocation algorithm proposed in [6] (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2), and the simulation methodology and tool (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' For more details, we refer the reader to [6], in particular sections II and IV, and references within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 3 end-to-end entanglement path Figure 1: Quantum network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' End-to-end entanglement can be established between two nodes s and d for which there is a path in G(V, E), where the intermediate nodes perform entanglement swapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' F is the fidelity with which the local link EPR pairs are generated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' q is the measurement success probability, which affects the entanglement swapping procedure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Cij is the capacity of edge eij, in EPR-pairs/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Quantum network model The quantum network model is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 1 as a graph G(V, E), where nodes represent quantum devices (repeaters or computers), and edges represent direct quantum communication links between them [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We assume that maximally entangled EPR (Einstein–Podolsky–Rosen) pairs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', |Φ+⟩, are generated periodically at each link, and they have initial fidelity equal to ¯F ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='5, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The fidelity is a measure of how close a given quantum state is to a reference state, with 1 meaning that they are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Every edge eij ∈ E has a given capacity Cij, which is the rate of generation of EPR-pairs, which in turn depends on the physical characteristics of the quantum network devices and links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Quantum networks will offer the capability to produce entangled EPR-pairs between remote nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', nodes that are interconnected through intermediate hops, which will perform the procedure of entanglement swapping to this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Such a procedure is stochastic in nature, in 1G quantum repeaters, and we assume that it succeeds with probability q [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' An end-to- end EPR-pair can only be used in a meaningful manner if all the entanglement 4 swaps along the path have succeeded, which leads to the following formula to compute the maximum net rate that can be used by a quantum application consuming resources along a path p = {(s, v1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' , (vN, d)}: r(p) ≤ mineij∈p{Cij} q|p|−1 , (1) where |p| is the length of the path p, in number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Furthermore, en- tanglement swapping reduces the fidelity of the end-to-end entangled EPR-pair according to the following formula [12]: F(p) = 1 4 + 3 4 �4 ¯F − 1 3 �|p| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (2) We can say that r(p) and F(p) define the effective rate at which two end-points can transfer EPR-pairs, which is the logical equivalent of throughput in classical networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In our previous work [6] we have classified the quantum applications in two categories: – Flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' They are characterized by the need for two specific nodes to exchange a constant flow of EPR pairs in a point-to-point manner for the whole dura- tion of a session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' If the rate of EPR pairs falls below the requested amount, then the application Quality of Service (QoS) degrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Examples of such ap- plications are: clock synchronization and Quantum Key Distribution (QKD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' – Apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In this category we find distributed quantum computing applications, each characterized by a given quantum computer (host) running an algorithm that pools the resources of a number of other quantum computers (peers or workers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' There is no required EPR-pair rate, but the application wishes to consume as many EPR-pairs as possible to complete the execution faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Such kind of service is equivalent to best-effort or elastic traffic in a classical data network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Since the network operator might provide a differentiated service, we foresee that each app is also assigned a weight (ρ), which is a relative indication of how much throughput (in EPR-pair/s) it should be given in the long-term compared to another app with a different weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 5 For both categories, we foresee the minimum fidelity (F min) can be also a user requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Since in this work we focus on distributed quantum computing applications, whose traffic is better modeled by apps than flows, in the rest of the paper we do not consider further the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' QDRR resource allocation algorithm We report below a recap of the apps’ resource allocation algorithm in [6], which in the following will be called Quantum DRR (QDRR) as it was inspired by the well-known DRR algorithm [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The basic idea of QDRR is to provide all applications with a fair chance to be allocated a fraction of the quantum network resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' This is enforced by visiting the applications in a round robin: at each visit, the application can be allocated capacity across multiple paths towards its peers, up until a given amount that is proportional to the application’s weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Shortest paths are always preferred to longer ones, because they are more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' A more detailed explanation follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The algorithm has two system parameters, which are set based on our pre- vious results: k = 4 and the round size φ = 10 EPR-pairs/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' For a set of apps i ∈ A, each defined by a host node {hi} and a set of candidate peers Wi, QDRR consists of the following steps: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' ∀i ∈ A, ∀j ∈ Wi: find k shortest paths from hi to wij and add them to Pi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' at the end, each Pi contains up to k paths for each possible peer of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Initialize the active list of apps L with the identifiers of all the apps A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Copy the graph G(V, E) into a temporary copy G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' If L = ∅ terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Otherwise, let a be the next application to be visited in L in round robin order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Set the residual capacity that can be used by the current app a in this round to δa = φ ρa � i∈A ρi , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', a fraction of the round size φ proportional to its priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Select the shortest path p ∈ Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The shortest path is the one that requires the least amount of resources among those available for the current app, 6 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (1), and gives the maximum fidelity, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' If p is not feasible anymore because it contains edges that have been removed from G′ discard it and move to the next shortest path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' If Pa = ∅ remove a from the active list L and continue from Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Determine the gross rate to be assigned to the current application a along path p at this round as R = min {δa, mine∈p Ce} ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The corresponding net rate will be r = R · q|p|−1, as per Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Remove R from the capacity of all the edges along the path p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Remove from G′ all the vanishing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Update δa ← δa − R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' If δa = 0 restart from Step 1, otherwise continue from Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Simulation methodology and tool We conclude the section by describing the methodology and tool adopted for the performance evaluation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 4 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Like in [13], we use a Poisson Point Process (PPP) to generate the position of an average of µ nodes in a flat square grid with edge size 60 km;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' a link is added between two nodes with probability plink = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='5 if their Euclidean distance is smaller than a threshold τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The capacity of each link is drawn from a r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' uniformly distributed between 1 Bell pair/s and 400 Bell pairs/s, as in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The initial fidelity of Bell pairs is ¯F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='95, which is widely used in the literature, and the entanglement swapping success probability is q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='5, which is the best value that can be obtained with linear optics components [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Based on previous results in [6] we have selected two representative topologies: – dense: µ = 100, τ = 20 km;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' – sparse: µ = 50, τ = 15 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The following metrics are used to evaluate the performance: – The net rate of the apps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', the number of EPR-pairs that the end-points can consume in the unit of time, which is a direct operational measure from the point of view of the end users;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 7 – The max-min fairness, which is the difference between the top and lowest net rates assigned to the apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' – The fidelity, weighted for each app on the net rate assigned to the correspond- ing peer, which impacts on the accuracy and convergence of the distributed QC applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' – The inter-class unfairness index, for a class of apps with R priority weights ρj, provided that each app is allocated net rate ri, defined as: � � � � R � i=2 � ri r1 − ρi ρ1 �2 (3) This measures the distance of the net allocations with respect to an ideal case where the proportions between rates is exactly the same as the proportions between priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We used a Monte Carlo approach: for any combination of the parameters under study, we simulated 6,000 drops with randomly generated networks and random workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Statistical significance has been verified for all the metrics in the experiments performed, but we seldom include error bars in plots for better readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The simulation tool used is a custom simulator, developed in C++ and using the Boost Graph Library, available as open source under a MIT license on GitHub: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='com/ccicconetti/quantum-routing/ For full reproducibility, the repository also includes the scripts to run the experiments, as well as the artifacts obtained and the Gnuplot files to produce the plots: see tag v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='5, experiments labeled 004 and 005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Related Work The literature on quantum networking and distributed QC is not vast: even though the basic ingredients have been known since a long time ago —consider 8 for instance the seminal paper by Bouwmeester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' on quantum telepor- tation [14] published on Nature in 1997— only recently there have been in- vestments in an order of magnitude sufficient for technology to take off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' This revamped interest has triggered new research activities in this area, briefly re- viewed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In general terms, the problem of quantum routing is formulated as follows: given a network of quantum nodes (repeaters or computers) and a set of traffic flows identified by their sources, destinations, and application requirements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', the minimum fidelity), find the “best” paths that fulfill the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Some works have studied the problem by reusing the findings in the area of routing in classical networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Van Meter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' proposed a quantum version of the famous Dijkstra’s shortest path algorithm, which was shown to give very good performance with an appropriate selection of the routing metric that considers the specific properties of quantum networks [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' More recently, Caleffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' have proposed a slightly less efficient variation of Dijkstra’s algorithm that can work with non-isotonic routing metrics, which they have advocated to provide superior performance in selected use cases [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Dijkstra’s algorithm is also the subject of [17], where the authors lay some mathematical foundations that allow them to derive upper bounds of performance in specific network topologies, including grid and ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' A different direction is explored by Pant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', who studied the distribution of routing information to the nodes [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' for this they propose a time-slotted approach: in the first part of the slot every repeater tries to create a local entan- glement with all its neighbors, then in the second part the paths are established as instructed by a centralized authority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' One interesting aspect of the paper is that multiple paths are selected for the same (source, destination) to maxi- mize the rate of end-to-end EPR-pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We have also adopted this time-slotted model in [19], where we have investigated the issue of “scheduling” of traffic flows, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', determining the order in which to assign paths to pending requests, in case the network resources are not sufficient to serve them all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' This prob- lem is called “distribution” in [13], where the authors formulate it as an Integer 9 Linear Programming (ILP), for which they derive closed formula performance bounds in the case of a homogeneous chain of quantum repeaters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The issue is also addressed in [10], where the authors have proposed to split the overall quantum routing problem in two to reduce the computational complexity: first, they determine the rates achievable by the traffic flows under the given network constraints using an approach based on multi-commodity flow optimization, then they map these rates to paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The paper adopts a network model using probabilistic entanglement swapping, which we reuse in this work (described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' An important reference for our study is [20], where the authors study the allocation strategy of traffic flows for which the paths have been pre-determined: they do so by borrowing the fairness concept from data networks and re-using traditional algorithms from the relevant literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In our paper, we also bor- row from the same literature, though we apply the concepts to a different class of applications, as it will be clear in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' As a matter of fact, all the scientific works cited above have focused on point-to-point traffic flows, while in this paper we focus on a different type of traffic that is more suitable to model distributed QC, with distinguishing features that do not al- low the reuse of state-of-the-art solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Rather, we claim that any existing routing/allocation/scheduling solutions should work in parallel to our proposed scheme to provide an effective resource allocation to each of the two traffic classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In addition to mere routing aspects, system-wide studies have also been published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We mention [21], which is a compendium of several previous studies from the same authors that illustrates an overall architecture of the Quantum Internet, also including application, protocol, and deployment aspects at a high level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' On the other hand, other works have focused on specific components, which are complementary to the research activity presented, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', [22] on con- gestion control in transport protocols and [23] on the link layer, with a focus on hardware and physical-layer considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Furthermore, some research groups have been working to define the basic 10 principles of distributed QC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Parekh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' have defined an elegant frame- work for the parallel execution of a broad class of quantum algorithms on mul- tiple nodes [5], both using remote entanglement and with Local Operations and Classical Communication (LOCC) only, also studying in depth three classes of algorithms: variational quantum eigensolver, low-depth quantum amplitude estimation, and quantum k-means clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In [24] the authors address the problem of the efficient compilation of circuits for distributed QC by considering that some gate operations will be executed remotely, hence with much different latency and reliability than on-chip operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The research of Dahlber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' moved in the same direction and went as far as defining a set of low-level in- structions (called NetQASM) for distributed QC systems seamlessly supporting local and remote gates [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' These works confirm that there is a growing interest in distributed QC, which is a motivation for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' On another line of research, solutions have been proposed to trade capacity for fidelity, by using purification (or distillation) techniques [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In brief, they consist in entangling multiple pairs of qubits with low fidelity and then collapsing them into a single one with high fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We do not consider network- level purification in this work to remain consistent with the positioning of our contribution within the realm of 1G-repeater quantum networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' End-to-end purification is also possible, that is the operation is performed by quantum computers after the qubits have been entangled all along the path(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' This is studied, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', in [27], where the authors propose a quantum routing algorithm that maximizes the rate of EPR-pairs, while deciding not only the paths but also the purification patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' These works complement our contributions since they operate on constant-rate point-to-point flows only and they do not take into account network provisioning issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Finally, in line with the vast majority of prior works, we only consider bi- partite entanglement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', made of two qubits, each situated in a quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' While there are some promising theoretical studies on repeater- assisted multi-partite entanglement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', involving more than two qubits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', [28]), the research in that area is in still its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' One noteworthy contribution 11 is [29], where the authors propose to adopt n-fusion of bi-partite entanglements to create higher level entanglements between n > 2 quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' An appealing property that they demonstrate, under some assumptions, is that the entanglement rate between nodes remains constant with increasing distance, in number of hops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Ways to exploit this phenomenal quality are still under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Multi-partite entanglement in a quantum network is generated starting from elementary bi-partite entanglements, which is the subject of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Service Differentiation In this section we extend the study in [6] by analyzing the QDRR algorithm along two directions which have remained so far uninvestigated: service differ- entiation by assigning apps different ρ values (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='1) and apps with different fidelity thresholds F min (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In all the simulations in this section the peers are selected as follows: for each app i on node v we draw at random be- tween 2 and 4 candidate nodes by sampling in a uniform manner from the set of all nodes that are reachable from v in 2–7 hops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' For benchmarking purposes, QDRR is compared to two baseline algorithms: random and best-fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The Step 0 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2 is the same for all the algorithms, that is for each app we find k = 4 shortest paths to reach any peer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' All the algorithms then loop through all the possible paths for all candidates for each app until there are no more feasible paths to be assigned, but they differ in how they do so: QDRR is descriped by Steps 1–6 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2 (and in far more details in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' IV-C in [6]), while: – Random: at each iteration one app with remaining paths is chosen at random and assigned its shortest path among any of its peers, which is allocated the maximum rate along the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Random is representative of allocation algorithms that provide fair access to the quantum network resources in a per-app manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' – Best-fit: at each iteration, select the app with the shortest path to reach one of its peers and allocate the maximum rate along that path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Best-fit is representative of allocation algorithms that strive to maximize the efficiency, 12 that is the ratio between net entanglement rate and the quantum network capacity allocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 0 10 20 30 40 50 60 70 0 100 200 300 400 500 600 700 800 900 1000 Iterations (x1000) Load (#apps) Dense|Random Dense|BestFit Dense|QDRR Sparse|Random Sparse|BestFit Sparse|QDRR Figure 2: Number of iterations (expect Step 0 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2) with random, best-fit, QDRR, in a dense vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' sparse topology, when increasing the number of apps with ρ ∈ {1, 2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Unlike QDRR, both random and best-fit can be considered greedy algo- rithms, since they never backtrack to a previously selected combination of (app, peer, path), which is always allocated as much throughput as possi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Therefore, their worst-case computational complexity (expect Step 0) is O (k|A| log |A|E[Wi]): k is the number of shortest paths selected in Step 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' |A| is the number of apps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' log |A| takes into account the random selection or the ex- traction from an sorted data structure, respectively for the random and best-fit resource allocation algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' and, E[Wi] is the average number of peers per app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The complexity of QDRR is discussed in [6] and it depends on the choice of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' To give an idea of the relative average complexity between QDRR and random/best-fit, we report their number of iterations in the simulations dis- cussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='1 below in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' As can be seen, in a sparse scenario the time complexity of QDRR is only marginally higher than that of greedy algorithms random and best-fit, but it becomes clearly higher in a dense scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' If this is an issue, the value of φ can always be tuned to reduce the number of iterations, trading off fairness for speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Different traffic priorities In a first batch of results we increase the load of the network from 10 to 1000 apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' For every app, its priority weight ρ is drawn randomly in {1, 2, 4}, 13 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='8 1 0 100 200 300 400 500 600 700 800 900 1000 Residual/total capacity Load (#apps) Dense|Random Dense|BestFit Dense|QDRR Sparse|Random Sparse|BestFit Sparse|QDRR Figure 3: Ratio between the residual capacity and the total capacity with random, best-fit, QDRR, in a dense vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' sparse topology, when increasing the number of apps with ρ ∈ {1, 2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' while F min is the same for all and equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 3, for all the allocation algorithms and topologies the relative residual capacity decreases with a sub-linear trend as the load increases, which is due to the exponential relation between the net rate, in EPR-pairs/s, and the number of hops as per Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The utilization is higher in a sparse topology, while the difference between the allocation algorithms is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In the following we report only the results in a dense topology, due to limited space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' the complete results can be retrieved from the public GitHub repo above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 120 140 160 180 200 220 240 260 280 300 320 0 100 200 300 400 500 600 700 800 900 1000 Max-min fairness (EPR pairs/s) Load (#apps) Random (class avg) Best-fit (class avg) QDRR (ρ = 1) QDRR (ρ = 2) QDRR (ρ = 4) Figure 4: Max-min fairness with random, best-fit, QDRR, in a dense topology, when increasing the number of apps with ρ ∈ {1, 2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We begin by showing the max-min fairness in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Since random and best-fit do not differentiate based on the ρ values, for them we show an aggre- gate average, while we keep separate curves for QDRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' With very low loads, the max-min fairness is good, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', low, for all allocation algorithms, because there is 14 little contention on resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' However, as the load increases, the max-min fair- ness increases steeply and the behavior is significantly affected by the allocation algorithm: with random the curve reaches a peak, which then slowly decreases towards high loads;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' best-fit performs worst, as expected, by continuing to in- crease, even though only slightly after the initial spur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' for all traffic categories, QDRR provides increasingly better fairness as the load increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The latter can be explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' When there are few apps, it is likely that there are not many shared nodes/paths, hence QDRR does not really have a chance to dis- tribute the resources proportional to the apps’ weights;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' on the other hand, with more apps, it is increasingly easier for QDRR to enforce priorities by regulating the resources in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='8 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2 0 100 200 300 400 500 600 700 800 900 1000 Inter-class unfairness Load (#apps) Random Best-fit QDRR Figure 5: Inter-class unfairness index with random, best-fit, QDRR, in a dense topology, when increasing the number of apps with ρ ∈ {1, 2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' To better show the service differentiating behavior of QDRR, we show the inter-class unfairness index, as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (3), in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' It is clear that QDRR is the only allocation algorithm providing the apps with a clear service differentiation, which improves as the load increases for the same reason above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 6 we show the net rate/app, in EPR-pairs/s: even though it slowly decreases for all the resource allocation algorithms, we can see that random and best-fit achieve better rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Therefore, QDRR is effective in dif- ferentiating service across apps with different priority weights, but this incurs a cost, in terms of net rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 15 0 5 10 15 20 25 30 35 40 0 100 200 300 400 500 600 700 800 900 1000 Net rate/app (EPR pairs/s) Load (#apps) Random (class avg) Best-fit (class avg) QDRR (ρ = 1) QDRR (ρ = 2) QDRR (ρ = 4) Figure 6: Net rate/app with random, best-fit, QDRR, in a dense topology, when increasing the number of apps with ρ ∈ {1, 2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 50 100 150 200 250 300 350 0 100 200 300 400 500 600 700 800 900 1000 Max-min fairness (EPR pairs/s) Load (#apps) Dense|Random Dense|Best-fit Dense|QDRR Sparse|Random Sparse|Best-fit Sparse|QDRR Figure 7: Max-min fairness with random, best-fit, QDRR, in dense vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' sparse topologies, when increasing the number of apps with F min ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Different fidelity thresholds We now report the results obtained in a new batch, which follows the same direction as above, but we set ρ = 1 for all apps and draw randomly the mini- mum fidelity F min from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='9}, instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We show the max-min fairness in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 7, for both topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Like in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='1, QDRR achieves significantly better performance than random and best-fit, the latter performing worst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 0 20 40 60 80 100 120 0 100 200 300 400 500 600 700 800 900 1000 Net rate/app (EPR pairs/s) Load (#apps) Dense|Random Dense|Best-fit Dense|QDRR Sparse|Random Sparse|Best-fit Sparse|QDRR Figure 8: Net rate/app with random, best-fit, QDRR, in dense vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' sparse topologies, when increasing the number of apps with F min ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' However, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 8, also with a mix of fidelity thresholds, the net rate/app of QDRR is slightly less than that with both greedy allocation strategies, which confirms the trade-off already identified in the previous batch of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' It is worth noting that such a trade-off is well-known also in different contexts: for instance, in cellular systems, greedy scheduling algorithms that prioritize user terminals with good channel conditions (often known as “max C/I”) are bound to provide a higher cell throughput at the cost of an inferior fairness compared to milder strategies such as Proportional Fair [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In conclusion, like for heterogeneous weights, QDRR can provide service dif- ferentiation to apps with different fidelity thresholds, but the net rate achievable is slightly reduced compared to alternatives that do not differentiate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Fair Sharing In this section we address a problem that is preliminary and complementary to that defined in our previous work [6] and investigated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 4 with differen- 17 tiated service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' So far we have assumed that all nodes are equal, and each host is wishing to cooperate with a given set of workers, without elaborating further on how such a set is selected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' for performance evaluation purposes, such a set was selected randomly from candidates depending only on the distance as per the specific scenario simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In the following, instead, we address specifically this issue: indeed, how does one decide which are the possible workers of a host node?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' end users data centers Figure 9: Example of quantum network with three end users, labeled from u1 to u3, wishing to host distributed quantum computing algorithms with data center nodes d1 or d2, represented with multiple co-located circles to indicate that they are expected to be more powerful than end users and, possibly, they might have a more complex internal structure that is not elaborated further in this paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' the other nodes in G(V, E) participate to the end-to-end entanglement of qubits as intermediate hops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Like in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='1/Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 1, the network is characterized by capacity Cij of the link between nodes i and j, initial generation fidelity F, and entanglement swapping success probability q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We adapt our system model to the new landscape by specializing the role of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' As illustrated in the example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 9, we assume that nodes can be of three types: (i) end users (ui): they act as the “home QCs” of customers wish- ing to run quantum algorithms on them, also exploiting quantum computation resources offered by other nodes through distributed quantum computing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' a cus- tomer operates the end user via a classical computer for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', input preparation, 18 circuit compilation, and quantum network resource reservations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (ii) data centers (dj): these are QCs that can provide customers with extra quantum computa- tion capacity to be added to their respective end users for the purpose of solving bigger instances of their problems via distributed quantum computing, exploit- ing end-to-end entanglement of qubits through an underlying quantum network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (iii) intermediate nodes: quantum repeaters who do not consume or offer QC capacity but contribute to the quantum network by performing entanglement swapping between links as instructed by the resource allocation algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', QDRR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' A node can play any combination of the three roles above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We assume that end user ui may perform distributed quantum computing with any combi- nation of data centers {dj}, following commercial agreements that are out of the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' All other quantum network assumptions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='1 remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='1 we formulate the problem in mathematical terms and propose a solution, which is then evaluated via simulation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2, compared to two alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Quantum Workers’ Assignment Problem In its most general formulation, the problem of how to best assign each host a set of workers depends on several factors that may be not known or under the control of the quantum network operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' They include, for instance: the quan- tum algorithms to be run, the different characteristics of the end user and data center QCs, the schedule of the task execution, not to mention administrative factors (contracts, partnerships, billing issues) and technical constraints (do the QCs need to have the same hardware or software?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Since both quantum net- working and distributed quantum computing are in their infancy, we consider unrealistic to address the problem under such general settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Rather, we focus on aspects that are captured by our network model (in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='1) with the goal of providing an initial understanding of the problem, to be used as a stepping stone by future studies as the technologies involved become more mature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Our formulation, in natural language, is the following: Quantum Workers’ Assignment Problem (QWAP): find the sets of workers, se- 19 lected from the data center nodes, to be assigned to each host, from the end user nodes, so that the overall profit of the hosts is maximum while balancing the load of data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The problem can be formulated in a formal manner once we define the no- tions of “profit” and “load balancing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Based on the prior works in the literature (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 3), we consider the net entanglement rate in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (1) as the profit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', between two possible data centers d1 and d2 considered as candidate workers for end user u, we prefer the one that potentially achieves the highest net en- tanglement rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' On the other hand, we introduce load balancing as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' First, we define the system parameter W as the target number of workers per end user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Then, we force each data center to be assigned as worker to at most B end users, where B is determined as the minimum value that allows this con- straint to be provided with given W, Nu end users, and Nd data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' This way, we force an even load across data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Note that this formulation can be trivially extended to the case where data centers have different capabilities by introducing appropriate weights, which we do not consider in this work to keep the notation succinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Assuming without loss of generality, again to simplify notation, that all end users have one and only one request to run distributed quantum computing, the QWAP can be expressed as an optimization problem with objective function: max Nu � u=1 Nd � d=1 πudxud (4) such that: 20 Nu � u=1 xud ≤ B d = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' , Nd (5) Nd � d=1 xud ≤ W u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' , Nu (6) xud ∈ {0, 1} (7) B = �Nu · W Nd � (8) where the profit πud between end user u and data center d is defined by: πud = � � � � � 0 if ∀p ∈ Pu,d : F(p) ≤ F min u rud¯p where ¯p = arg maxp � rudp|F(p) ≥ F min u � , (9) where F(p) is the fidelity of the end-to-end entanglement along the path p, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (2), F min u is the minimum fidelity requested by end user u, Pu,d is the set of all paths between u and d in G(V, E), and the net rate rudp between end user u and data center d along path p is as follows: rudp = max(i,j)∈p {Cij} q|p|−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (10) The output assignment integer variables, as per Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (8), are the xud, with the constraints as follows: Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (5) ensures that no data center is assigned more than its fair share of B users, where B is computed via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (6) ensures that no end user is assigned more than W workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The profit πud, defined through Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (9–10), corresponds to the maximum net rate of EPR-pairs/s that can assigned to end user u along any path towards data center d that fulfils its minimum fidelity requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Before proceeding, we state two key observations about the QWAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Observation#1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In a general graph G(V, E), the number of paths between two nodes can be exponential with the size of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' For instance, in a complete graph this number is ⌊(V − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='e⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Therefore, the preparation of the problem input in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (9) might be very computation-intensive in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In 21 the evaluation below, we adopt a reasonable approximation: rather than finding all the paths Pud between nodes u and d, we use a reduced set P¯k ud that consists of the ¯k shortest paths (¯k = 10 in the simulations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The rationale is that long paths have a small chance of being selected as ¯p in the second branch of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (9), because the net rate decreases exponentially with the path length as per Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Observation#2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' When W = 1, then Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (4–10) above can be trivially trans- formed into an assignment problem, whose optimum solution can be found ef- ficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' With W > 1, however, the QWAP becomes a “multiple knapsack problem”, which is NP-hard, but for which several efficient heuristics are well- known in the operations research literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Based on the two observations above, we propose our load balancing al- gorithm to solve the QWAP, which we implemented in our simulator and eval- uated in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The idea of the load balancing algorithm is to find the optimal allocation for the first worker of each end user using the Hungar- ian algorithm [32], which finds the best (exact) solution in assignment problem instances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' then, it progresses by considering one more worker at a time, until W, each time invoking the Hungarian algorithm again on the data centers with residual slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The algorithm is greedy because it never backtracks prior deci- sions and it always terminates after a fixed number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' More formally, the load balancing algorithm consists of the following three steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Prepare the problem input, in particular the profits πud, by finding up to ¯k shortest paths between u and d in G(E, V ) using Yen’s algorithm [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Determine B based on Nu, Nd, and W using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Arrange the output of Step 1 in a profit matrix where the rows are the end users and there are B columns for each data center, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', the matrix size is Nu × BNd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Then run W iterations of the Hungarian algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' After each iteration, set πud ← 0 in all columns where a data center has been assigned (avoids that the constraint Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' (5) is violated) and in all cells that refer to the same pair u, d that has been assigned (avoids that a user is assigned multiple 22 times the same data center).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=" Evaluation Yen's algorithm using Dijkstra with Fibonacci heap preparation execution for each end user and data center number of shortest paths to search Hungarian algorithm number of iterations Dijkstra with Fibonacci heap for each end user and worker for each end user and worker random selection of data center a) b) c) Figure 10: Worst case time complexity of a) the load balancing algorithm to solve the QWAP in Sec." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' the comparison algorithms b) random and c) shortest path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In this section we evaluate the load balancing algorithm defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='1 using the simulator described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' End users and data centers are selected randomly from the set of nodes V , with the following composition of the nodes: 10% end users, 10% data centers, 80% intermediate nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' As comparison algorithms, we define: – Random, which assigns each end user W data centers at random, thus it maximizes fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' – Shortest path, which assigns each end user the W data centers that are closest in G(E, V ), in number of hops, thus it maximizes the net rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' After the assignment, the resources are allocated using QDRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We report in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 10 the worst case time complexity of the three algorithms, where we assume that the shortest path is computed using Dijkstra’s algorithm with the help of a Fibonacci heap to keep edges sorted [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' As can be seen, the 23 50 100 150 200 250 300 350 400 450 500 1 2 3 4 5 Net rate/app (EPR-pairs/s) W Random Shortest path Load balancing 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='5 1 2 3 4 5 Max-min num users per data center W Random Shortest path Load balancing Figure 11: Net rate/app (top) and max-min number of users per data center (bottom) with random, shortest path, and load balancing, in a dense topology, with 10 apps, when increasing W from 1 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' load balancing algorithm is far more complex than random and shortest-path, in both the preparation and the execution phases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' random, in particular, does not even depend on the graph size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' However, in the following we will see that the added complexity brings benefits that can be of potential interest to the future quantum network operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 11 (top) we show the net rate/app with Nu = 10 and W increasing from 1 to 5, in a dense topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' As can be seen, the random algorithm per- forms poorly, because it does not consider at all the network topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' On the other hand, shortest path and load balancing perform similarly, with the latter exhibiting a higher net rate with small values of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 11 (bottom) we plot a measure of the spread of resources, as the max-min number of users per data center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Load balancing performs consistently and significantly better than both random and shortest path, with the latter exhibiting the highest unfair- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' We note that our conclusions are limited to the settings of the scenarios 24 simulated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' in particular, different topologies or link capacity distributions can lead to situations where the gap between load balancing and either random or shortest path is reduced significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' However, a crucial advantage of our pro- posed solution, compared to its alternatives under test, is that it can adapt to different settings, thus it can perform well even when the scenario is not known or changes dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='96 load balancing random shortest-path load balancing random shortest-path Fidelity Dense|10 apps Dense|20 apps Sparse|10 apps Sparse|20 apps Figure 12: Fidelity with random, shortest path, and load balancing, in dense vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' sparse topologies, with 10 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 20 apps and W = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The fidelity is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 12, with Nu = {10, 20} and in dense vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' sparse topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The number of end users/apps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=', Nu, does not affect the perfor- mance in a noticeable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' On the other hand, the fidelity is generally lower in sparse topologies, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Random performs worse, while load balanc- ing and shortest path give similar results, with the former performing slightly worse only in the sparse case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' This can be explained by following the same line of reasoning for the net rate/app above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' To conclude the analysis, we modify the mix of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In one batch of simulations we increase the ratio of end users from 10% (like in the results so far) to 50%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' in another one we do the same for data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Results are shown only for load balancing in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 13, in terms of the net rate, with Nu = 15 and W = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' As the ratio of data centers increases (green curves), the net rate/app increases almost linearly, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' On the other hand, increasing the number of users (blue curves), only provides a sub-linear performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' This is because the former case corresponds to increasing the physical resources 25 50 100 150 200 250 300 350 400 450 500 550 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content='5 Net rate per app (EPR-pairs/s) Ratio of (end users | data centers) [end users]|Dense [end users]|Sparse [data centers]|Dense [data centers]|Sparse Figure 13: Net rate/app with load balancing, in dense vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' sparse topologies, with 15 apps and W = 3, when increasing the fraction of nodes as either end users or data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' provided to the users, while the latter only to a more uniform distribution of the same resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The conclusions are the same for dense and spare topologies, though the net rate for the latter is significantly lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' In conclusion, assigning data centers to end users through the load balancing algorithm, which provides an approximate solution of the QWAP, achieves an even distribution of resources, better than both random and shortest path as- signment, without compromising on the net rate and fidelity of the end-to-end entanglement paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Conclusions In this paper we have studied two open issues in quantum networking for distributed quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' First, we have assessed through simulation the effectiveness of the QDRR resource allocation algorithm [6] in handling sce- narios where applications have different priority weights or minimum fidelity requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Second, we have defined a novel problem involving the selection of workers for a set of nodes hosting computation, called the Quantum Work- ers’ Assignment Problem (QWAP), which we have modeled as an optimization problem and for which we have proposed a heuristic called “load balancing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The latter has been evaluated in comparison to alternatives seeking to maxi- mize only fairness and the net rate of end-to-end entanglement, respectively, and the results have shown that load balancing achieves an excellent compromise in 26 terms of the two metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The source code and simulation scripts are publicly available to the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Further open research areas are: the use of purification to increase fidelity at the expense of capacity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' modeling distributed QC applications to understand their characteristic time scales and requirements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' integration with link layer pro- tocols;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' incorporation in the simulation of more realistic models for the quantum channel and repeaters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' Acknowledgment Work co-funded by EU, PON Ricerca e Innovazione 2014–2020 FESR/FSC Project ARS01_00734 QUANCOM, and European High-Performance Comput- ing Joint Undertaking (JU) under grant agreement No 101018180 HPCQS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' The paper reflects only the authors’ view and the funding agencies are not respon- sible for any use that may be made of its content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfkQe0/content/2301.03977v1.pdf'} +page_content=' References [1] J.' metadata={'source': 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b/A9E1T4oBgHgl3EQf9QZb/content/tmp_files/2301.03554v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0083a42a5c8feff504914d7680b4251a7f427b28 --- /dev/null +++ b/A9E1T4oBgHgl3EQf9QZb/content/tmp_files/2301.03554v1.pdf.txt @@ -0,0 +1,3502 @@ +Highlights +A systematic literature review of capstone courses in software engineering +Saara Tenhunen*,Tomi Männistö*,Matti Luukkainen,Petri Ihantola +• Our taxonomy of course features is based on ACM/IEEE guide for capstone courses +• There is a vast diversity in how capstone courses in SE are implemented +• More research is needed to compare different course implementation strategies +• Many of the capstone courses are shorter than the recommended two semesters +• Many of the capstone courses are missing an external client +arXiv:2301.03554v1 [cs.SE] 9 Jan 2023 + +A systematic literature review of capstone courses in software +engineering +Saara Tenhunen*, Tomi Männistö*, Matti Luukkainen and Petri Ihantola +University of Helsinki, , , Finland +A R T I C L E I N F O +Keywords: +capstone +project course +computer science education +software engineering education +A B S T R A C T +Context: +Tertiary education institutions aim to prepare their computer science and software +engineering students for working life. +While much of the technical principles are covered in +lower-level courses, team-based capstone projects are a common way to provide students with +hands-on experience and teach soft skills. +Objective: This paper explores the characteristics of software engineering capstone courses presented +in the literature. The goal of this work is to understand the pros and cons of different approaches by +synthesising the various aspects of software engineering capstone courses and related experiences. +Method: In a systematic literature review for 2007–2007, we identified 127 primary studies. These +studies were analysed based on their presented course characteristics and the reported course +outcomes. +Results: The characteristics were synthesised into a taxonomy consisting of duration, team sizes, +client and project sources, project implementation, and student assessment. +We found out that +capstone courses generally last one semester and divide students into groups of 4–5 where they work +on a project for a client. For a slight majority of courses, the clients are external to the course staff +and students are often expected to produce a proof-of-concept level software product as the main end +deliverable. The courses also offer versatile assessments for students throughout the project. +Conclusions: This paper provides researchers and educators with a classification of characteristics +of software engineering capstone courses based on previous research. We also further synthesise +insights on the reported outcomes of capstone courses. Our review study aims to help educators to +identify various ways of organising capstones and effectively plan and deliver their own capstone +courses. The characterisation also helps researchers to conduct further studies on software engineer- +ing capstones. +1. Introduction +Universities and other tertiary education institutions should +provide their students with sufficient skills and abilities be- +fore the students enter working life. In software engineering- +related programs, this entails having an understanding of the +common principles and theory in computer science (ACM/IEEE, +2013, 2014) and technical competencies and knowledge de- +manded by the industry (Radermacher et al., 2014; Garousi +et al., 2019). Any recent graduate should also be able to ap- +ply this technical knowledge in practice (ACM/IEEE, 2013). +While much of the technical knowledge and theories are +covered in lower-level courses, many institutions hold team- +based capstone project courses to ensure students are ready +to apply the knowledge in a workplace environment. A “cap- +stone course“ usually means a course that finishes an aca- +demic degree (Ikonen and Kurhila, 2009). The main goal +of a capstone project is to provide hands-on experience in +applying the tools, techniques, principles and best practices +that are taught more theoretically in previous courses (Ziv +and Patil, 2010; Majanoja and Vasankari, 2018; Panicker +et al., 2020). Capstone projects are also regarded as crucial +in teaching students the necessary soft skills such as team- +∗Corresponding author +saaraten@gmail.com (S. Tenhunen*); tomi.mannisto@helsinki.fi +(T. Männistö*); matti.luukkainen@helsinki.fi (M. Luukkainen); +petri.ihantola@helsinki.fi (P. Ihantola) +ORCID(s): 0000-0002-4894-8365 (S. Tenhunen*) +work (Keogh et al., 2007; Venson et al., 2016), verbal and +written communication (Watkins and Barnes, 2010), time +management (Dupuis et al., 2010), problem solving (Ma- +janoja and Vasankari, 2018) and project management (Had- +dad, 2013). In computer science (CS) and software engi- +neering (SE) programs, capstone courses generally last one +or two semesters, and they include assigning students into +teams and having them work on various kinds of software +engineering projects (Ikonen and Kurhila, 2009; Bowring +and Burke, 2016; Paasivaara et al., 2019). In these projects, +they are expected to experience stages of the software devel- +opment life-cycle from requirements solicitation to software +maintenance (Keogh et al., 2007). +Given the general acceptance of capstones as a practical +way of teaching industry-relevant skills, a high number of +institutions have implemented their own capstone courses. +This has resulted in a great deal of research done on cap- +stone courses and their outcomes. In order to provide a co- +herent and compact view of software engineering capstones, +this research synthesises the current body of knowledge on +the topic in a systematic manner. We believe that such a +review gives educators an effective tool for planning and +implementing their own capstone courses. Researchers can +also benefit from a systematic review of capstones to con- +duct further comparative studies on the impact of the varying +course forms. +This study is organised as follows. The next section fo- +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 1 of 22 + +aCapstone courses in software engineering +Table 1 +Searches for systematic reviews on software engineering capstones +Database +Search term +Hits +Scopus +TITLE-ABS-KEY( software AND engineering AND capstone AND literature AND review ) +10 +Scopus +TITLE-ABS-KEY( software AND engineering AND capstone AND review ) +61 +Scopus +TITLE-ABS-KEY( education AND “software engineering“ AND literature AND mapping ) +40 +Scopus +TITLE-ABS-KEY( project AND course AND software AND engineering AND systematic AND review ) +20 +Scopus +TITLE-ABS-KEY( “computer science“ AND capstone AND literature AND review ) +8 +Scopus +TITLE-ABS-KEY( software AND engineering AND project AND course AND systematic AND literature AND review ) +17 +Scopus +TITLE-ABS-KEY( “software engineering“ AND education AND literature AND review ) +182 +Google Scholar +software engineering capstone systematic review +20 300 +Google Scholar +software engineering capstone characteristics +31 400 +Google Scholar +computer science capstone literature review +53 400 +Google Scholar +capstone literature review +94 800 +cus on the previous literature reviews on SE capstones, as +well as general characteristics of such courses. Section 3 +describes the research questions and the related methods, in- +cluding how the articles were selected. Section 4 presents +the results of the literature review. The main findings and +their validity are discussed in Section 5. Suggestions for fu- +ture research are also given. Finally, Section 6 concludes the +research. +2. Previous work +2.1. Systematic literature reviews of SE capstones +Many literature reviews have been written in the general +area of software engineering education (SEE). Usually, they +focus on specific sub-areas of SEE such as teaching meth- +ods in software engineering (Anicic and Stapic, 2022), prac- +tical approaches to SEE (Marques et al., 2014), trends in +SEE (Cico and Jaccheri, 2019; Cico et al., 2021) or teach- +ing global software engineering (Fortaleza et al., 2012). +As the focus of this research is especially on project- +based capstone courses in software engineering, we carefully +sought any earlier systematic reviews done on them. The +search was conducted on May 17th, 2022, first in the cita- +tion database Scopus and secondly in Google Scholar. Table +1 lists the search terms used in both databases. All search +results produced by Scopus were checked to see whether +they include an SLR of SE capstones, whereas for Google +Scholar, each search produced tens of thousands of hits, so +we went through the first 20 pages of each search (200 hits). +At this point, the results started to become highly irrele- +vant, and often repetitive. Based on our search, we believe +that the three review papers presented in Table 2 are the +ones that have been published so far on this topic. Table +2 also presents the characteristics of capstone courses each +of these reviews investigated. Next, we will briefly present +these studies and discuss the necessity of this review. +Dugan Jr (2011) presents a survey done on the literature +related to undergraduate computer science capstone courses. +The survey is comprehensive, comprising 200 papers on the +subject and summarising them under two major themes: course +issues and project issues (Table 2). Out of these, course +issues include aspects related to the general course organi- +sation, such as course models, learning theories present in +the course and student evaluation. Project issues, on the +other hand, categorise and describe the projects and how +they are implemented. The category includes things like +software process phases, project type and documentation of +the projects. +Martin (2019) has provided an abstract of a systematic +literature review for designing an IT capstone course. The +review plans to provide answers to several questions relating +to capstone course design, such as the optimal team size for +project teams, identifying and selecting suitable projects and +determining the correct duration for the course (Table 2). We +are not aware that the research proposed in the abstract would +have been completed. +Trevisan et al. (2006) have performed a systematic re- +view on the assessment practices in capstone engineering +design courses. They were especially interested in discov- +ering the extent to which classroom assessment has received +attention in the capstone literature. The paper included 32 +journal articles and conference proceedings presenting vary- +ing assessment techniques and their use. +In addition to the presented three literature reviews, a +study on the dimensions of SE and CS capstone projects has +been conducted by Burge and Gannod (2009). Said dimen- +sions are roughly divided into two groups: project dimen- +sions such as customer identity and development dimensions +such as project type and source code visibility. The purpose +of their study is to provide a framework for analysing cap- +stone courses, especially in terms of risk and realism. While +their categorisation is versatile, their study does not, how- +ever, include a thorough systematic literature review. The +categorisation presented is more of an experience-based pro- +posal, and therefore the study is left out of Table 2. +To the best of our knowledge, there is no extensive, re- +cent literature review done on software engineering capstone +courses. A survey conducted by Dugan Jr (2011) is com- +prehensive but dated to 2011 and therefore does not cover +the large number of primary studies published in the past +decade. It also does not provide any quantitative statistics of +the course characteristics, which would enable educators or +researchers to assess how common some aspect in reality is. +Trevisan et al. (2006) provide a review on capstone literature, +but it is limited to continuous assessment techniques and is +dated to 2006. Martin (2019) aims to provide a systematic +literature review on IT capstone design and characteristics, +but as of now, the paper has not proceeded beyond the orig- +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 2 of 22 + +Capstone courses in software engineering +Table 2 +Systematic reviews of software engineering capstones +Title +Year +Ref +Course characteristics examined in the survey +A survey of computer science capstone +course literature +2011 +(Dugan Jr, 2011) +Course-related: models, learning theories, goals, top- +ics, student evaluation, evaluation. +Project-related: +software process models, phases, +type, documentation, tools, groups, instructor admin- +istration. +Designing the IT capstone course +2019 +(Martin, 2019) +Course duration, learning of new skills, project iden- +tification and selection, teams sizes, team formation, +followed methodologies, assessment of learning out- +comes, team and project supervision* +A +review +of +literature +on +assessment +practices in capstone engineering design +courses: Implications for formative assess- +ment +2006 +(Trevisan et al., 2006) +Connection to student achievement +*For the survey by Martin (2019), these are characteristics, which would have been examined in the actual survey. +inal abstract. In light of this, current research does not pro- +vide an up-to-date view of how SE capstone courses gener- +ally are organised and with what kind of outcomes. Such a +view on the software engineering capstones would not only +provide educators with an important tool for planning their +own capstone courses but also give researchers a basis for +performing comparative studies on these courses. +2.2. Background: Capstone course characteristics +ACM/IEEE Curriculum Guidelines for Software Engi- +neering (SE) Degree Programs (ACM/IEEE, 2014) view the +capstone project as an essential element of a SE degree pro- +gramme and state that the main goal of a capstone course +is to ensure that the curriculum has a significant real-world +basis. According to ACM/IEEE (2014), incorporating real- +world elements into the curriculum is necessary to enable ef- +fective learning of software engineering skills and concepts. +The ACM/IEEE Curriculum Guidelines for Computer Sci- +ence (CS) degree programs (ACM/IEEE, 2013) align with +these views and state that all graduates of CS programs should +have been involved in at least one substantial project. Such +projects should challenge students by being integrative, re- +quiring evaluation of potential solutions and working on a +larger scale than typical course projects. For students, a cap- +stone project typically represents a culmination of their stud- +ies and is one of the last milestones before graduation (ACM/IEEE, +2013; ACM, 2009). Indeed, since the 1970s, hundreds of +primary studies have been written on this large, final-year +project course (Dugan Jr, 2011). +The ACM/IEEE (2014) also lists a set of key recommen- +dations that a capstone course should follow. The recom- +mendations are listed word by word in Table 3. We decided +to use these recommendations as the basis for formulating +our research questions. They give a general outline of cap- +stone courses and therefore provide a valid starting point for +the categorisation done in this research. +Thus according to these guidelines, there are some basic +characteristics that capstone courses have. They can be char- +acterised as long and substantial projects (CR1, CR2) that +should preferably be completed in a team (CR2). Projects +Table 3 +ACM/IEEE recommendations for SE capstones +CR # +Recommendation +CR 1 +The project should span a full academic year, giving +students adequate time to reflect upon experiences and +retry solutions as appropriate. +CR 2 +Where possible, this should preferably be undertaken +as a group project. If such factors as assessment make +this difficult, it is essential that there should be a sep- +arate group project of substantial size. +CR 3 +Where possible, a project should have a “customer” +other than the supervisor so that the student gains +fuller experience with product development life-cycle +activities. +CR 4 +A project should have some form of implementation +as its end deliverable so that the students can experi- +ence a wide set of software development activities and +adequately evaluate these experiences. Theory-based +projects such as the development of formal specifica- +tions is therefore inappropriate for this role. +CR 5 +Evaluation of project outcomes should go beyond con- +cept implementation (“we built it, and it worked” +(Glass et al., 2004)), using walkthroughs, interviews, +or simple experiments to assess the effectiveness and +limitations of the deliverables. +CR 6 +Assessment of a capstone project should consider how +effectively software engineering practices and processes +have been employed, including the quality of student +reflection on the experience, and not be based only on +the delivery of a working system. +should have customers (CR3) for whom the students are ex- +pected to deliver some form of real implementation at the +end of the course (CR4). Students should therefore engage +in real software development activities and not just complete +simple, theory-based assignments provided by the teacher +(CR4). Evaluation of the project outcomes should focus not +only on the fact that the project “works“, but also assess the +deliverables on how well they have been completed (CR5). +Finally, the focus of the course and its assessment should +be on software engineering practices and processes and stu- +dents should give adequate opportunities to reflect on the ex- +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 3 of 22 + +Capstone courses in software engineering +perience (CR6). The next section describes in more detail +the process of how we derived the research questions based +on these basic characteristics. +3. Research questions and method +3.1. Research questions +Characteristics of capstone courses (described in Sec- +tion 2.2) can be achieved in many ways. The main goal of +this research was to understand these differences in how cap- +stone courses are implemented in universities and other ter- +tiary education institutions, and thus provide a holistic view +over the various capstone course implementations. +We decided to use the ACM/IEEE Curriculum Guide- +lines for Undergraduate SE Degree Programmes (ACM/IEEE, +2014) as the basis for starting to explore these characteris- +tics. The recommendations are listed in Table 3. Although +some of these aspects, e.g., team formation, have been ad- +dressed in the previous reviews, previous literature reviews +are slightly outdated. Moreover, we are not aware of any +study covering all the aspects mentioned in the ACM/IEEE +recommendations. +Related to CR1, we were interested in the duration of +the courses and what rationale primary studies provide for +choosing a specific course duration if any: +RQ1 What is the duration of SE capstone courses, and what +advantages or disadvantages are related to a certain +duration? +Related to CR2, we wanted to find out if these projects +are conducted in teams, how teams are composed and what +is the rationale behind choosing a certain team size: +RQ2 What team sizes do SE capstone courses have, and +how are team sizes justified? +Based on the ACM/IEEE recommendations (i.e., CR3), +a project should have a customer other than the teacher of the +course. An alternative approach to bringing an outside view +to a project is to outsource project topics. Thus, our third +research question, how are the project and client sourcing +handled in SE capstone courses, was divided into two sub- +questions: +RQ3.1 Who acts as the client for capstone projects? +RQ3.2 How are the ideas for projects sourced? +Related to CR4, we asked: How are the projects in cap- +stone courses implemented (RQ4). We wanted to uncover +what students do in these courses and therefore, we looked +into the actual project implementation. As ‘project imple- +mentation‘ can mean a multitude of things, we decided to di- +vide this research question into smaller, more concrete sub- +questions: +RQ4.1 What artefacts are students expected to produce on +capstone courses? +RQ4.2 What is the software life-cycle gone through during +these projects? +RQ4.3 How are the implementation technologies chosen for +capstone projects? +With RQ4.1 we aimed to find out what students actu- +ally produce in these courses and whether any software is +being developed. RQ4.2 helped us to find out if the cap- +stone project is as integrative experience on software en- +gineering practices as curriculum guidelines (ACM/IEEE, +2014; ACM, 2009) suggest. Finally, finding out how educa- +tors make the choices for implementation technologies and +what implications these choices have, gave some insight into +project implementation. +As CR6 speaks about assessment, our last research ques- +tion also asked: How is the student assessment conducted on +SE capstone courses? (RQ5). As assessment can be divided +into continuous feedback and final grading, RQ5 was also +split into two: +RQ5.1 How are the students assessed at the end of SE cap- +stone courses? +RQ5.2 How are students guided, if at all, during SE cap- +stone courses? +The rationalisation here was that we wanted to uncover +whether the evaluation is based on a multitude of factors like +(ACM/IEEE, 2014) suggests and whether students are given +adequate possibilities to reflect on their experiences (Hattie +and Timperley, 2007). +In order to get a comprehensive representation of how +project-based capstone courses are generally organised, rel- +evant research articles were searched. One could argue that +the characteristics and any organisational details of these courses +could be derived from the web pages of universities and other +tertiary institutions. However, we wanted not only to pro- +duce a list of characteristics such as the duration and work- +load of the courses but also to reveal more about the con- +tents of these courses. An important part of the research was +also to provide educators with insights related to the various +characteristics. Without any evaluation or assessment of the +chosen structure and characteristics, this would have been +impossible to achieve. +3.2. Search strategy +The method used in this study follows the SLR method +by Kitchenham and Charters (2007). The initial data col- +lection was done by finding relevant sources from scientific +databases: Scopus, ACM Digital Library, IEEE Xplore and +ScienceDirect. Some preliminary searches were conducted +on these databases to find out to which extent research ar- +ticles use the word “capstone“ and its synonyms when de- +scribing large, degree-culminating project courses in soft- +ware engineering-related programs. It turned out that the +term “capstone“ is well-known and widely used in research +articles. It was also used by Dugan Jr (2011) in their ear- +lier work. Therefore, the first search string was simply con- +structed as: +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 4 of 22 + +Capstone courses in software engineering +Table 4 +Initial search results +Database +Search strings +Hits +Scopus +TITLE-ABS-KEY ( software AND capstone ) +762 +Scopus +TITLE-ABS-KEY ( software AND “project course“ ) +262 +ACM Digital Library +[Title: software] AND [Title: capstone] +24 +ACM Digital Library +[Keywords: software] AND [Keywords: capstone] +32 +ACM Digital Library +[Abstract: software] AND [Abstract: capstone] +130 +ACM Digital Library +[Title: software] AND [Title: “project course“] +6 +ACM Digital Library +[Keywords: software] AND [Keywords: “project course“] +7 +ACM Digital Library +[Abstract: software] AND [Abstract: “project course“] +44 +ScienceDirect +(TITLE ABS KEY: SOFTWARE CAPSTONE) +22 +ScienceDirect +(TITLE ABS KEY: SOFTWARE PROJECT COURSE) +12 +IEEE Xplore +(“All Metadata“:Software) AND (“All Metadata“:capstone) +223 +IEEE Xplore +(“All Metadata“:software) AND (“All Metadata“:“project course“) +86 +software AND capstone +In order to have a complete picture of the project course +landscape in software engineering, a second search was per- +formed using the second search string: +software +AND 'project course' +This was deemed necessary as not all sources had the +word “capstone“ present in the metadata even though they +clearly were describing courses relevant to this research. Searches +with the two search strings were conducted sequentially in +each database. +Dugan Jr (2011) used “software engineering course“ as +another search term in their study, but we did not want to +limit ourselves to the SE discipline, as relevant software- +related courses might be presented, for instance, in computer +science. Using only the words “software“ and “course“ on +the other hand, provided too many irrelevant hits. Scopus +alone produced nearly 30 000 hits of which only a small frac- +tion would have been relevant to our study. +A total of 981 unique papers were found after combin- +ing the papers found from all four databases using the search +strings and removing duplicates. The databases were searched +on June 11th and June 12th 2022, one after the other, starting +with Scopus, moving on to ACM Digital Library, followed +by ScienceDirect and finishing with IEEE Xplore. As the +search fields and filters are slightly different in each of these +databases, the search strings were adjusted to match each +specific set-up. They were, however, kept semantically the +same across the searches. Exact search strings and initial +search results are listed in Table 4. As we wanted to identify +current ways of organising capstone courses, the searches in +all four databases were limited to the years 2007 to the search +day in June 2022. This time period was regarded as suffi- +ciently long to provide a holistic view of the current capstone +courses. It overlaps with Dugan Jr (2011) by a few years but +also uncovers 11 years of research done on the area that has +not been systematically reviewed since. Three stages of se- +lection were applied to this initial set, after which 127 pri- +mary studies remained. Fig. 1 summarises the search and se- +lection process, and the following subsections will describe +it in greater detail. +Figure 1: Search strategy +3.3. Paper selection +The paper selection was conducted from the initial set +of 981 sources by the first author (Fig. 1). The details of +inclusion and exclusion are explained next. +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 5 of 22 + +Research questions +(Section 3.) +Initial search +1. Identty suitable search terms by conducting +Used databases +preliminary searches +1. Scopus +2. Construct the search strings +2. ACM Digital Library +3. Conduct search to four databases +3. IEEE Xplore +(Section 3.1.) +4. ScienceDirect +981 unique papers +First stage +Read abstracts, tites and keywords and apply +inclusion criteria to them +(Section 3.2.1.) +398 primary studies +Second stage +Read full text and apply exclusion criteria +(Section 3.2.2.) +171 primary studies +Third stage +Remove studies of courses with newer +publications and studies of poor quality +(Section 3.2.3.) +Final set: +127 primary studiesCapstone courses in software engineering +3.3.1. The first stage - Inclusion criteria +The titles, abstracts and keywords of the initial papers +were read and evaluated against the inclusion criteria pre- +sented below (IC1-IC3). After the first stage, 398 papers +remained. +IC1 The title or abstract strongly hints that the study presents +frameworks or case studies of software engineering capstones +or other large, project-based courses in software engineering +IC2 Based on the title or abstract, the study describes real ex- +periences of implementing a software engineering capstone +course +IC3 The title or abstract indicates that the study assesses the +outcomes of the course or its characteristics +The first inclusion criterion was developed to set the fo- +cus on software engineering courses in particular. A large +number of the articles in the initial set were ruled out due to +the first criterion (IC1). The papers were found to research, +for instance, mechanical engineering courses, which were +out of the scope of this research. We also wanted to rule +out any purely hypothetical papers, where the researchers +show no course that follows the frameworks or structures +presented. The second inclusion criterion (IC2) aimed to +ensure that all included papers would present a real-world +course. The final inclusion criterion (IC3) was generated so +that all papers would also evaluate the outcomes of the var- +ious course implementations. +3.3.2. The second stage - Exclusion criteria +The second stage was performed on the 398 papers re- +maining from the first stage. Any article that, based on read- +ing the full paper, met at least one of the presented exclusion +criteria was excluded at this stage. After this selection, 171 +articles remained for the final evaluation. The used exclu- +sion criteria were: +EC1 The length of the study is less than four pages +EC2 The study is not published in conference proceedings +or as a journal article +EC3 The study does not have full text available in English +EC4 The study turned out not to describe a software engi- +neering capstone course in a tertiary institution +EC5 The study is not able to provide answers to most of the +research questions +Exclusion criteria from EC1 through EC3 aimed to en- +sure that the study was of sufficient quality. According to +Kitchenham and Charters (2007) workshop proceedings of- +ten do not provide sufficient input for the purposes of an +SLR. Additionally, quite many of the papers that were first +published as short workshop proceedings or abstracts were +also found to have a conference proceeding or a journal arti- +cle published later on. EC3 relates to the language skills of +the authors as well as the status of English as the primary +language in software engineering-related research. These +exclusion criteria led to some papers being rejected before +reading their entire content. +For exclusion criteria EC4–EC5, the content of the ar- +ticle was examined more carefully and, in most cases, read +in its entirety to make a justified decision. Exclusion crite- +ria EC4 and EC5 relate to our research goal. For instance, +many articles were found to describe courses in computer en- +gineering or mini-projects conducted prior to SE capstones +which meant that they were out of the scope of this research +and excluded based on EC4. Most of the papers left out dur- +ing this stage met EC4. As for EC5, some studies were, for +example, found to describe a whole curriculum with cap- +stone courses playing only a minor part in the research, and +they could therefore not provide answers to our research ques- +tions. A number of studies also evaluated a tool, method or +framework relevant to the software engineering industry, not +the capstone course itself. In many of these studies, the cap- +stone course presented the researchers merely a convenient +way of gaining study participants, which is why they did not +fit this research. This led them to be excluded due to EC5. +3.3.3. The third stage - Removal of duplicates and +studies of poor quality +The third stage was included mainly to rule out any du- +plicate data and studies of poor quality from our research. +After the third stage, the final set of 127 papers remained. +Duplicate data – All 171 primary studies remaining af- +ter the second stage describe real-life software engineering +capstones. As educators often like to modify their courses +over time to find the best ways of teaching, studies here too +reflect on the changes done to the courses. Some authors +also have written multiple articles based on the same cap- +stone course. In such cases, the most recent article was cho- +sen. Similarly, if a study describes several instances of the +course in one paper, the principal characteristics of the most +recent course instance were chosen for the data extraction. +Choosing the latest instance of each course stems from the +goal of this research to synthesise the current state of knowl- +edge on capstone implementations. In addition, the decision +of whether two descriptions of the same course are different +enough for them to be included as their own capstone courses +would have been too ambiguous and open for interpretation. +Kitchenham and Charters (2007) also state that it is impor- +tant not to include multiple publications of the same data, as +it would seriously bias any results. Due to this procedure, 42 +studies were removed from the final set. +Quality assessment – In addition to inclusion/exclusion +criteria, Kitchenham and Charters (2007) state that it is crit- +ical to perform a quality assessment on the primary studies. +We also conducted a such assessment and used it to ensure +that our final data set is of sufficient quality. At this stage, +two studies were filtered out. Any tables or graphs presented +from here on do not include these two excluded studies, and +therefore represent the final set of 127 studies. +Table 5 lists the set of questions used by Dybå and Dingsøyr +(2008); Ali et al. (2010); Mahdavi-Hezavehi et al. (2013) +which we also used to determine the quality of primary stud- +ies. Originally the questions were supposed to be graded on +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 6 of 22 + +Capstone courses in software engineering +Figure 2: Quality scores of the final set of studies +a dichotomous (“Yes“ = 1 or “No“ = 0) scale (Dybå and +Dingsøyr, 2008), but we decided to use a three-point scale +of “Yes“ (= 1), “To some extent“ (= 0.5) and “No“ (= 0). +This three-point scale has also been adopted by Ali et al. +(2010) and Mahdavi-Hezavehi et al. (2013) and allowed us +better to assess the studies where authors only provided some +answers to the question. The two articles filtered out had a +quality score of less than 4. +In our assessment, we decided to group the first eight +questions to represent the quality of reporting and rigour of +the studies and final three questions to represent the credibil- +ity of evidence, similarly to Ali et al. (2010). The grouped +scores are presented in Fig. 2 and individual scores for each +study can be found at https://github.com/article-additions. +Regarding the quality of reporting, the selected primary stud- +ies performed fairly well. It was mostly clear how the data +had been collected, and the relevance of the study was ex- +plicitly discussed. However, the aspect most studies were +lacking was providing justifications either for the sample se- +lection or research designs. Regarding the credibility of ev- +idence, the studies performed fairly poorly. Interestingly, +many of the otherwise well-established studies did not in- +clude a section for explicitly discussing the limitations of the +study or the author’s role in data and sample selection. This +is indicated in the low averages of the credibility category. +Table 5 +Questions for quality assessment +No. +Question +Q1 +Is there a rationale for why the study was undertaken? +Q2 +Is there an adequate description of the context (e.g. indus- +try, laboratory setting, products used, etc.) +in which the +research was carried out? +Q3 +Is there a justification and description for the research de- +sign? +Q4 +Has the researcher explained how the study sample (partic- +ipants or cases) was identified and selected, and what was +the justification for such selection? +Q5 +Is it clear how the data was collected (e.g. through inter- +views, forms, observation, tools, etc.)? +Q6 +Does the study provide a description and justification of the +data analysis approaches? +Q7 +Has ‘sufficient’ data been presented to support the findings? +Q8 +Is there a clear statement of the findings? +Q9 +Did the researcher critically examine their own role, poten- +tial bias and influence during the formulation of research +questions, sample recruitment, data collection, and analysis +and selection of data for presentation? +Q10 +Do the authors discuss the credibility of their findings? +Q11 +Are limitations of the study discussed explicitly? +3.3.4. Overview of the final papers +The three stages taken resulted in 127 primary studies, +published between 2007 and June 2022. Research activity +in this area has been fairly steady over the years, as depicted +in Figure 3. It is worth noting, that we searched for papers in +June 2022, making the study amount for 2022 partial. Also, +as explained in Section 3.3.3, 42 earlier studies, which other- +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 7 of 22 + +2.5 +2 +Credibility +1.5 +0.5 +8 +12 +19 +2.5 +3 +3.5 +4.5 +5.5 +6 +6.5 +7 +7.5 +8 +Quality of reporting and rigorCapstone courses in software engineering +wise would have been valid for this research, were excluded +from the final set of papers as there was a newer study of the +same course available. This procedure skews the year distri- +bution towards the end of the scale. The figure also shows +the distribution by study type. In total, of studies 73% were +conference proceedings, and 27% of studies were published +as journal articles. +Figure 3: Timeline and types of primary studies +All the articles included for further analysis are listed in +Appendix A, Table 13, and referenced later in this section +with their publication ID in the table (i.e., S1–S127.) +3.4. Data extraction and synthesis +After applying the study selection process, the properties +presented in Table 6 were extracted from the remaining 127 +studies to a common datasheet. Table 6 defines how each +extracted field relates to the research questions of this study. +Table 6 +Data extraction form +Identifier +Field +RQ +F1 +Title +metadata +F2 +Author(s) +metadata +F3 +Year +metadata +F4 +Publication venue +metadata +F5 +Duration of the course +RQ1 +F6 +Course workload +RQ1 +F7 +Team sizes +RQ2 +F8 +Clients +RQ3.1 +F9 +Project sources +RQ3.2 +F10 +Artefacts produced +RQ4.1 +F11 +Project phases +RQ4.2 +F12 +Technologies +RQ4.3 +F13 +Student assessment +RQ5 +F14 +Outcomes of the course +RQ1–RQ5 +F15 +Quality score +QA +Values F1–F4 were extracted for basic documentation +purposes. Items F5–F14 concern the course and its organisa- +tion presented in the study. Two of the studies present multi- +ple separate capstone courses from different institutions. For +these two studies (S8 and S93 in Table 13), the items F5–F14 +were extracted for each of the courses they presented. For +F5–F14, we were not only interested in quantifying these +characteristics into statistics but also in providing implica- +tions of different course design choices. Therefore, if the +study stated, for example, that they had a two-semester cap- +stone course because it provided students adequate time to +learn, we recorded both of these information pieces: the +quantifiable duration as well as any such insight relating to +the characteristic. This enabled us to analyse and discuss the +course characteristics better in Section 4. A data-driven the- +matic analysis was applied to synthesise the qualitative data +extracted as part of F5–F14 (Castleberry and Nolen, 2018). +F5 and F6 were considered essential in assessing the gen- +eral workload and duration of the course from the student’s +perspective. Few sources have given the duration of their +course (F5) as months or weeks. These were rounded to +the nearest amount of semesters. Courses lasting less than 4 +months were categorised as “less than one semester“, 4 to 6 +months as “one semester“ and anything more than 6 but less +than 10 months as “two semesters“. +Team sizes (F7) included the number of students per team. +The courses were also examined on whether the projects in +the course were done for a client (F8). The client could be ex- +ternal to the course staff, the role of a client could be played +by the course staff, and some projects did not have clients at +all. Some sources present a mix of these categories, in which +case, the source was labelled by the client category, which +we thought was the most prevalent in the course. +We were also interested in how the project topics were +generated (F9). Three main sources for projects were identi- +fied during the data extraction: course staff, external clients +and the students themselves. The projects were also found +to vary regarding whether the students were working on the +same project idea or whether each team had their own initial +problem to solve. +We extracted all the artefacts that students were expected +to produce during the course (F10). This included both the +deliverables used for grading the course as well as artefacts +produced for project management reasons, as in most cases, +it was hard to draw a distinction between the two. Evidence +of project phases (F11) was extracted to find out which soft- +ware life-cycle activities are gone through in these courses. +F12 describes the technologies used in the course. We found +that most of the studies do not explicitly specify all the tech- +nologies used for the projects in their courses, and more- +over, these technologies could potentially include any soft- +ware technologies available. We, therefore, categorised these +into two categories, based on whether the main technology +selections are made team-wise, or all use a common technol- +ogy stack. +We extracted information on how the students learning +process was assessed and improved throughout the course +and how the student’s progress and achievement were as- +sessed at the end of the course (F13). The key outcomes in +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 8 of 22 + +16 +14 +12 +10 +8 +6 +4 +2 +■Conference papers +■ Journal articlesCapstone courses in software engineering +Table 7 +Duration of capstone courses +Category +Number of studies +Percentage +Study identifiers +Less than one semester +10 +8% +S16, S18, S40, S46, S55, S61, S78, S94, S99, S113 +One semester +87 +66% +S1, S2, S3, S4, S5, S8b, S8d, S9, S11, S13, S15, S20, +S21, S22, S23, S24, S25, S26, S27, S30, S31, S32, S34, +S35, S36, S37, S38, S42, S43, S44, S45, S47, S48, S51, +S53, S54, S56, S57, S58, S60, S62, S63, S64, S65, S66, +S67, S68, S69, S70, S73, S74, S75, S76, S77, S79, S80, +S81, S82, S83, S84, S86, S89, S90, S92, S93a, S93b, S95, +S97, S100, S102, S103, S104, S105, S106, S107, S112, +S115, S116, S117, S119, S120, S121, S123, S124, S125, +S126, S127 +Two semesters +32 +24% +S6, S7, S8a, S8c, S10, S12, S14, S17, S19, S28, S33, S39, +S41, S50, S52, S59, S71, S72, S85, S87, S88, S91, S96, +S98, S101, S108, S109, S110, S111, S114, S118, S122 +More than two semesters +1 +1% +S49 +Not specified +1 +1% +S29 +the study (F14) were also extracted to assess the advantages +and disadvantages of the presented capstone characteristics. +F15 was extracted as presented in Section 3.3.3 for quality +assessment and study filtering. +4. Results and analysis +This section represents quantitative statistics and qual- +itative outcomes of capstone characteristics extracted from +the primary studies. The characterisation enables us to an- +swer our research questions and ultimately helps educators +when they are planning their capstone courses. +4.1. Duration (RQ1) +Regarding the actual course characteristics, we first looked +into the reported duration of these courses (F5). A clear +majority of institutions conduct capstone courses that last +one semester (Table 7). Interestingly, this is in conflict with +the ACM/IEEE (2014) recommendations for undergraduate +capstone courses, which propose having capstones lasting +the entire academic year. However, the unfortunate reality +is that not all curricula can absorb a full-year implementa- +tion [S117]. The capstone courses often are very labour- +intensive for the teaching staff, with many teams to man- +age and evaluate throughout the projects [S39], [S112]. Stu- +dents might have full- or part-time work, which makes the +longer courses harder to arrange [S49], [S58], [S73]. Stu- +dents also perceive two-semester capstone courses as labo- +rious [S73], [S109], and some even the one-semester ones +[S23], [S41]. In order to provide an intensive and realis- +tic experience, many of these courses take up at least half a +work-week [S18], [S28], [S41], [S69], [S73], [S109], [S127]. +This again might make other courses taken simultaneously +suffer [S41], which limits the possibilities for an intensive, +year-long capstone. +However, the educators who had experiences with both, +shorter and longer duration, had shifted to the longer du- +ration since they felt it was impossible to reach the wanted +depth in just a few months. S6 describes how they switched +to a two-semester capstone as they found the one-semester +projects inadequate in skill coverage and depth. S70 is writ- +ten by students of the course, and they strongly recommend +that their course be lengthened into one academic year from +the current duration of one semester. S33 have had experi- +ences with one-period, i.e. a quarter of an academic year, and +three-period courses, and stated that the change to a longer +version received overwhelmingly positive feedback from all +the participating parties. Students were able to gain more +hands-on experience in applying new and familiar tools and +project management. Additionally, they learned to act when +faced with unanticipated events as the teams experienced +surprises – regarding both technologies and people – mul- +tiple times during the year. Industrial clients received more +ambitious and polished products as a result of the course, +and the course staff felt that the learning objectives for the +course were finally truly met. +4.2. Team sizes (RQ2) +To find out how many students there generally are in a +project group, we extracted the reported team sizes (F7) for +each course in Fig. 4. If a study refers to their course having +teams of 4–5 students, this is thus reflected in both columns +4 and 5. By looking at Fig. 4, it is evident that capstone +courses are almost always conducted as group projects. Only +three institutions in our research allow their capstone or se- +nior project courses to be completed as single-student en- +deavours [S11], [S89], [S111]. +Team sizes vary a great deal, ranging from 1 to 35. Re- +search has found that in very small groups, e.g., 2–3 stu- +dents, the teams are unlikely to generate the dynamics and +issues that are common in collaborative software develop- +ment [S36], [S53], [S56], [S58]. Such a small team size +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 9 of 22 + +Capstone courses in software engineering +Figure 4: Team sizes in capstone courses +does not present enough of a challenge [S36], and smaller +groups are unable to complete substantial projects in a typ- +ical one-semester course [S53]. Having very small teams +might also be unmaintainable in large programs with hun- +dreds of, or even a thousand, students due to the extra or- +ganisational overhead each team causes [S19]. Going to the +other extreme, larger groups with 7 or more students have +often been found to be facing other kinds of problems, such +as the inability to meet all together and other management +and coordination issues [S30], [S36], [S39], [S53], [S56], +[S78]. “Free-rider“ problem is also reportedly common in +larger teams, where it is possible for few students to take the +bigger responsibility for ensuring the overall success, and the +small contribution of others might go unnoticed [S9], [S56], +[S58], [S69], [S87], [S121]. In larger teams, ensuring fair +grading and an equal balance of work and responsibilities +requires more attention from the course staff [S56], [S87]. +The course conducted in S106 had one of the largest team +sizes found in our research. The course had 15 students, all +working on the same game project in one team. The idea was +to simulate what large-scale game development in a diverse +team feels like and what it takes to create production-quality +games. The authors share that their approach was not en- +tirely successful. In the aftermath of the course, it came up +that some students wanted explicit direction while others felt +that they wanted more autonomy and control. According to +the authors, for the latter group of students, it was clear that +they were uncomfortable following the leadership of the vi- +sion team and would have preferred to work on a project of +their own design. However, the authors also mention that +getting to work with your own project vision is a very un- +likely case for any recent graduate, which is why they did +try to come up with such a real-world teamwork scenario. +The majority of educators do seem to opt for the middle +ground regarding team sizes and have 4 to 5 people working +in a single group (Fig. 4). This size is perceived as the sweet +spot, cancelling out the negatives of the two extremes [S36], +[S52], [S53], [S56], [S58]. Students themselves have also +reported being satisfied with such a team size [S56]. Addi- +tional measures for combating any non-productive and op- +portunistic group behaviour, such as social loafing and free +riding, have also been proposed. Conducting peer reviews +has been proven to mitigate the risk of such behaviour [S72], +[S98]. Some periodic monitoring should also be done by the +course staff to ensure working team dynamics [S69]. Both +of these will be discussed further in Section 4.5. +4.3. Clients and project ideas (RQ3) +4.3.1. Clients (RQ3.1) +We also looked at who is in the role of a client for these +projects (F8) and how the project ideas are sourced (F9). Al- +most half of the studies (42 %) report conducting their cap- +stone courses without clients that are external to the course +(Table 8). In these courses, the course staff may act as clients +or Product Owners for the projects or alternatively, the stu- +dent teams work on their own and only report progress reg- +ularly to the course staff. S6 explains that they have instruc- +tors playing clients due to the difficulty of finding suitable +clients. Being a small program in a rural institution makes +the businesses and organisations suited for such collabora- +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 10 of 22 + +70 +62 +60 +56 +50 +40 +Occurences of team sizes +34 +33 +30 +20 +19 +18 +15 +10 +10. +6 +3 +2 +2 +1 +1 +1 +1 +1 +1 +1 +0 +8 +1 +06 +9 +30 +3 +V +VotCapstone courses in software engineering +Table 8 +Clients of capstone courses +Category +Number of studies +Percentage +Study identifiers +Clients +external +to +the +course staff +76 +58% +S2, S3, S4, S5, S7, S8a, S8d, S9, S10, S11, S14, S15, +S17, S18, S19, S23, S24, S27, S28, S29, S30, S31, S33, +S34, S35, S37, S38, S39, S41, S46, S48, S49, S50, S52, +S55, S58, S59, S60, S61, S62, S63, S66, S67, S69, S71, +S73, S78, S80, S81, S84, S85, S86, S87, S88, S90, S91, +S92, S93a, S93b, S94, S96, S97, S98, S104, S108, S110, +S112, S113, S114, S117, S120, S121, S122, S124, S126, +S127 +Course staff acts as clients +16 +12% +S1, S6, S12, S21, S40, S43, S53, S54, S57, S68, S76, S82, +S99, S115, S116, S123 +No clients +38 +29% +S8b, S8c, S13, S20, S22, S25, S26, S32, S36, S42, S44, +S45, S47, S51, S56, S64, S65, S70, S72, S74, S75, S77, +S79, S83, S89, S95, S100, S101, S102, S103, S105, S106, +S107, S109, S111, S118, S119, S125 +Not specified +1 +1% +S16 +tion far and scarce. Also, the institution’s wish to own the +intellectual property rights for the developed products puts +off potential clients. For institutions, that would have suit- +able clients available, there is always the upfront investment +in time and effort that the course staff has to make to contact +said clients, guiding them through creating project propos- +als and assigning the students to these projects [S118]. S42 +aimed to create a course with students from five different +technical and non-technical disciplines, such as computer +science and business informatics. S42 mentions that they +need to be careful in how they organise the course so that +it would suit the needs of all disciplines. Bringing an ex- +ternal client into the mix might not fulfil the learning goals +for all students. Some studies also explain how the course +outcomes are less predictable with multiple external clients. +S114 have experienced several cases when the project spon- +sors did not show up for the bi-weekly meetings with the stu- +dents. Such client behaviour caused very low motivation in +the student teams, and some capstone projects failed due to +client unavailability. S3, S23, S78 and S122 have made sim- +ilar observations and stress the importance of finding com- +mitted clients to ensure a good experience for the students. +Despite these risks having real, external clients other than +the course supervisor for the projects is recommended for +both undergraduate and graduate capstones (ACM/IEEE, 2014; +ACM, 2009). These clients can be from other units within +the university [S7], [S9], [S14], [S52], [S58], [S86], [S87], +[S94], local businesses [S7], [S9], [S86], [S87], [S98], [S110], +[S122] or various non-profit organisations [S7], [S9], [S11], +[S16], [S58], [S110]. Graduates of the program who already +work in the industry are also a convenient way to find clients +[S35]. Working closely with real-world clients has also of- +ten received highly positive feedback from students [S14], +[S15], [S52], [S73], [S98], [S117] and organising staff alike +[S14], [S35], [S84], [S98], [S117]. It has been found to in- +crease the motivation and commitment of students, when +there is an actual client with a real need behind the project +[S9], [S14], [S15], [S35], [S66], [S73], [S84], [S121]. It has +helped to keep the experience more realistic and credible in +the students’ eyes [S19]. Having industry clients improves +the students’ technical and nontechnical skills and better pre- +pares them for the challenges they will face in the work-life +[S14]. +The collaboration has been reported to have benefits for +the client too. S35 conducted a study to find out the rea- +sons why clients participate in such project courses. The rea- +sons included getting a tailored software product, research- +ing new technologies and, as a clear number one, recruit- +ment. Recruiting students could happen directly from the +team or more indirectly by adding visibility among the stu- +dents as potential employers. Others have noticed this ben- +efit, too; it is not uncommon for students to get hired by the +industry partner who sponsored their capstone project [S15], +[S28], [S35], [S73], [S84], [S114], [S121]. S73 report hav- +ing at least 60 out of a few hundred students gaining full- or +part-time job offers based on the capstone project outcomes, +in mere few years. Keeping the experience positive also for +the clients, might make them come back with further project +ideas [S35]. This helps to reduce the client acquisition over- +head for years to come. Some organising institutions have +even managed to attract more external clients than there are +student teams, which has enabled them to collect a small fee +from the ones participating in the course [S35], [S121]. +4.3.2. Project sources (RQ3.2) +In addition to finding out the clients for these projects, +we also looked into how the projects for these courses are +sourced (F9). Three main ways for project sourcing were +identified (Table 9). As the majority of courses have multiple +external clients, the project ideas in these courses are mainly +derived from the needs of the customer. In these cases, the +organising staff often performs some pre-screening and scop- +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 11 of 22 + +Capstone courses in software engineering +Table 9 +Sources for projects in capstone courses +Category +Number of courses +Percentage +Study identifiers +External +stakeholders +pro- +pose project ideas +81 +62% +S2, S3, S4, S5, S7, S8a, S8b, S8c, S8d, S9, S10, S11, +S14, S15, S17, S18, S19, S23, S24, S27, S28, S29, S30, +S31, S33, S34, S35, S37, S38, S39, S41, S46, S48, S49, +S50, S52, S55, S58, S59, S60, S61, S62, S63, S66, S67, +S69, S71, S72, S73, S77, S78, S80, S81, S84, S85, S86, +S87, S88, S90, S91, S92, S93a, S93b, S94, S96, S97, S98, +S104, S108, S110, S112, S113, S114, S117, S120, S121, +S122, S124, S125, S126, S127 +Course staff provides project +ideas +27 +21% +S1, S6, S12, S13, S16, S20, S21, S25, S40, S42, S43, S45, +S54, S57, S65, S68, S74, S75, S79, S82, S99, S105, S106, +S109, S115, S116, S123 +Students generate their own +project ideas +22 +17% +S22, S26, S36, S44, S47, S51, S53, S56, S64, S70, S76, +S83, S89, S95, S100, S101, S102, S103, S107, S111, S118, +S119 +Not specified +1 +1% +S32 +ing in collaboration with the clients, to ensure that the ex- +pectations for the projects are realistic and that the project +scopes suit the intended learning outcomes [S9], [S24], [S35], +[S52], [S61], [S87], [S90], [S94], [S98], [S121]. Capstone +projects should generally not be on the critical path of any +external organisation, as the course is intended to remain +a safe learning place for the students [S35], [S52], [S87], +[S90], [S121]. Some studies also emphasise that students +are not supposed to be working for these clients, but be in +collaboration with them [S9], [S52], [S87], [S121]. Thus, +the projects need to remain such that the students can have a +say in how the project will be developed. +For 17% of the courses, the students themselves are the +main source for project ideas. These studies state that stu- +dents are more motivated if they get to choose the project +idea rather than have teachers assigning the projects [S36], +[S53]. According to S53, if the team selects and defines +the projects, their level of commitment and excitement to +the project rises as the software system grows. At the end +of the semester, the students have a strong sense of owner- +ship towards the project, rather than feeling that they have +just done one additional assignment [S36], [S53]. However, +there are some potential pitfalls with this approach that edu- +cators should be aware of. S114 states that students should +not be allowed to bring project ideas from the companies +they work at or from their own businesses. S114 have found +that it causes a conflict of interests for the student with the +proposal and creates an unfair situation for the rest of the +team. S36 lets students form their own teams and generate +their own project ideas, but states this might not accurately +reflect the situation in the students’ future professional lives. +If the project idea comes from the team itself, all complex- +ities associated with requirements elicitation and analysis +are eliminated [S73], making the experience less realistic. +Real-life projects come with challenges relating to contra- +dicting expectations coming from various external and in- +ternal stakeholders [S73]. +For 21% of the courses, the course staff provides the +project specifications. Some educators have assigned the +same project idea to all the student teams [S40], [S45], [S72], +[S82] or even in some cases, all the students work on the ex- +act same project in one team [S106], [S112]. Having the +same project has the benefit of giving the course staff a con- +sistent basis for grading and teaching [S40], [S59], [S72] and +providing technical assistance to the students [S40], [S53]. +In such cases, all teams will need to deal with the same +complexities, project management issues and technology de- +mands as in a typically constructed course, which makes the +experience more predictable [S72]. Having one project idea +also opens up the possibility for competition amongst the +teams, e.g., which team will create the best design and im- +plementation [S5], [S30], [S37], [S42], [S54], [S72], [S106] +potentially even for an external client [S30]. It also possi- +bly allows the course to focus more on the quality of the +developed software [S5]. S5 has experience with both ap- +proaches, having multiple project ideas and having only one +project. They have found it more productive and rewarding +to focus on doing one project really well rather than juggling +multiple projects and obtaining partial results. +4.4. Project implementation (RQ4) +We also aimed to uncover information on how capstone +projects are implemented and what students are expected to +do in these courses. For this, we extracted various informa- +tion on the project implementation (F10–F12). +4.4.1. Produced artifacts (RQ4.1) +We were interested in finding out what kind of artefacts +students are expected to produce on the course (F10). It was +often difficult to determine which artefacts were used for the +final student assessment (i.e. grading) and which were only +produced to manage the project in some way. Therefore we +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 12 of 22 + +Capstone courses in software engineering +Figure 5: Most commonly mentioned artefacts produced by students in capstone courses +could not produce a list specifically of graded artefacts. Ta- +ble 5 provides a rough view of the explicitly mentioned arte- +facts that students are expected to produce. +We were, however, able to determine that all but three +courses [S16], [S17], [S103] expect some form of a software +prototype, a software product or source code as the end de- +liverable. Out of the courses which did not include produc- +ing software, S16 conducted a course where the process and +end deliverables are focused on students completing various +research items, such as testing new team-based technolo- +gies (e.g. pair programming). For S17 the end deliverable +was often a solution proposal for the client organisation’s IT +department, and S103 delivered an inter-disciplinary course +which might result in software deliverables or alternatively +in reports describing a software-based solution to the defined +problems, e.g. using 3D engines for art. +Most studies mention requiring some documentation for +the software project (Fig. 5). The actual number of courses +that require documentation might be larger than reported here, +as we only counted the times the study explicitly mentions +that project documentation is done on the course. Studies +at the beginning of our time range are often following more +“plan-first“ software development approaches, such as the +waterfall model, and as a result, the quantity and detail of +non-software artefacts are substantial [S2], [S11], [S22], [S70]. +The documentation usually starts heavily upfront by students +producing project plans [S2], [S68], [S70], detailed designs +[S2], [S22], [S52], [S68], architecture plans [S22], [S68] and +test plans [S2], [S70]. In later, more agile, courses, there +is less evidence of extensive documentation and planning. +With agile projects, the system documentation is largely de- +veloped as the project evolves [S14], [S65], [S94]. Students +in these courses often also produce agile artefacts, such as +product backlogs, sprint backlogs and burndown charts for +project management and planning purposes [S6], [S65], [S94], +[S124]. +A large share of the studies (60%) explicitly mention that +students must present or demonstrate their projects to wider +audiences than just the immediate project team. This is a +way to teach students how to present and explain their work +also to a non-technical audience [S6]. The most common +time for presentations is at the end of the semester, and typ- +ically these presentations are given at fairs or class sessions +where all stakeholders of the course are invited [S2], [S4], +[S6], [S33], [S95]. The format of final presentations varies +from poster sessions [S6], [S28], [S34], [S44] to live demon- +stration sessions of the software [S89], [S95] to different +kinds of demo videos [S1], [S73], [S118]. Students are some- +times also expected to draw up a project proposal or a pitch +and then present it to the teachers or the class before start- +ing to work on the implementation [S87]. In these cases, +the purpose is to offer the students a chance to practice their +pitching skills [S100]. +Software requirements are something that students are +often expected to detail during the course. These can be writ- +ten down as a Software Requirements Specification created +before the implementation phase begins [S30]. For courses +with agile methodologies, software requirements are often +documented in an initial backlog with user stories, and the +backlog is then updated as the project continues [S124]. Stu- +dents are also quite often expected to write some form of a +report at the end of the experience, either individually or as +a group. The reports generally involve students reflecting +on the learning done and development processes employed +throughout the course [S6], [S19], [S21], [S24], [S27], [S41], +[S63], [S65]. +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 13 of 22 + +% of courses +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Software or prototype +97% +System documentation (technical/architecture/design) +70 % +Presentations and demos +60 % +Requirements specification +40 % +Final and/or mid-term reports +36% +29% +Agile artifacts +Regular progress reports +14%Capstone courses in software engineering +The balance between too little and too many other arte- +facts is a delicate one. Having more documentation and de- +liverables presents teachers with more opportunities to grade +and assess students’ understanding of software development +processes [S110]. On the other hand, more documentation +means less product which might not be in the interests of +project clients [S49]. Indeed, some educators mandate only +basic time tracking and reflective reporting from the students +and have left the majority of the deliverables for the external +client and team to decide [S24]. +4.4.2. Project phases (RQ4.2) +We sought evidence of the project phases and software +life-cycle gone through on these courses (F11). Software +life-cycle models include, with varying frequency and order, +phases such as requirements gathering or solicitation, plan- +ning and designing, developing, testing and maintaining the +product (Mishra and Dubey, 2013). Out of the studies that +discuss the development process and end product quality, +the projects generally proceed from ideas to robust proof- +of-concepts or products with few core requirements imple- +mented [S1], [S6], [S7], [S11], [S12] [S16], [S18], [S20], +[S21], [S22], [S26], [S40], [S42], [S45], [S52], [S54], [S55], +[S62], [S64], [S65], [S68], [S79], [S70], [S72], [S75], [S77], +[S78], [S87], [S90], [S94], [S95], [S99], [S100], [S106], +[S107], [S109], [S115], [S118], [S122], [S123], [S124], [S125]. +Students thus get to get experience the phases of planning, +designing, developing and testing the products in these projects. +In courses with clients, either external or internal, the stu- +dents usually have to solicit the requirements from the clients +(Table 9). However, sometimes the teachers provide stu- +dents with ready-made feature or requirements lists [S12], +[S21], [S45], [S79], [S82] and in some courses, students gen- +erate their own project proposals (Table 4.3.2). The experi- +ence of requirements gathering is somewhat diminished in +these cases. +Additionally, as the projects proceed from ideas to proofs- +of-concept or simple, handed-off products, the projects gen- +erally do not include developing existing products, especially +ones that are in production-use during the course. There are +some courses where some of the projects have been production- +ready at the end of the course, but these too were then handed +over to the customer [S5], [S9], [S117]. This practice leaves +students without the experience of working with existing prod- +ucts or products in the true maintenance phase of their soft- +ware life-cycle. Assigning students to contribute to Free +and Open Source Software (FOSS) projects is an emerging +approach to remedy these shortcomings. The idea is to al- +low students to deal with existing codebases, often large and +complex, such as the one they will face when working in the +industry [S8b], [S8c], [S112], [S113]. Some courses have +also had a continuation of earlier projects in the course to ex- +pose students to code generated by other people [S14], [S15], +[S19], [S38], [103]. Both of these approaches allow students +to maintain existing code, but they still present a minority in +our research. +4.4.3. Project technologies (RQ4.3) +We also looked into the development technologies used +in capstone courses and how they are selected (F12). Com- +monly in multi-customer courses or in courses with other- +wise very differing project ideas, the technology choices are +made based on the project (Table 10). In these cases, the +course staff does not impose an entirely common technology +stack for all the projects. For some of these projects, the tech- +nology stack is based on the client’s infrastructure [S35] and +in some cases, the students get to make manager-like deci- +sions on the suitable development technologies [S6], [S84]. +Having the teams decide on the tools and technologies makes +the students explore available options and justify their se- +lections [S6], [S56], [S84]. S84 states that not only give +them autonomy but also make them responsible for their own +successes and failures. However, even though the majority +of technologies would be selected based on the project and +client, some studies recommend having some shared infras- +tructural tools and technologies [S12], [S19], [S36], [S65], +[S102]. Version-control [S6], [S12], [S19], [S29], [S36], +[S67], [S102], project management and communication tools +[S19], [S29], [S65], [S67], [S80] and tools for continuous in- +tegration and delivery are examples of these [S32], [S78]. It +has been found to make the management and evaluation of +projects easier [S19], [S29]. Then again, having common +development technologies for all projects is fairly common +in cases, where the teachers provide students with the project +requirements [S42]. In some cases, the evaluation meth- +ods focus heavily on the technical implementation, and the +course graders might, for example, have sets of tests they like +to run on each project to determine the quality [S109]. Some +educators have the students compete on the same project +proposal, which makes choosing a common stack justifiable +[S37]. +4.5. Assessment of students (RQ5) +We were also interested in finding out how the assess- +ment is conducted in these courses (F13), and to which ex- +tent students are given possibilities to reflect on their experi- +ences. We looked at both the end-of-course student assess- +ment (Section 4.5.1) and any continuous guidance and feed- +back students are given during the course (Section 4.5.2). +4.5.1. End of course student assessment (RQ5.1) +All studies which explicitly described the course evalu- +ation process had course teachers involved in it (Table 11). +Teachers are generally the ones assessing any artefacts pro- +duced by students, such as the software product itself, reports +and documentation done of the project [S118]. However, +due to the shift from traditional development approaches to +more agile ones, the quantity of delivered artefacts has de- +creased over time, leaving teachers with fewer data points +for assessing students’ comprehension of software develop- +ment processes [S110]. In addition, many of the learning +goals in capstone courses relate to soft skill development, +such as the ability to work in a team or with a client and be- +ing able to manage a software project [S6], [S86], [S122]. +These skills are generally employed when the teaching staff +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 14 of 22 + +Capstone courses in software engineering +Table 10 +Development technologies in capstone courses +Category +Number of courses +Percentage +Study identifiers +Projects use common tech- +nologies +33 +25% +S1, S5, S8a, S8d, S11, S13, S22, S32, S37, S38, S42, S43, +S45, S46, S48, S53, S54, S57, S59, S72, S76, S79, S82, +S83, S93b, S99, S105, S106, S107, S109, S112, S113, +S124 +Choices +done +primarily +team-wise +65 +50% +S2, S3, S4, S8b, S8c, S9, S10, S12, S14, S19, S24, S26, +S29, S31, S33, S35, S36, S39, S44, S47, S50, S51, S52, +S56, S58, S61, S62, S65, S66, S68, S69, S71, S73, S74, +S75, S77, S80, S84, S85, S86, S87, S88, S89, S90, S92, +S95, S96, S98, S100, S101, S102, S103, S104, S110, S111, +S114, S116, S117, S118, S119, S120, S121, S122, S125, +S127 +Not specified +33 +25% +S6, S7, S15, S16, S17, S18, S20, S21, S23, S25, S27, S28, +S30, S34, S40, S41, S49, S55, S60, S63, S64, S67, S70, +S78, S81, S91, S93a, S94, S97, S108, S115, S123, S126 +is not present, and teams work on the projects on their own, +making the evaluation of these skills harder [S52], [S56], +[S86]. For these reasons, several studies report having ad- +ditional sources for student assessment beyond the teachers’ +evaluation of produced artefacts. Different kinds of (anony- +mous) peer evaluations are fairly common (31%). They give +course staff a look into the team dynamics during the course +and help in detecting social loafing or free-rider behaviour +[S12], [S86], [S112]. Similarly, self-evaluation is often done +either in combination with peer evaluations [S56] or as a part +of the reflection done in a project’s final report [S65]. +Some studies report utilising the client’s opinion in the +course assessment process. Clients can fill out a question- +naire considering each student’s performance during the course +[S86] or only evaluate the team’s deliverables or presenta- +tions and their value from the client’s point-of-view [S35], +[S52], [S89], [S127]. As with self- and peer-reviews, edu- +cators use the client’s opinion as a complementary source of +assessment when grading students (Table 11). +4.5.2. Continuous student assessment and guidance +(RQ5.2) +Many studies specifically mention that the teams should +not be left entirely on their own to complete the course project +and should be guided along the way [S6], [S9], [S10], [S16], +[S18], [S53], [S69], [S72], [S86]. Three main ways for con- +ducting such continuous assessment and guidance were found +based on our research (Table 12). Studies where there is +no mention of the teacher, or anyone else, having an active +role in how the teams work during the course, fall into the +category “Not specified“. If the course staff only passively +receives reports of the student’s progress and evaluates the +course outcomes after its completion, these are not the active +guidance of the teams we were looking for. Some courses +have several types of guidance present, in which case the +study has been listed under each corresponding category in +Table 12. Only 11% of the studies do not explicitly specify +having any ongoing feedback and guidance system present +during the course. +The most convenient way is to have the course staff, such +as the responsible teacher or hired teaching assistants, act- +ing in an advisory role (76%). The intensity of the guid- +ance given by course staff varies a great deal between these +courses, or even within these courses. Sometimes course +staff provides oversight in a more supervisory role and inter- +venes in the team’s work if any conflicts arise or the team in- +cludes clearly non-contributing students [S6]. On the other +end, some instructors have weekly meetings with the stu- +dents where the teachers actively propose solutions and guide +the teams with technical and non-technical issues and team +dynamics [S6], [S88]. Some teachers prefer even to prac- +tically manage the team [S6]. S38 explains that the most +successful changes made on the course were those that al- +lowed the course staff to take a more active role in each team. +The grades of students improved, and the teams were able to +complete more functionality of the software products. +Another, often complementary, guidance form is to have +industry experts occasionally participate in the course (16%). +This can be seen as especially relevant when the course projects +are focused around a common theme, for instance, the gam- +ing industry [S25], [S106]. However, finding the correct bal- +ance in this type of assessment, without a client relationship, +has sometimes proven to be tricky. S106 had industry ex- +perts from the gaming and software industries participating +as advisers on their course. The advisers’ feedback on the +students’ game product was mainly positive and encourag- +ing. While the staff took their remarks to mean that the game +concept and development for the moment were commend- +able, the students took the feedback to mean that the proto- +type was, as presented, worthy of praise. This presented a +dichotomy that never really resolved: students felt that the +project was near-complete, whereas the instructors felt that +the project was, at best, a rough sketch. +Many authors have noticed the upsides of having more +experienced students outside of the course staff, mentoring +the students in the course (18%). These can be, for instance, +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 15 of 22 + +Capstone courses in software engineering +Table 11 +End of course assessment +Category +Number of courses +Percentage +Study identifiers +Course staff +93 +71% +S1, S2, S4, S6, S7, S8a, S8b, S8c, S8d, S9, S12, S13, +S17, S18, S19, S20, S22, S23, S24, S25, S26, S27, S28, +S29, S30, S31, S33, S35, S36, S37, S39, S40, S41, S42, +S43, S44, S46, S47, S48, S51, S52, S53, S56, S58, S62, +S63, S64, S66, S67, S68, S69, S71, S72, S73, S74, S77, +S80, S81, S82, S84, S85, S86, S87, S88, S89, S90, S94, +S95, S96, S97, S98, S99, S100, S101, S102, S103, S104, +S105, S106, S107, S108, S109, S110, S111, S112, S116, +S117, S118, S120, S123, S124, S126, S127 +Students’ peer-evaluations +40 +31% +S2, S4, S7, S8a, S8b, S12, S13, S27, S30, S31, S33, S37, +S41, S42, S46, S48, S51, S52, S56, S58, S63, S64, S67, +S72, S73, S74, S81, S84, S86, S87, S90, S98, S106, S107, +S108, S110, S112, S117, S124, S127 +Students’ self-evaluations +24 +18% +S1, S2, S4, S22, S23, S27, S37, S41, S42, S47, S51, S56, +S62, S64, S69, S73, S74, S80, S81, S86, S87, S116, S126, +S127 +External project clients +17 +13% +S4, S20, S24, S29, S33, S35, S37, S42, S58, S73, S84, +S85, S86, S89, S96, S98, S127 +Others +1 +1% +S17 +Not specified +38 +29% +S3, S5, S10, S11, S14, S15, S16, S21, S32, S34, S38, S45, +S49, S50, S54, S55, S57, S59, S60, S61, S64, S65, S70, +S75, S76, S78, S79, S83, S91, S92, S93a, S93b, S113, +S114, S115, S119, S121, S122 +Table 12 +Continuous student assessment and guidance +Category +Number of courses +Percentage +Study identifiers +Course staff +99 +76% +S2, S3, S4, S6, S8a, S8b, S8d, S9, S12, S15, S16, S17, +S18, S19, S20, S22, S23, S24, S25, S27, S28, S29, S30, +S31, S32, S35, S36, S37, S39, S40, S41, S43, S44, S45, +S46, S47, S48, S49, S50, S51, S52, S53, S54, S55, S56, +S57, S59, S60, S62, S63, S64, S65, S66, S67, S68, S69, +S71, S73, S74, S75, S76, S77, S78, S80, S81, S82, S84, +S85, S86, S87, S88, S90, S91, S92, S93a, S94, S95, S96, +S97, S98, S99, S100, S101, S102, S103, S105, S106, S108, +S109, S110, S112, S113, S114, S115, S116, S122, S123, +S124, S126, S127 +More experienced students +23 +18% +S5, S8a, S12, S14, S17, S19, S35, S40, S48, S58, S61, +S68, S81, S85, S92, S95, S104, S105, S112, S113, S118, +S119, S121 +Industry advisers (other than +project clients) +22 +17% +S5, S8a, S8b, S8c, S10, S20, S25, S52, S58, S65, S71, +S72, S73, S77, S80, S83, S89, S93b, S98, S106, S118, +S120 +Not specified +15 +11% +S1, S7, S11, S13, S21, S26, S33, S34, S38, S42, S70, S79, +S107, S111, S117 +students who have completed the project course themselves +in the past year [S122]. This has been found to benefit both +the project implementation and group dynamics: an active +and knowledgeable coach can, for example, help students +ask clarifying questions of the customer, overcoming fear of +these being stupid and saving days or weeks [S122]. +Interestingly, forming a capstone team of the final year +students with similar skill levels are in accordance with the +ACM/IEEE Curriculum Guidelines for Undergraduate SE +Degree Programmes (ACM/IEEE, 2014), but leaves out an +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 16 of 22 + +Capstone courses in software engineering +integral part of the real software development team experi- +ence: junior and senior positions. This discrepancy has been +noted in some studies [S19], [S35], [S58], [S98], where the +course implementation has gone beyond having senior stu- +dents just as advisors. In these capstones, less-experienced +students work as junior developers and more-experienced +students as senior developers or team leaders. S19 organ- +ised their capstone course in a way that students are required +to work two course units on the same project, one unit as +a junior member and one unit as a senior member. Each +unit lasts one period (a quarter of an academic year), but +the periods do not have to be consecutive to allow some +flexibility for students in organising their studies. In order +for such an arrangement to work, the projects in the course +are large, long-term products, which undergo enhancements +over a number of semesters. S19 found that for junior stu- +dents, this setup allowed a smooth transition to the project, +up-skilling on relevant skills and acquiring the necessary ori- +entation from senior students. Senior students, on the other +hand, were enthusiastic about mentoring junior students and +finding answers to their questions ranging from project re- +quirements to the technology stack. S98 have similarly split +their capstone project into two parts with junior and senior +positions. They also had faculty mentors with industrial ex- +perience mentoring the student teams working with exter- +nal clients. According to S98, having this course design en- +abled them to create an effective industrial simulation. S98 +reports that students used tools and practices prevalent in the +industry but frequently not taught in university and were able +to develop professional and team working skills more inten- +sively. +5. Discussion +In this research, the main objective was to understand +how tertiary education institutions conduct their SE capstone +courses. This was done by looking at the characteristics +of capstone courses through an extensive literature review. +Firstly, we summarise the main findings of this study and +compare them to the findings of previous systematic litera- +ture reviews, whenever appropriate (Section 5.1). Secondly, +we present suggestions for further research in this area in +Section 5.2. Finally, we discuss the validity of the results in +Section 5.3. +5.1. Main findings +5.1.1. Duration (RQ1) +Despite ACM/IEEE (2014) recommending that under- +graduate SE capstones should span the whole academic year, +most of courses identified here last only one semester. This +is in line with findings regarding course models by Dugan Jr +(2011). In our research, the studies presenting two-semester +capstones often found one-semester courses inadequate in +depth and breadth of skills they can provide. These stud- +ies reported that a longer course better prepares students for +the experiences they can expect in their working life in soft- +ware engineering. There are, however, some real-world con- +straints to why the courses generally are shorter, such as +cramped curricula and the time and effort capstones require +from both, the staff and the students. +5.1.2. Team sizes (RQ2) +We found that capstone courses are generally conducted +as large-scale group projects. The team sizes varied greatly +between courses, ranging from 1 to 35 students in one team. +Dugan Jr (2011) found no agreement in the literature on the +appropriate team sizes. Our results were somewhat contra- +dictory to this. In our research, several studies that reported +experiences with different team sizes had found the optimal +to be 4–6 students per group. This was also reflected in av- +erage team sizes for capstones we found based on our re- +search. For a group of 2–3 students, there is no communi- +cation challenge to solve, and smaller groups often are not +able to accomplish larger projects. In contrast, larger teams +often present too many issues for communication, coordina- +tion and fair grading of students. Larger teams also require +more effort from the teaching staff to ensure an even distri- +bution of work. +5.1.3. Clients and project ideas (RQ3) +In our research, 58% of the studies reported having ex- +ternal clients for student projects (RQ3.1) and projects are +based on the real needs of external stakeholders in 62% of +the courses (RQ3.2). Having clients outside the immediate +course staff presents more work for the teachers, but is often +rewarding for students when they get to work for real clients +on real projects. The motivation boost in students, as well +as the positive implications in their skills and employment +after the course, were found to be among the top reasons for +having real clients with real projects. Using external clients +is also recommended for both undergraduate and graduate +degree programmes in software engineering (ACM/IEEE, +2014; ACM, 2009). Despite these benefits, there still is a +considerable number of capstone courses (42%), where the +course staff acts as the client for these projects, or there is +no client for students to interact with regularly. In addition, +there are quite many courses where the teacher provides the +students with project specifications (21%) or students them- +selves generate project proposals (17%). Such courses gen- +erally are less burdensome for teachers who do not have to +get involved in sourcing multiple clients and projects. Stu- +dents are often also motivated when they get to choose a +project topic of their own. However, in these cases, students +do not get to experience requirement solicitation and plan- +ning the project with an external stakeholder, who might not +be technically knowledgeable at all. The taxonomy used by +Dugan Jr (2011) relates to project topics and not particularly +project sources or clients making it difficult to assess whether +there has been changes in this over the years. Moreover, +our work demonstrates that external stakeholders can get in- +volved in many ways. In the future, it would be important to +explore this in more detail, and evaluate the consequences as +demonstrated by Steghöfer et al. (2018). +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 17 of 22 + +Capstone courses in software engineering +5.1.4. Project implementation (RQ4) +To understand the project implementation, we first looked +into the artefacts that students are expected to produce through- +out the course (RQ4.1). We found that in 97% of the courses, +students were involved in developing some form of a soft- +ware product as an end deliverable. Additionally, students +were often required to produce agile development artefacts +(e.g. product and sprint backlogs), project plans and soft- +ware documentation. The produced artefacts, therefore, sup- +ported the idea of a well-rounded software development ex- +perience. In our research, the role of documentation was +slightly different from the role it had in the survey done by +Dugan Jr (2011). In their classification, the core written doc- +uments involve project proposals, requirements documents, +project plans, designs, test plans and user manuals. We, on +the other hand, found that when courses had shifted towards +more agile development approaches, the number of written +assignments had reduced, and the documentation was being +generated more throughout the course rather than as detailed +plans up front. +Secondly, we investigated the phases that these projects +generally go through (RQ4.2). We found that projects often +proceeded from an idea or a ready-made list of requirements +into project delivery. For a large number of courses, the stu- +dents start from scratch and produce a prototype or a soft- +ware product that is handed off to the clients or teachers at +the end of the course. Therefore, the maintenance of exist- +ing software products and working with existing codebases +is often not experienced. In contrast Dugan Jr (2011) states +that regardless of the software process model, the common +phases were requirements, design, implementation, testing, +presentation and maintenance. However, they do make the +same finding we made that, maintenance was frequently men- +tioned in the literature with little supporting detail of its ac- +tual implementation. Maintenance thus still remains an is- +sue that is left with little attention in SE capstone litera- +ture, despite its high relevance in the industry. Some ed- +ucators have solved this by involving large projects, which +go through various incremental improvements over the years +in their courses. Others have students contributing to large +Open Source projects, but both of these approaches were still +found to present a minority. Dugan Jr (2011) makes no re- +marks on Open Source projects being used as a solution to +this problem like we did, which would indicate that it is an +upcoming solution to this problem. Indeed, the studies in +our research covering large Open Source projects were writ- +ten after the survey by Dugan Jr (2011). +Finally, we investigated what technologies are used in +these courses and how the selections are made (RQ4.3). We +found that studies mostly do not explicitly describe all the +technologies used in these courses. We also found that for +the majority of courses, the technology selections are made +based on the project specifications and needs. This often en- +tails students learning new technologies and having to justify +their selections. +5.1.5. Assessment of students (RQ5) +Regarding end-of-course student assessment (RQ5.1) we +aimed to find out what constitutes the course grade in the +end. A large number of studies gave inconclusive answers +in this regard and did not describe grading rubrics in detail. +Therefore, we were unable to draw any conclusions about +what artefacts or assignments formed the final grades. How- +ever, we were able to determine fairly well, who does the +final assessment and how well students are given a chance to +reflect on their experiences. In all studies that discussed the +student assessment, the teachers were involved in determin- +ing the final grades. Any sort of concrete deliverables (pro- +duced software, plans, agile artefacts, reports) were gener- +ally graded by the teacher. These artefacts provided teachers +with some understanding of how well students understood +the phases of software development. A minority of studies +also mention including students in the assessment process, in +the form of self- and peer-reviews. These both have proven +not only to hold the individual students more accountable +during project work but also to give valuable insight for the +teacher into the soft skill development of an individual stu- +dent. +Continuous assessment and guidance during the course +(RQ5.2) were explicitly addressed in most studies (89%). +The survey by Trevisan et al. (2006) focused on similar sort +of assessment practices when they sought to find out, how of- +ten engineering capstones implement classroom assessment. +They found only 32 articles in all engineering disciplines be- +tween the years 1994 and 2006 which discussed classroom +assessment schemes. While their scope is tighter, it seems +that the guidance of students during the course would have +increased in the past 15 years or so. Trevisan et al. (2006) +also reflects on this, stating that the importance of classroom +assessment has gained traction in recent years. +Our research showed that any sort of mentoring or coach- +ing was found to be highly beneficial for the students. It in- +creased the success rate of projects and helped teachers to +identify problems early on. Course staff were the ones that +most often guided students during their projects, and sev- +eral courses had hired teaching assistants for such positions. +Some studies also found that having more experienced stu- +dents advising the capstone participants was a rewarding ex- +perience for both groups. Mentoring activities are also com- +mon in real-life companies where graduates generally join +an existing team with various skill levels. +5.2. Implications for practitioners and researchers +A large amount of research found on software engineer- +ing capstones shows that capstones are a common way for +educators to prepare students for varying aspects of working +life. For an educator to find ways to implement a capstone +course, it would be too a time-consuming task to go through +all the published primary studies and distil the experience +and evidence into concrete suggestions. For such situations, +we have provided an overview of the most common char- +acteristics of capstone courses, and what kinds of choices +regarding each of these can be made. +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 18 of 22 + +Capstone courses in software engineering +The overriding guideline set by the ACM/IEEE (2014) +for undergraduate SE capstone courses is that they should +help to ensure that the curriculum has a significant real-world +basis. Capstones are expected to be the culminating expe- +rience that ties everything learned so far together and pre- +pares students for the working life in software engineering +(ACM/IEEE, 2014; ACM, 2009). Interestingly, our research +revealed that primary studies on capstone courses only rarely +included experiences, of course, alumni or industry clients. +This would be important to see how well these courses reflect +what students are expected to do in real life. We would like +to see more research done on how well these courses capture +what students face later on in their careers. It would also +be worthwhile to see more controlled, comparative studies +where one of the presented characteristics is changed and its +impact on the course outcomes. Most of the research iden- +tified here does not provide controlled, comparative results +on the capstone characteristics. +5.3. Threats to validity +5.3.1. Deviations from the procedures for systematic +reviews +Although we aimed to use the guidelines provided in +Kitchenham and Charters (2007) to perform our systematic +review, we had deviations from their procedures. In our re- +search, the study selection and data extraction were carried +out by the first author rather than by a group of researchers. +This means that some relevant papers might have been ex- +cluded or that some of the collected data may be erroneous. +5.3.2. Inaccuracy and bias in selected papers for +review +One of the main limitations of any review is the possible +bias in the study selection process. In our case, we included +only studies considering software-related capstone courses +with a relatively tight scope; for instance, we did not include +any studies with courses on embedded systems or computer +engineering. However, we were clear in our goals of describ- +ing only software capstones. A similar kind of search has +also been conducted in the earlier survey done by Dugan Jr +(2011), whose search strings were “capstone“ and “software +engineering course“ into a selected set of journals in SEE +and CS education. And to ensure that the selection pro- +cess was as unbiased as possible, we described the employed +search strategy and the inclusion and exclusion criteria in +detail. This way, we aimed to make the selection process as +visible to the reader as possible. +The primary studies only represent the capstone courses +with some aspects or outcomes worthy of publication. There- +fore the study sample in our research might be skewed to- +ward successful, well-planned courses or easy to research +courses. We believe this is not a problem in describing how +versatile the projects can be. However, quantitative aspects +of the data (e.g., the portion of the courses having an ex- +ternal client) should be addressed with caution as certain +kinds of courses may be more likely in real life than in SE +education research. Finally, we also acknowledge that there +are similar courses organised under other related disciplines, +such as data science and computer engineering. We, how- +ever, knowingly chose to leave other disciplines out of the +scope of this research as we wanted to provide a classifica- +tion and insights specifically on software-related capstones. +The ACM/IEEE (2014) recommendations we derived our re- +search questions on, were also provided specifically for soft- +ware engineering capstones. +5.3.3. Inaccuracy and bias in data extraction +As with any systematic review, one of the main limita- +tions is the potential bias and inaccuracy of the data extrac- +tion procedure. This is also the most likely step with inac- +curacy in our research. For example, the quality assessment +was done by the first author, whose interpretation of quality +might differ from that of another researcher. The distinction +between whether a study has explicitly discussed limitations +(“Yes“) or they have only shortly referred to a limitation of +the study (“Somewhat“) is something that another researcher +might view differently. However, the two summed-up cate- +gories presented, rigour and credibility, aimed to diminish +the impact of a single quality assessment question and eval- +uate the study rather as a whole. +It is also worth noting that the primary studies presented +in this review are not exclusively written to provide course +descriptions or general course evaluations. Some studies +have a section dedicated to the course overview, which might +have provided all the details of the course structure we needed. +Then again, some studies had the relevant details scattered +across various sections and might not have been explicitly re- +ferred to as our categories suggest. Especially regarding the +produced artefacts and student assessment, the descriptions +varied greatly in terms of detail and clarity. In situations +like these, some interpretation was needed. To mitigate this +problem, we tried to keep the categories generic and descrip- +tive, so that it would be easy to grasp the general outline of +each course. We also refrained from reading too much into +the text itself. For instance, if the study mentioned that the +student teams were composed of “at most 5 students“, we +left these courses in the category “not specified“. +5.3.4. Lack of third party assessment of capstone +courses +Usually, at least one of the authors of the study was some- +how involved in organising the course in question. Addi- +tionally, quite a large portion of these reports lacked an hon- +est evaluation of the author bias, as can be seen in Section +3.3.3. Therefore there is an inherent lack of truly objective +third-party assessment of these SE capstone courses in the +literature. This is something that we were unable to affect +but is worth noting. We would welcome more research on +capstone courses, or on SE education in general, where the +author is an unbiased third party. +5.3.5. Evaluation of review +For evaluating any SLR, ? present criteria based on four +quality assessment (QA) questions. We will briefly provide +answers to each one of these. +QA1 — Are the review’s inclusion and exclusion criteria +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 19 of 22 + +Capstone courses in software engineering +described and appropriate? Yes, we have explicitly defined +and described the inclusion and exclusion criteria. The foun- +dation for the criteria stems from our research objectives and +aims to ensure that the studies included in the review are of +sufficient quality and help to answer our research questions. +QA2 – Is the literature search likely to have covered all +relevant studies? ? state that if the authors have searched 4 or +more digital libraries and included additional search strate- +gies, this criterion is met. In this research, we did search +4 digital libraries and included a description of our search +strategies, so this criterion is fulfilled. +QA3 – Did the reviewers assess the quality/validity of +the included studies? We did use a question set used by +many similar SLRs to assess the quality and validity of the +included studies. Therefore this criterion is also met. +QA4 – Were basic data/studies adequately described? +We provided bibliographical references to each of the studies +used, described from various viewpoints the target of their +research (i.e. the capstone course presented in each of them), +described how the data was collected in each of the studies +and synthesised the reported outcomes. Therefore it is safe +to say, that this criterion was met as well. +6. Conclusions +This research aimed to understand how software engi- +neering capstone courses are organised in tertiary education +institutions. For this purpose, we conducted a systematic +literature review, including 127 primary studies on SE cap- +stone courses. The characteristics were synthesised into a +taxonomy consisting of duration, team sizes, clients and project +sources, project implementation and student assessment. Based +on the synthesised justifications and outcomes for these char- +acteristics, we provided suggestions on how the courses can +be organised and what the trade-offs are to be weighted re- +garding each characteristic. +The main curriculum guideline that capstones should help +to accomplish is “The curriculum should have a significant +real-world basis“ (ACM/IEEE, 2014). In our research, we +focused on the concrete recommendations given to accom- +plish this goal and formulated our research questions based +on them. We found out that the courses have a software im- +plementation as the main deliverable, the students are as- +sessed based on various factors, not just the delivery of a +working system, and the projects in these courses are almost +always completed as group assignments. Students were also +often given guidance and continuous assessment throughout +the course via written and oral feedback on their progress +and deliverables. The area which educators should pay at- +tention to is the duration of the course which in practice is +one semester, whilst for instance, ACM/IEEE (2014) recom- +mends having two-semester courses to reach adequate depth +and breadth in skills and experiences. A considerable num- +ber of courses also did not have a client external to the course +staff, despite external clients being recommended for under- +graduate and graduate capstones (ACM/IEEE, 2014; ACM, +2009). In these cases, the project specifications were gen- +erated by the course staff or the students themselves. Such +arrangements tend to leave students without the experience +of having to solicit, negotiate and implement requirements +set by a real client. In addition, the projects usually progress +from idea to product, and often do not include maintenance, +especially that of pre-existing projects. These characteris- +tics somewhat diminish the real-world compatibility of the +course. +References +ACM, 2009. 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Primary studies +Table 13 +Included sources for data extraction +ID +Author(s) +Year +Title +Source title +S1 +Marzolo, P., Guazzaloca, M., +Ciancarini, P. +2021 +“Extreme Development” as a Means for Learning +Agile +International Conference on Frontiers in +Software Engineering +S2 +Tan, J., Jones, M. +2008 +A case study of classroom experience with +client-based team projects +Journal of Computing Sciences in Colleges +S3 +Wong, W., Pepe, J., Stahl, +J., Englander, I. +2013 +A collaborative capstone to develop a mobile +hospital clinic application through a student +team competition +Information Systems Education Journal +S4 +Tappert, C. C., Stix, A. +2011 +A decade review of a masters-level real-world- +projects capstone course +Info. +Systems Educators Conf., ISECON +2011 +S5 +Gotel, O., Kulkarni, V., Say, +M., Scharff, C., Sunetnanta, +T. +2009 +A global and competition-Based model for fos- +tering technical and soft skills in software engi- +neering education +22nd Conference on Software Engineering +Education and Training, CSEE&T 2009 +S6 +Scott, +A., +Kreahling, +W., +Holliday, M., Barlowe, S. +2017 +A holistic capstone experience: Beyond techni- +cal ability +18th Annual Conference on Information +Technology Education +S7 +Koolmanojwong, S., Boehm, +B. +2013 +A look at software engineering risks in a team +project course +26th +International +Conference +on +Soft- +ware Engineering Education and Training, +CSEE&T 2013 +S8abcd +Braught, G., et al. +2018 +A multi-institutional perspective on H/FOSS +projects in the computing curriculum +ACM Transactions on Computing Educa- +tion +S9 +Mertz, J., Quesenberry, J. +2019 +A scalable model of community-based experien- +tial learning through courses and international +projects +2018 World Engineering Education Forum - +Global Engineering Deans Council, WEEF- +GEDC 2018 +S10 +Bloomfield, A., Sherriff, M., +Williams, K. +2014 +A Service Learning Practicum capstone +45th ACM technical symposium on Com- +puter science education +S11 +Brazier, P., Garcia, A., Vaca, +A. +2007 +A software engineering senior design project in- +herited from a partially implemented software +engineering class project +37th Annual Frontiers in Education Confer- +ence - Global Engineering +S12 +Morales-Trujillo, M.E., Gal- +ster, M., Gilson, F., Math- +ews, M. +2021 +A Three-Year Study on Peer Evaluation in a +Software Engineering Project Course +IEEE Transactions on Education +S13 +Liang, +Z., +Chapa-Martell, +M.A. +2019 +A Top-Down Approach to Teaching Web Devel- +opment in the Cloud +IEEE International Conference on Teach- +ing, Assessment, and Learning for Engineer- +ing, TALE 2018 +S14 +Murphy, C., Sheth, S., Mor- +ton, S. +2017 +A Two-Course Sequence of Real Projects for +Real Customers +Conference on Integrating Technology into +Computer Science Education, ITiCSE 2017 +S15 +Rusu, A., Rusu, A., Docimo, +R., Santiago, C., Paglione, +M. +2009 +Academia-academia-industry collaborations on +software engineering projects using local-remote +teams +40th ACM Technical Symposium on Com- +puter Science Education, SIGCSE’09 +S16 +Stettina, +C.J., +Zhao, +Z., +Back, T., Katzy, B. +2013 +Academic education of software engineering +practices: towards planning and improving cap- +stone courses based upon intensive coaching and +team routines +26th +International +Conference +on +Soft- +ware Engineering Education and Training, +CSEE&T 2013 +S17 +Venson, E., Figueiredo, R., +Silva, W., Ribeiro, L.C.M. +2016 +Academy-industry collaboration and the effects +of the involvement of undergraduate students in +real world activities +IEEE Frontiers in Education Conference, +FIE 2016 +S18 +Eloe, N., Hoot, C. +2020 +Accommodating Shortened Term Lengths in a +Capstone Course using Minimally Viable Proto- +types +IEEE Frontiers in Education Conference, +FIE 2020 +S19 +Schneider, +J.-G., +Eklund, +P.W., +Lee, +K., +Chen, +F., +Cain, A., Abdelrazek, M. +2020 +Adopting industry agile practices in large-scale +capstone education +42nd International Conference on Software +Engineering: Software Engineering Educa- +tion and Training, ICSE-SEET 2020 +S20 +Ye, H. +2009 +An +academia-industry +collaborative +teaching +and learning model for software engineering ed- +ucation +21st International Conference on Software +Engineering and Knowledge Engineering, +SEKE 2009 +S21 +Demuth, B., Kandler, M. +2017 +An Approach for Project Task Approximation in +a Large-Scale Software Project Course +30th IEEE Conference on Software Engi- +neering Education and Training, CSEE&T +2017 +S22 +Ellis, H.J.C. +2007 +An assessment of a self-directed learning ap- +proach in a graduate web application design and +development course +IEEE Transactions on Education +S23 +Anslow, C., Maurer, F. +2015 +An experience report at teaching a group based +agile software development project course +46th ACM Technical Symposium on Com- +puter Science Education +S24 +Bareiss, R., Katz, E. +2011 +An exploration of knowledge and skills transfer +from a formal software engineering curriculum +to a capstone practicum project +24th +IEEE-CS +Conference +on +Software +Engineering +Education +and +Training, +CSEE&T 2011 +S25 +Stephenson, B., James, M., +Brooke, N., Aycock, J. +2016 +An Industrial Partnership Game Development +Capstone Course +17th Annual Conference on Information +Technology Education +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 22 of 22 + +Capstone courses in software engineering +Table 13 +Continued from previous page +ID +Author(s) +Year +Title +Source title +S26 +Bell, J.T., Prabhu, A. +2015 +An innovative approach to Software Engineer- +ing term projects, coordinating student efforts +between multiple teams over multiple semesters +IEEE Frontiers in Education Conference, +FIE 2014 +S27 +Vasilevskaya, +M., +Broman, +D., Sandahl, K. +2015 +Assessing large-project courses: Model, activi- +ties, and lessons learned +ACM Transactions on Computing Educa- +tion, TOCE +S28 +von Konsky, B.R., Ivins, J. +2008 +Assessing the capability and maturity of cap- +stone software engineering projects +Tenth conference on Australasian comput- +ing education - Volume 78 +S29 +Fontao, A., Gadelha, B., Ju- +nior, A.C. +2019 +Balancing Theory and Practice in Software En- +gineering Education - A PBL, toolset based ap- +proach +IEEE Frontiers in Education Conference, +FIE 2019 +S30 +Harding, T. +2007 +Benefits and struggles of using large team +projects in capstone courses +ASEE Annual Conference and Exposition +S31 +Engelsma, J. R. +2014 +Best practices for industry-sponsored CS cap- +stone courses +Journal of Computing Sciences in Colleges +S32 +Matthies, +C., +Teusner, +R., +Hesse, G. +2019 +Beyond Surveys: Analyzing Software Develop- +ment Artifacts to Assess Teaching Efforts +IEEE Frontiers in Education Conference, +FIE 2018 +S33 +Ziv, H., Patil, S. +2010 +Capstone project: From software engineering to +“Informatics“ +23rd IEEE Conference on Software Engi- +neering Education and Training, CSEE&T +2010 +S34 +Anderson, Ruth E.; Borriello, +Gaetano; +Martin, +Hélène; +Black, Leonard +2009 +Capstone projects as community connectors +Journal of Computing Sciences in Colleges +S35 +Paasivaara, +M., +Vanhanen, +J., Lassenius, C. +2019 +Collaborating with industrial customers in a cap- +stone project course: The customers’ perspec- +tive +IEEE/ACM 41st International Conference +on Software Engineering: +Software En- +gineering Education and Training, ICSE- +SEET 2019 +S36 +Adams, R., Kleiner, C. +2016 +Collaboration support in an international com- +puter science capstone course +International Conference on Social Com- +puting and Social Media +S37 +Watkins, K.Z., Barnes, T. +2010 +Competitive and agile software engineering ed- +ucation +IEEE SoutheastCon, SoutheastCon 2010 +S38 +Gustavsson, H., Brohede, M. +2019 +Continuous assessment in software engineering +project course using publicly available data from +GitHub +15th International Symposium on Open +Collaboration, OpenSym 2019 +S39 +Hadfield, Steven M.; Jensen, +Nathan A. +2007 +Crafting a software engineering capstone project +course +Journal of Computing Sciences in Colleges +S40 +Rong, G., Shao, D. +2012 +Delivering +software +process-specific +project +courses in tertiary education environment: Chal- +lenges and solution +25th IEEE Conference on Software Engi- +neering Education and Training, CSEE&T +2012 +S41 +Nguyen, D.M., Truong, T.V., +Le, N.B. +2013 +Deployment of capstone projects in software en- +gineering education at Duy Tan university as +part of a university-wide project-based learning +effort +Learning and Teaching in Computing and +Engineering, LaTiCE 2013 +S42 +Lago, P., Schalken, J., Vliet, +H.V. +2009 +Designing a multi-disciplinary software engineer- +ing project +22nd IEEE Conference on Software Engi- +neering Education and Training, CSEE&T +2009 +S43 +Angelov, S., de Beer, P. +2017 +Designing and applying an approach to software +architecting in agile projects in education +Journal of Systems and Software +S44 +Anderson, R.E., Kolko, B. +2011 +Designing technology for resource-constrained +environments: A multidisciplinary capstone se- +quence +Frontiers in Education, FIE 2012 +S45 +Leilde, V., Ribaud, V. +2017 +Does Process Assessment Drive Process Learn- +ing? the Case of a Bachelor Capstone Project +30th IEEE Conference on Software Engi- +neering Education and Training, CSEE&T +2017 +S46 +Brown, Q., Lee, F., Alejan- +dre, S. +2009 +Emphasizing soft skills and team development +in an educational digital game design course +4th International Conference on the Foun- +dations of Digital Games, FDG 2009 +S47 +Takala, T. M., Malmi, L., +Pugliese, R., Takala, T. +2016 +Empowering students to create better virtual re- +ality applications: A longitudinal study of a VR +capstone course +Informatics in Education +S48 +Marques, M., Ochoa, S.F., +Bastarrica, M.C., Gutierrez, +F.J. +2018 +Enhancing the Student Learning Experience in +Software Engineering Project Courses +IEEE Transactions on Education +S49 +De Souza, R.T., Zorzo, S.D., +Da Silva, D.A. +2015 +Evaluating capstone project through flexible and +collaborative use of Scrum framework +Frontiers in Education Conference, +FIE +2015 +S50 +Vu, J.H., Frojd, N., Shenkel- +Therolf, C., Janzen, D.S. +2009 +Evaluating +test-driven +development +in +an +industry-sponsored capstone project +6th International Conference on Informa- +tion Technology: New Generations, ITNG +2009 +S51 +Laplante, +P.A., +Defranco, +J.F., Guimaraes, E. +2019 +Evolution of a graduate software engineering +capstone course - A course review +International Journal of Engineering Educa- +tion +S52 +Lederman, Timoth C. +2010 +Evolution of capstone-courses in software engi- +neering a finishing school +Journal of Computing Sciences in Colleges +S53 +Delgado, +D., +Velasco, +A., +Aponte, J., Marcus, A. +2017 +Evolving a Project-Based Software Engineering +Course: A Case Study +30th IEEE Conference on Software Engi- +neering Education and Training, CSEE&T +2017 +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 23 of 22 + +Capstone courses in software engineering +Table 13 +Continued from previous page +ID +Author(s) +Year +Title +Source title +S55 +Ras, Eric and Carbon, Ralf +and Decker, Björn and Rech, +Jörg +2007 +Experience Management Wikis for Reflective +Practice in Software Capstone Projects +IEEE Transactions on Education +S56 +Schorr, R. +2020 +Experience Report on Key Success Factors for +Promoting Students’ Engagement in Software +Development Group Projects +4th IEEE World Conference on Engineering +Education, EDUNINE 2020 +S57 +Longstreet, +C. +Shaun; +Cooper, Kendra +2013 +Experience report: A sustainable serious educa- +tional game capstone project +CGAMES’2013 USA +S58 +Dupuis, R., Champagne, R., +April, A., Séguin, N. +2010 +Experiments with Adding to the Experience that +Can be Acquired from Software Courses +7th International Conference on the Quality +of Information and Communications Tech- +nology, QUATIC 2010 +S59 +Burge, J. +2007 +Exploiting Multiplicity to Teach Reliability and +Maintainability in a Capstone Project +20th IEEE Conference on Software Engi- +neering Education and Training, CSEE&T +2007 +S60 +Marshall, +L., +Pieterse, +V., +Thompson, L., Venter, D.M. +2016 +Exploration of Participation in Student Software +Engineering Teams +ACM Transactions on Computing Educa- +tion, TOCE +S61 +Ganci, +A., +Ramnath, +R., +Ribeiro, B., Stone, R.B. +2011 +Exploring collaboration between computer sci- +ence engineers and visual communication de- +signers in educational settings +13th International Conference on Engineer- +ing and Product Design Education, E&PDE +2011 +S62 +Burden, H., Steghöfer, J.-P., +Hagvall Svensson, O. +2019 +Facilitating entrepreneurial experiences through +a software engineering project course +41st International Conference on Software +Engineering: Software Engineering Educa- +tion and Training, ICSE-SEET 2019 +S63 +Basholli, +A., +Baxhaku, +F., +Dranidis, D., Hatziapostolou, +T. +2013 +Fair assessment in software engineering cap- +stone projects +6th Balkan Conference in Informatics +S64 +Magana, A. J., Seah, Y. Y., +Thomas, P. +2018 +Fostering cooperative learning with Scrum in +a semi-capstone systems analysis and design +course +Journal of Information Systems Education +S65 +Sievi-Korte, +O., +Systä, +K., +Hjelsvold, R. +2015 +Global vs. local – Experiences from a distributed +software project course using agile methodolo- +gies +Frontiers in Education, FIE 2015 +S66 +Hebig, R., Ho-Quang, T., Jo- +lak, R., Schröder, J., Linero, +H., Ågren, M., Maro, S.H. +2020 +How do students experience and judge software +comprehension techniques? +28th International Conference on Program +Comprehension +S67 +Verdicchio, Michael +2021 +Hurricanes and pandemics: an experience report +on adapting software engineering courses to en- +sure continuity of instruction +Journal of Computing Sciences in Colleges +S68 +Włodarski, R., Poniszewska- +Marańda, A., Falleri, J.-R. +2022 +Impact of software development processes on +the outcomes of student computing projects: A +tale of two universities +Information and Software Technology +S69 +Izu, Cruz +2018 +Improving Outcomes for a Masters Capstone IT +Project +IEEE International Conference on Teach- +ing, Assessment, and Learning for Engineer- +ing, TALE 2018 +S70 +Flowers, J.G. +2008 +Improving the Capstone project experience: a +case study in software engineering +46th Annual Southeast Regional Confer- +ence on XX +S71 +Gannod, +Gerald C.; +Bach- +man, Kristen M.; Troy, Dou- +glas A.; Brockman, Steve D. +2010 +Increasing alumni engagement through the cap- +stone experience +Frontiers in Education, FIE 2010 +S72 +Zilora, S.J. +2015 +Industry-emulated projects in the classroom +16th Annual ACM Conference on Informa- +tion Technology Education, SIGITE 2015 +S73 +Spichkova, M. +2019 +Industry-oriented project-based learning of soft- +ware engineering +24th +International +Conference +on +Engi- +neering of Complex Computer Systems, +ICECCS 2019 +S74 +Carvalho, J.A., Sousa, R.D., +Sá, J.O. +2010 +Information systems development course: Inte- +grating business, IT and IS competencies +2010 IEEE Transforming Engineering Edu- +cation: Creating Interdisciplinary Skills for +Complex Global Environments +S75 +Palacin-Silva, M.V., Seffah, +A., Porras, J. +2018 +Infusing sustainability into software engineer- +ing education: Lessons learned from capstone +projects +Journal of Cleaner Production +S76 +Kumar, S., Wallace, C. +2015 +Instruction in software project communication +through guided inquiry and reflection +Frontiers in Education, FIE 2015 +S77 +Zeid, A. +2012 +Integrating international students’ contests with +computer science capstone: Lessons learned and +best practices +Frontiers in Education, FIE 2012 +S78 +Lundqvist, K., Ahmed, A., +Fridman, D., Bernard, J.-G. +2019 +Interdisciplinary Agile Teaching +Frontiers in Education, FIE 2019 +S79 +Santoso, H.B., Lawanto, O., +Purwandari, B., Isal, R.Y.K., +Fitriansyah, R. +2018 +Investigating +Students’ +Metacognitive +Skills +while Working on Information Systems Devel- +opment Projects +7th World Engineering Education Forum, +WEEF 2017 +S80 +Christensen, +E.L., +Paasi- +vaara, M. +2022 +Learning Soft Skills through Distributed Soft- +ware Development +International Conference on Software and +System Processes and Internation Confer- +ence on Global Software Engineering +S81 +Rout, Terence P.; Seagrott, +John +2007 +Maintaining High Process Capability in a Stu- +dent Project Course +20th Conference on Software Engineering +Education & Training, CSEE&T 2007 +S82 +Rodriguez, +G., +Soria, +A., +Campo, M. +2016 +Measuring the Impact of Agile Coaching on Stu- +dents’ Performance +IEEE Transactions on Education +S83 +Linhoff, J., Settle, A. +2009 +Motivating and evaluating game development +capstone projects +4th International Conference on Founda- +tions of Digital Games +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 24 of 22 + +Capstone courses in software engineering +Table 13 +Continued from previous page +ID +Author(s) +Year +Title +Source title +S84 +Haddad, H.M. +2013 +One-semester CS capstone: A 40-60 teaching +approach +10th International Conference on Informa- +tion Technology: New Generations, ITNG +2013 +S85 +Fan, Xiaocong +2018 +Orchestrating Agile Sprint Reviews in Under- +graduate Capstone Projects +Frontiers in Education, FIE 2018 +S86 +Fagerholm, +F., +Vihavainen, +A. +2013 +Peer assessment in experiential learning: Assess- +ing tacit and explicit skills in agile software en- +gineering capstone projects +Frontiers in Education, FIE 2013 +S87 +Vasankari, T., Majanoja, A.- +M. +2019 +Practical Software Engineering Capstone Course +– Framework for Large, Open-Ended Projects to +Graduate Student Teams +Internation Conference on Computer Sup- +ported Education +S88 +Karunasekera, S., Bedse, K. +2007 +Preparing software engineering graduates for an +industry career +20th Conference on Software Engineering +Education & Training, CSEE&T 2007 +S89 +Weerawarana, S.M., Perera, +A.S., Nanayakkara, V. +2012 +Promoting creativity, innovation and engineer- +ing excellence: A case study from Sri Lanka +IEEE International Conference on Teach- +ing, Assessment, and Learning for Engineer- +ing, TALE 2012 +S90 +Fornaro, +R.J., +Heil, +M.R., +Tharp, A.L. +2007 +Reflections on 10 years of sponsored senior de- +sign projects: Students win-clients win! +Journal of Systems and Software +S91 +Roach, S. +2011 +Retrospectives in a software engineering project +course: Getting students to get the most from +a project experience +24th +IEEE-CS +Conference +on +Software +Engineering +Education +and +Training, +CSEE&T 2011 +S92 +Mäkiaho, P., Poranen, T. +2018 +Risks management in software development cap- +stone projects +19th International Conference on Computer +Systems and Technologies +S93(a,b) +MacKellar, B. K., Sabin, M., +Tucker, A. +2013 +Scaling +a +framework +for +client-driven +open +source software projects: A report from three +schools +Journal of Computing Sciences in Colleges +S94 +Yuen, T.T. +2015 +Scrumming with educators: Cross-departmental +collaboration for a summer software engineering +capstone +International Conference on Learning and +Teaching in Computing and Engineering, +LaTiCE 2015 +S95 +Isomöttönen, V., Daniels, M., +Cajander, Å., Pears, A., Mc- +Dermott, R. +2019 +Searching for global employability: Can students +capitalize on enabling learning environments? +ACM Transactions on Computing Educa- +tion +S96 +Maxim, B. +2008 +Serious games as software engineering capstone +projects +ASEE Annual Conference and Exposition +S97 +Krogstie, B.R., Divitini, M. +2009 +Shared timeline and individual experience: Sup- +porting retrospective reflection in student soft- +ware engineering teams +22nd Conference on Software Engineering +Education and Training, CSEE&T 2009 +S98 +Johns-Boast, L., Flint, S. +2013 +Simulating industry: An innovative software en- +gineering capstone design course +Frontiers in Education, FIE 2013 +S99 +Boti, E., Damasiotis, V., Fit- +silis, P. +2021 +Skills Development Through Agile Capstone +Projects +International Conference on Frontiers in +Software Engineering +S100 +Paiva, S.C., Carvalho, D.B.F. +2018 +Software creation workshop: A capstone course +for business-oriented software engineering teach- +ing +XXXII Brazilian Symposium on Software +Engineering +S101 +Saeedi, K., Visvizi, A. +2021 +Software development methodologies, HEIs, and +the digital economy +Education Sciences +S102 +Smith, T., Cooper, K.M.L., +Longstreet, C.S. +2011 +Software engineering senior design course: Ex- +periences with agile game development in a cap- +stone project +International Conference on Software Engi- +neering +S103 +Jaccheri, L., Sindre, G. +2007 +Software engineering students meet interdisci- +plinary project work and art +11th International Conference on Informa- +tion Visualisation, IV 2007 +S104 +Krusche, S., Dzvonyar, D., +Xu, H., Bruegge, B. +2018 +Software +Theater—Teaching +Demo-Oriented +Prototyping +ACM Transactions on Computing Educa- +tion, TOCE +S105 +Budd, A.J., Ellis, H.J.C. +2008 +Spanning the gap between software engineering +instructor and student +Frontiers in Education, FIE 2008 +S106 +Decker, +A., +Egert, +C.A., +Phelps, A. +2016 +Splat! er, shmup? A postmortem on a capstone +production experience +Frontiers in Education, FIE 2008 +S107 +Kerbs, R. +2007 +Student teamwork: A capstone course in game +programming +Frontiers in Education, FIE 2007 +S108 +Tadros, Ibrahem; Hammami, +Samir; Al-Zoubi, Khaled +2008 +Systems Development Projects +3rd +International +Conference +on +Infor- +mation and Communication Technologies: +From Theory to Applications +S109 +Jarzabek, S. +2013 +Teaching advanced software design in team- +based project course +26th +IEEE +International +Conference +on +Software Engineering Education and Train- +ing, CSEE&T 2013 +S110 +Lu, Baochuan; DeClue, Tim +2011 +Teaching agile methodology in a software engi- +neering capstone course +Journal of Computing Sciences in Colleges +S111 +Cagiltay, N.E. +2007 +Teaching software engineering by means of +computer-game development: +Challenges and +opportunities +British Journal of Educational Technology +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 25 of 22 + +Capstone courses in software engineering +Table 13 +Continued from previous page +ID +Author(s) +Year +Title +Source title +S112 +Tafliovich, A., Caswell, T., +Estrada, F. +2019 +Teaching software engineering with free open +source software development: An experience re- +port +Annual Hawaii International Conference on +System Sciences +S113 +Paasivaara, +M., +Lassenius, +C., +Damian, +D., +Raty, +P., +Schroter, A. +2013 +Teaching students global software engineering +skills using distributed Scrum +35th International Conference on Software +Engineering, ICSE 2013 +S114 +Khmelevsky, Y. +2016 +Ten years of capstone projects at Okanagan Col- +lege: A retrospective analysis +21st +Western +Canadian +Conference +on +Computing Education +S115 +Mahnič, V. +2015 +The capstone course as a means for teaching ag- +ile software development through project-based +learning +World Transactions on Engineering and +Technology Education +S116 +Broman, +D., +Sandahl, +K., +Baker, M.A. +2012 +The company approach to software engineering +project courses +IEEE Transactions on Education +S117 +Khakurel, J., Porras, J. +2020 +The Effect of Real-World Capstone Project in an +Acquisition of Soft Skills among Software Engi- +neering Students +32nd IEEE Conference on Software Engi- +neering Education and Training, CSEE&T +2020 +S118 +Iacob, C., Faily, S. +2020 +The impact of undergraduate mentorship on +student satisfaction and engagement, teamwork +performance, and team dysfunction in a software +engineering group project +51st ACM Technical Symposium on Com- +puter Science Education, SIGCSE 2020 +S119 +Hoar, R. +2014 +The real world web: +How institutional IT af- +fects the delivery of a capstone web development +course +19th +Western +Canadian +Conference +on +Computing Education, WCCCE 2014 +S120 +Yue, +K. B., +Damania, +Z., +Nilekani, R., Abeysekera, K. +2011 +The use of free and open source software in real- +world capstone projects +Journal of Computing Sciences in Colleges +S121 +Isomöttönen, V., Kärkkäinen, +T. +2008 +The value of a real customer in a capstone +project +21st Conference on Software Engineering +Education and Training, CSEE&T 2008 +S122 +Mohan, S., Chenoweth, S., +Bohner, S. +2012 +Towards a better capstone experience +43rd ACM Technical Symposium on Com- +puter Science Education, SIGCSE’12 +S123 +Rico, D.F., Sayani, H.H. +2009 +Use of agile methods in software engineering ed- +ucation +Agile Conference, AGILE 2009 +S124 +Tribelhorn, B., Nuxoll, A.M. +2021 +Using Agile and Active Learning in Software De- +velopment Curriculum +ASEE Virtual Annual Conference and Ex- +position +S125 +McDonald, J., Wolfe, R. +2008 +Using computer graphics to foster interdisci- +plinary collaboration in capstone courses +Journal of Computing Sciences in Colleges +S126 +Ju, A., Hemani, A., Dimitri- +adis, Y., Fox, A. +2020 +What agile processes should we use in software +engineering course projects? +51st ACM Technical Symposium on Com- +puter Science Education, SIGCSE 2020 +S127 +Bastarrica, +M.C., +Perovich, +D., Samary, M.M. +2017 +What can students get from a software engi- +neering capstone course? +39th IEEE/ACM International Conference +on Software Engineering: +Software Engi- +neering and Education Track, ICSE-SEET +2017 +Saara Tenhunen et al.: Preprint submitted to Elsevier +Page 26 of 22 + diff --git a/A9E1T4oBgHgl3EQf9QZb/content/tmp_files/load_file.txt b/A9E1T4oBgHgl3EQf9QZb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3daafd3fc76f4300c63f1bbadb5463477cc5c942 --- /dev/null +++ b/A9E1T4oBgHgl3EQf9QZb/content/tmp_files/load_file.txt @@ -0,0 +1,2579 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf,len=2578 +page_content='Highlights A systematic literature review of capstone courses in software engineering Saara Tenhunen*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Tomi Männistö*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Matti Luukkainen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Petri Ihantola Our taxonomy of course features is based on ACM/IEEE guide for capstone courses There is a vast diversity in how capstone courses in SE are implemented More research is needed to compare different course implementation strategies Many of the capstone courses are shorter than the recommended two semesters Many of the capstone courses are missing an external client arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='03554v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='SE] 9 Jan 2023 A systematic literature review of capstone courses in software engineering Saara Tenhunen*, Tomi Männistö*, Matti Luukkainen and Petri Ihantola University of Helsinki, , , Finland A R T I C L E I N F O Keywords: capstone project course computer science education software engineering education A B S T R A C T Context: Tertiary education institutions aim to prepare their computer science and software engineering students for working life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' While much of the technical principles are covered in lower-level courses, team-based capstone projects are a common way to provide students with hands-on experience and teach soft skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Objective: This paper explores the characteristics of software engineering capstone courses presented in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The goal of this work is to understand the pros and cons of different approaches by synthesising the various aspects of software engineering capstone courses and related experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Method: In a systematic literature review for 2007–2007, we identified 127 primary studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' These studies were analysed based on their presented course characteristics and the reported course outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Results: The characteristics were synthesised into a taxonomy consisting of duration, team sizes, client and project sources, project implementation, and student assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We found out that capstone courses generally last one semester and divide students into groups of 4–5 where they work on a project for a client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For a slight majority of courses, the clients are external to the course staff and students are often expected to produce a proof-of-concept level software product as the main end deliverable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The courses also offer versatile assessments for students throughout the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Conclusions: This paper provides researchers and educators with a classification of characteristics of software engineering capstone courses based on previous research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We also further synthesise insights on the reported outcomes of capstone courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Our review study aims to help educators to identify various ways of organising capstones and effectively plan and deliver their own capstone courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The characterisation also helps researchers to conduct further studies on software engineer- ing capstones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Introduction Universities and other tertiary education institutions should provide their students with sufficient skills and abilities be- fore the students enter working life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In software engineering- related programs, this entails having an understanding of the common principles and theory in computer science (ACM/IEEE, 2013, 2014) and technical competencies and knowledge de- manded by the industry (Radermacher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Garousi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Any recent graduate should also be able to ap- ply this technical knowledge in practice (ACM/IEEE, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' While much of the technical knowledge and theories are covered in lower-level courses, many institutions hold team- based capstone project courses to ensure students are ready to apply the knowledge in a workplace environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' A “cap- stone course“ usually means a course that finishes an aca- demic degree (Ikonen and Kurhila, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The main goal of a capstone project is to provide hands-on experience in applying the tools, techniques, principles and best practices that are taught more theoretically in previous courses (Ziv and Patil, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Majanoja and Vasankari, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Panicker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Capstone projects are also regarded as crucial in teaching students the necessary soft skills such as team- ∗Corresponding author saaraten@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='com (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Tenhunen*);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' tomi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='mannisto@helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='fi (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Männistö*);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' matti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='luukkainen@helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='fi (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Luukkainen);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' petri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='ihantola@helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='fi (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Ihantola) ORCID(s): 0000-0002-4894-8365 (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Tenhunen*) work (Keogh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Venson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', 2016), verbal and written communication (Watkins and Barnes, 2010), time management (Dupuis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', 2010), problem solving (Ma- janoja and Vasankari, 2018) and project management (Had- dad, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In computer science (CS) and software engi- neering (SE) programs, capstone courses generally last one or two semesters, and they include assigning students into teams and having them work on various kinds of software engineering projects (Ikonen and Kurhila, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Bowring and Burke, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Paasivaara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In these projects, they are expected to experience stages of the software devel- opment life-cycle from requirements solicitation to software maintenance (Keogh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Given the general acceptance of capstones as a practical way of teaching industry-relevant skills, a high number of institutions have implemented their own capstone courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This has resulted in a great deal of research done on cap- stone courses and their outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In order to provide a co- herent and compact view of software engineering capstones, this research synthesises the current body of knowledge on the topic in a systematic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We believe that such a review gives educators an effective tool for planning and implementing their own capstone courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Researchers can also benefit from a systematic review of capstones to con- duct further comparative studies on the impact of the varying course forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This study is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The next section fo- Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Page 1 of 22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='aCapstone courses in software engineering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Table 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Searches for systematic reviews on software engineering capstones ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Database ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Search term ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Hits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Scopus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='TITLE-ABS-KEY( software AND engineering AND capstone AND literature AND review ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Scopus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='TITLE-ABS-KEY( software AND engineering AND capstone AND review ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='61 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Scopus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='TITLE-ABS-KEY( education AND “software engineering“ AND literature AND mapping ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Scopus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='TITLE-ABS-KEY( project AND course AND software AND engineering AND systematic AND review ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Scopus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='TITLE-ABS-KEY( “computer science“ AND capstone AND literature AND review ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Scopus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='TITLE-ABS-KEY( software AND engineering AND project AND course AND systematic AND literature AND review ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Scopus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='TITLE-ABS-KEY( “software engineering“ AND education AND literature AND review ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='182 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Google Scholar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='software engineering capstone systematic review ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='20 300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Google Scholar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='software engineering capstone characteristics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='31 400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Google Scholar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='computer science capstone literature review ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='53 400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Google Scholar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='capstone literature review ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='94 800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='cus on the previous literature reviews on SE capstones,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' as well as general characteristics of such courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Section 3 describes the research questions and the related methods, in- cluding how the articles were selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Section 4 presents the results of the literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The main findings and their validity are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Suggestions for fu- ture research are also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Finally, Section 6 concludes the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Previous work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Systematic literature reviews of SE capstones Many literature reviews have been written in the general area of software engineering education (SEE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Usually, they focus on specific sub-areas of SEE such as teaching meth- ods in software engineering (Anicic and Stapic, 2022), prac- tical approaches to SEE (Marques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', 2014), trends in SEE (Cico and Jaccheri, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Cico et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', 2021) or teach- ing global software engineering (Fortaleza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' As the focus of this research is especially on project- based capstone courses in software engineering, we carefully sought any earlier systematic reviews done on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The search was conducted on May 17th, 2022, first in the cita- tion database Scopus and secondly in Google Scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Table 1 lists the search terms used in both databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' All search results produced by Scopus were checked to see whether they include an SLR of SE capstones, whereas for Google Scholar, each search produced tens of thousands of hits, so we went through the first 20 pages of each search (200 hits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' At this point, the results started to become highly irrele- vant, and often repetitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Based on our search, we believe that the three review papers presented in Table 2 are the ones that have been published so far on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Table 2 also presents the characteristics of capstone courses each of these reviews investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Next, we will briefly present these studies and discuss the necessity of this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Dugan Jr (2011) presents a survey done on the literature related to undergraduate computer science capstone courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The survey is comprehensive, comprising 200 papers on the subject and summarising them under two major themes: course issues and project issues (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Out of these, course issues include aspects related to the general course organi- sation, such as course models, learning theories present in the course and student evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Project issues, on the other hand, categorise and describe the projects and how they are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The category includes things like software process phases, project type and documentation of the projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Martin (2019) has provided an abstract of a systematic literature review for designing an IT capstone course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The review plans to provide answers to several questions relating to capstone course design, such as the optimal team size for project teams, identifying and selecting suitable projects and determining the correct duration for the course (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We are not aware that the research proposed in the abstract would have been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Trevisan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' (2006) have performed a systematic re- view on the assessment practices in capstone engineering design courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' They were especially interested in discov- ering the extent to which classroom assessment has received attention in the capstone literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The paper included 32 journal articles and conference proceedings presenting vary- ing assessment techniques and their use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In addition to the presented three literature reviews, a study on the dimensions of SE and CS capstone projects has been conducted by Burge and Gannod (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Said dimen- sions are roughly divided into two groups: project dimen- sions such as customer identity and development dimensions such as project type and source code visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The purpose of their study is to provide a framework for analysing cap- stone courses, especially in terms of risk and realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' While their categorisation is versatile, their study does not, how- ever, include a thorough systematic literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The categorisation presented is more of an experience-based pro- posal, and therefore the study is left out of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' To the best of our knowledge, there is no extensive, re- cent literature review done on software engineering capstone courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' A survey conducted by Dugan Jr (2011) is com- prehensive but dated to 2011 and therefore does not cover the large number of primary studies published in the past decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' It also does not provide any quantitative statistics of the course characteristics, which would enable educators or researchers to assess how common some aspect in reality is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Trevisan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' (2006) provide a review on capstone literature, but it is limited to continuous assessment techniques and is dated to 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Martin (2019) aims to provide a systematic literature review on IT capstone design and characteristics, but as of now, the paper has not proceeded beyond the orig- Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 2 of 22 Capstone courses in software engineering Table 2 Systematic reviews of software engineering capstones Title Year Ref Course characteristics examined in the survey A survey of computer science capstone course literature 2011 (Dugan Jr, 2011) Course-related: models, learning theories, goals, top- ics, student evaluation, evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Project-related: software process models, phases, type, documentation, tools, groups, instructor admin- istration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Designing the IT capstone course 2019 (Martin, 2019) Course duration, learning of new skills, project iden- tification and selection, teams sizes, team formation, followed methodologies, assessment of learning out- comes, team and project supervision* A review of literature on assessment practices in capstone engineering design courses: Implications for formative assess- ment 2006 (Trevisan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', 2006) Connection to student achievement For the survey by Martin (2019), these are characteristics, which would have been examined in the actual survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' inal abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In light of this, current research does not pro- vide an up-to-date view of how SE capstone courses gener- ally are organised and with what kind of outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Such a view on the software engineering capstones would not only provide educators with an important tool for planning their own capstone courses but also give researchers a basis for performing comparative studies on these courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Background: Capstone course characteristics ACM/IEEE Curriculum Guidelines for Software Engi- neering (SE) Degree Programs (ACM/IEEE, 2014) view the capstone project as an essential element of a SE degree pro- gramme and state that the main goal of a capstone course is to ensure that the curriculum has a significant real-world basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' According to ACM/IEEE (2014), incorporating real- world elements into the curriculum is necessary to enable ef- fective learning of software engineering skills and concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The ACM/IEEE Curriculum Guidelines for Computer Sci- ence (CS) degree programs (ACM/IEEE, 2013) align with these views and state that all graduates of CS programs should have been involved in at least one substantial project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Such projects should challenge students by being integrative, re- quiring evaluation of potential solutions and working on a larger scale than typical course projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For students, a cap- stone project typically represents a culmination of their stud- ies and is one of the last milestones before graduation (ACM/IEEE, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' ACM, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Indeed, since the 1970s, hundreds of primary studies have been written on this large, final-year project course (Dugan Jr, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The ACM/IEEE (2014) also lists a set of key recommen- dations that a capstone course should follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The recom- mendations are listed word by word in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We decided to use these recommendations as the basis for formulating our research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' They give a general outline of cap- stone courses and therefore provide a valid starting point for the categorisation done in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Thus according to these guidelines, there are some basic characteristics that capstone courses have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' They can be char- acterised as long and substantial projects (CR1, CR2) that should preferably be completed in a team (CR2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Projects Table 3 ACM/IEEE recommendations for SE capstones CR # Recommendation CR 1 The project should span a full academic year, giving students adequate time to reflect upon experiences and retry solutions as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' CR 2 Where possible, this should preferably be undertaken as a group project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' If such factors as assessment make this difficult, it is essential that there should be a sep- arate group project of substantial size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' CR 3 Where possible, a project should have a “customer” other than the supervisor so that the student gains fuller experience with product development life-cycle activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' CR 4 A project should have some form of implementation as its end deliverable so that the students can experi- ence a wide set of software development activities and adequately evaluate these experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Theory-based projects such as the development of formal specifica- tions is therefore inappropriate for this role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' CR 5 Evaluation of project outcomes should go beyond con- cept implementation (“we built it, and it worked” (Glass et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', 2004)), using walkthroughs, interviews, or simple experiments to assess the effectiveness and limitations of the deliverables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' CR 6 Assessment of a capstone project should consider how effectively software engineering practices and processes have been employed, including the quality of student reflection on the experience, and not be based only on the delivery of a working system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' should have customers (CR3) for whom the students are ex- pected to deliver some form of real implementation at the end of the course (CR4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Students should therefore engage in real software development activities and not just complete simple, theory-based assignments provided by the teacher (CR4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Evaluation of the project outcomes should focus not only on the fact that the project “works“, but also assess the deliverables on how well they have been completed (CR5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Finally, the focus of the course and its assessment should be on software engineering practices and processes and stu- dents should give adequate opportunities to reflect on the ex- Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 3 of 22 Capstone courses in software engineering perience (CR6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The next section describes in more detail the process of how we derived the research questions based on these basic characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Research questions and method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Research questions Characteristics of capstone courses (described in Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2) can be achieved in many ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The main goal of this research was to understand these differences in how cap- stone courses are implemented in universities and other ter- tiary education institutions, and thus provide a holistic view over the various capstone course implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We decided to use the ACM/IEEE Curriculum Guide- lines for Undergraduate SE Degree Programmes (ACM/IEEE, 2014) as the basis for starting to explore these characteris- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The recommendations are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Although some of these aspects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', team formation, have been ad- dressed in the previous reviews, previous literature reviews are slightly outdated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Moreover, we are not aware of any study covering all the aspects mentioned in the ACM/IEEE recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Related to CR1, we were interested in the duration of the courses and what rationale primary studies provide for choosing a specific course duration if any: RQ1 What is the duration of SE capstone courses, and what advantages or disadvantages are related to a certain duration?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Related to CR2, we wanted to find out if these projects are conducted in teams, how teams are composed and what is the rationale behind choosing a certain team size: RQ2 What team sizes do SE capstone courses have, and how are team sizes justified?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Based on the ACM/IEEE recommendations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', CR3), a project should have a customer other than the teacher of the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' An alternative approach to bringing an outside view to a project is to outsource project topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Thus, our third research question, how are the project and client sourcing handled in SE capstone courses, was divided into two sub- questions: RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1 Who acts as the client for capstone projects?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2 How are the ideas for projects sourced?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Related to CR4, we asked: How are the projects in cap- stone courses implemented (RQ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We wanted to uncover what students do in these courses and therefore, we looked into the actual project implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' As ‘project imple- mentation‘ can mean a multitude of things, we decided to di- vide this research question into smaller, more concrete sub- questions: RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1 What artefacts are students expected to produce on capstone courses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2 What is the software life-cycle gone through during these projects?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3 How are the implementation technologies chosen for capstone projects?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' With RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1 we aimed to find out what students actu- ally produce in these courses and whether any software is being developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2 helped us to find out if the cap- stone project is as integrative experience on software en- gineering practices as curriculum guidelines (ACM/IEEE, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' ACM, 2009) suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Finally, finding out how educa- tors make the choices for implementation technologies and what implications these choices have, gave some insight into project implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' As CR6 speaks about assessment, our last research ques- tion also asked: How is the student assessment conducted on SE capstone courses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' (RQ5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' As assessment can be divided into continuous feedback and final grading, RQ5 was also split into two: RQ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1 How are the students assessed at the end of SE cap- stone courses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' RQ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2 How are students guided, if at all, during SE cap- stone courses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The rationalisation here was that we wanted to uncover whether the evaluation is based on a multitude of factors like (ACM/IEEE, 2014) suggests and whether students are given adequate possibilities to reflect on their experiences (Hattie and Timperley, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In order to get a comprehensive representation of how project-based capstone courses are generally organised, rel- evant research articles were searched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' One could argue that the characteristics and any organisational details of these courses could be derived from the web pages of universities and other tertiary institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' However, we wanted not only to pro- duce a list of characteristics such as the duration and work- load of the courses but also to reveal more about the con- tents of these courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' An important part of the research was also to provide educators with insights related to the various characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Without any evaluation or assessment of the chosen structure and characteristics, this would have been impossible to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Search strategy The method used in this study follows the SLR method by Kitchenham and Charters (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The initial data col- lection was done by finding relevant sources from scientific databases: Scopus, ACM Digital Library, IEEE Xplore and ScienceDirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some preliminary searches were conducted on these databases to find out to which extent research ar- ticles use the word “capstone“ and its synonyms when de- scribing large, degree-culminating project courses in soft- ware engineering-related programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' It turned out that the term “capstone“ is well-known and widely used in research articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' It was also used by Dugan Jr (2011) in their ear- lier work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Therefore, the first search string was simply con- structed as: Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Page 4 of 22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Capstone courses in software engineering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Table 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Initial search results ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Database ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Search strings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Hits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Scopus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='TITLE-ABS-KEY ( software AND capstone ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='762 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Scopus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='TITLE-ABS-KEY ( software AND “project course“ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='262 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='ACM Digital Library ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='[Title: software] AND [Title: capstone] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='ACM Digital Library ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='[Keywords: software] AND [Keywords: capstone] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='ACM Digital Library ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='[Abstract: software] AND [Abstract: capstone] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='130 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='ACM Digital Library ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='[Title: software] AND [Title: “project course“] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='ACM Digital Library ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='[Keywords: software] AND [Keywords: “project course“] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='ACM Digital Library ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='[Abstract: software] AND [Abstract: “project course“] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='ScienceDirect ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='(TITLE ABS KEY: SOFTWARE CAPSTONE) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='ScienceDirect ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='(TITLE ABS KEY: SOFTWARE PROJECT COURSE) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='IEEE Xplore ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='(“All Metadata“:Software) AND (“All Metadata“:capstone) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='223 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='IEEE Xplore ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='(“All Metadata“:software) AND (“All Metadata“:“project course“) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='86 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='software AND capstone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='In order to have a complete picture of the project course ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='landscape in software engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=" a second search was per- formed using the second search string: software AND 'project course' This was deemed necessary as not all sources had the word “capstone“ present in the metadata even though they clearly were describing courses relevant to this research." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Searches with the two search strings were conducted sequentially in each database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Dugan Jr (2011) used “software engineering course“ as another search term in their study, but we did not want to limit ourselves to the SE discipline, as relevant software- related courses might be presented, for instance, in computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Using only the words “software“ and “course“ on the other hand, provided too many irrelevant hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Scopus alone produced nearly 30 000 hits of which only a small frac- tion would have been relevant to our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' A total of 981 unique papers were found after combin- ing the papers found from all four databases using the search strings and removing duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The databases were searched on June 11th and June 12th 2022, one after the other, starting with Scopus, moving on to ACM Digital Library, followed by ScienceDirect and finishing with IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' As the search fields and filters are slightly different in each of these databases, the search strings were adjusted to match each specific set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' They were, however, kept semantically the same across the searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Exact search strings and initial search results are listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' As we wanted to identify current ways of organising capstone courses, the searches in all four databases were limited to the years 2007 to the search day in June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This time period was regarded as suffi- ciently long to provide a holistic view of the current capstone courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' It overlaps with Dugan Jr (2011) by a few years but also uncovers 11 years of research done on the area that has not been systematically reviewed since.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Three stages of se- lection were applied to this initial set, after which 127 pri- mary studies remained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 1 summarises the search and se- lection process, and the following subsections will describe it in greater detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Figure 1: Search strategy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Paper selection The paper selection was conducted from the initial set of 981 sources by the first author (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The details of inclusion and exclusion are explained next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 5 of 22 Research questions (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=') Initial search 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Identty suitable search terms by conducting Used databases preliminary searches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Scopus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Construct the search strings 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' ACM Digital Library 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Conduct search to four databases 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' IEEE Xplore (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' ScienceDirect 981 unique papers First stage Read abstracts, tites and keywords and apply inclusion criteria to them (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=') 398 primary studies Second stage Read full text and apply exclusion criteria (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=') 171 primary studies Third stage Remove studies of courses with newer publications and studies of poor quality (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=') Final set: 127 primary studiesCapstone courses in software engineering 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The first stage - Inclusion criteria The titles, abstracts and keywords of the initial papers were read and evaluated against the inclusion criteria pre- sented below (IC1-IC3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' After the first stage, 398 papers remained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' IC1 The title or abstract strongly hints that the study presents frameworks or case studies of software engineering capstones or other large,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' project-based courses in software engineering IC2 Based on the title or abstract,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' the study describes real ex- periences of implementing a software engineering capstone course IC3 The title or abstract indicates that the study assesses the outcomes of the course or its characteristics The first inclusion criterion was developed to set the fo- cus on software engineering courses in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' A large number of the articles in the initial set were ruled out due to the first criterion (IC1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The papers were found to research, for instance, mechanical engineering courses, which were out of the scope of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We also wanted to rule out any purely hypothetical papers, where the researchers show no course that follows the frameworks or structures presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The second inclusion criterion (IC2) aimed to ensure that all included papers would present a real-world course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The final inclusion criterion (IC3) was generated so that all papers would also evaluate the outcomes of the var- ious course implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The second stage - Exclusion criteria The second stage was performed on the 398 papers re- maining from the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Any article that, based on read- ing the full paper, met at least one of the presented exclusion criteria was excluded at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' After this selection, 171 articles remained for the final evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The used exclu- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='sion criteria were: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='EC1 The length of the study is less than four pages ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='EC2 The study is not published in conference proceedings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='or as a journal article ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='EC3 The study does not have full text available in English ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='EC4 The study turned out not to describe a software engi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='neering capstone course in a tertiary institution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='EC5 The study is not able to provide answers to most of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='research questions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Exclusion criteria from EC1 through EC3 aimed to en- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='sure that the study was of sufficient quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' According to Kitchenham and Charters (2007) workshop proceedings of- ten do not provide sufficient input for the purposes of an SLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Additionally, quite many of the papers that were first published as short workshop proceedings or abstracts were also found to have a conference proceeding or a journal arti- cle published later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' EC3 relates to the language skills of the authors as well as the status of English as the primary language in software engineering-related research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' These exclusion criteria led to some papers being rejected before reading their entire content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For exclusion criteria EC4–EC5, the content of the ar- ticle was examined more carefully and, in most cases, read in its entirety to make a justified decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Exclusion crite- ria EC4 and EC5 relate to our research goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For instance, many articles were found to describe courses in computer en- gineering or mini-projects conducted prior to SE capstones which meant that they were out of the scope of this research and excluded based on EC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Most of the papers left out dur- ing this stage met EC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' As for EC5, some studies were, for example, found to describe a whole curriculum with cap- stone courses playing only a minor part in the research, and they could therefore not provide answers to our research ques- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' A number of studies also evaluated a tool, method or framework relevant to the software engineering industry, not the capstone course itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In many of these studies, the cap- stone course presented the researchers merely a convenient way of gaining study participants, which is why they did not fit this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This led them to be excluded due to EC5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The third stage - Removal of duplicates and studies of poor quality The third stage was included mainly to rule out any du- plicate data and studies of poor quality from our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' After the third stage, the final set of 127 papers remained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Duplicate data – All 171 primary studies remaining af- ter the second stage describe real-life software engineering capstones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' As educators often like to modify their courses over time to find the best ways of teaching, studies here too reflect on the changes done to the courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some authors also have written multiple articles based on the same cap- stone course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In such cases, the most recent article was cho- sen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Similarly, if a study describes several instances of the course in one paper, the principal characteristics of the most recent course instance were chosen for the data extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Choosing the latest instance of each course stems from the goal of this research to synthesise the current state of knowl- edge on capstone implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In addition, the decision of whether two descriptions of the same course are different enough for them to be included as their own capstone courses would have been too ambiguous and open for interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Kitchenham and Charters (2007) also state that it is impor- tant not to include multiple publications of the same data, as it would seriously bias any results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Due to this procedure, 42 studies were removed from the final set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Quality assessment – In addition to inclusion/exclusion criteria, Kitchenham and Charters (2007) state that it is crit- ical to perform a quality assessment on the primary studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We also conducted a such assessment and used it to ensure that our final data set is of sufficient quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' At this stage, two studies were filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Any tables or graphs presented from here on do not include these two excluded studies, and therefore represent the final set of 127 studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Table 5 lists the set of questions used by Dybå and Dingsøyr (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Mahdavi-Hezavehi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' (2013) which we also used to determine the quality of primary stud- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Originally the questions were supposed to be graded on Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 6 of 22 Capstone courses in software engineering Figure 2: Quality scores of the final set of studies a dichotomous (“Yes“ = 1 or “No“ = 0) scale (Dybå and Dingsøyr, 2008), but we decided to use a three-point scale of “Yes“ (= 1), “To some extent“ (= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5) and “No“ (= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This three-point scale has also been adopted by Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' (2010) and Mahdavi-Hezavehi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' (2013) and allowed us better to assess the studies where authors only provided some answers to the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The two articles filtered out had a quality score of less than 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In our assessment, we decided to group the first eight questions to represent the quality of reporting and rigour of the studies and final three questions to represent the credibil- ity of evidence, similarly to Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The grouped scores are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2 and individual scores for each study can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='com/article-additions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Regarding the quality of reporting, the selected primary stud- ies performed fairly well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' It was mostly clear how the data had been collected, and the relevance of the study was ex- plicitly discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' However, the aspect most studies were lacking was providing justifications either for the sample se- lection or research designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Regarding the credibility of ev- idence, the studies performed fairly poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Interestingly, many of the otherwise well-established studies did not in- clude a section for explicitly discussing the limitations of the study or the author’s role in data and sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This is indicated in the low averages of the credibility category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Table 5 Questions for quality assessment No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Question Q1 Is there a rationale for why the study was undertaken?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Q2 Is there an adequate description of the context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' indus- try, laboratory setting, products used, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=') in which the research was carried out?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Q3 Is there a justification and description for the research de- sign?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Q4 Has the researcher explained how the study sample (partic- ipants or cases) was identified and selected, and what was the justification for such selection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Q5 Is it clear how the data was collected (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' through inter- views, forms, observation, tools, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' )?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Q6 Does the study provide a description and justification of the data analysis approaches?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Q7 Has ‘sufficient’ data been presented to support the findings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Q8 Is there a clear statement of the findings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Q9 Did the researcher critically examine their own role, poten- tial bias and influence during the formulation of research questions, sample recruitment, data collection, and analysis and selection of data for presentation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Q10 Do the authors discuss the credibility of their findings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Q11 Are limitations of the study discussed explicitly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Overview of the final papers The three stages taken resulted in 127 primary studies, published between 2007 and June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Research activity in this area has been fairly steady over the years, as depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' It is worth noting, that we searched for papers in June 2022, making the study amount for 2022 partial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Also, as explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3, 42 earlier studies, which other- Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 7 of 22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5 2 Credibility 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5 8 12 19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5 8 Quality of reporting and rigorCapstone courses in software engineering wise would have been valid for this research, were excluded from the final set of papers as there was a newer study of the same course available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This procedure skews the year distri- bution towards the end of the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The figure also shows the distribution by study type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In total, of studies 73% were conference proceedings, and 27% of studies were published as journal articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Figure 3: Timeline and types of primary studies All the articles included for further analysis are listed in Appendix A, Table 13, and referenced later in this section with their publication ID in the table (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', S1–S127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=') 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Data extraction and synthesis After applying the study selection process, the properties presented in Table 6 were extracted from the remaining 127 studies to a common datasheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Table 6 defines how each extracted field relates to the research questions of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Table 6 Data extraction form Identifier Field RQ F1 Title metadata F2 Author(s) metadata F3 Year metadata F4 Publication venue metadata F5 Duration of the course RQ1 F6 Course workload RQ1 F7 Team sizes RQ2 F8 Clients RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1 F9 Project sources RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2 F10 Artefacts produced RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1 F11 Project phases RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2 F12 Technologies RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3 F13 Student assessment RQ5 F14 Outcomes of the course RQ1–RQ5 F15 Quality score QA Values F1–F4 were extracted for basic documentation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Items F5–F14 concern the course and its organisa- tion presented in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Two of the studies present multi- ple separate capstone courses from different institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For these two studies (S8 and S93 in Table 13), the items F5–F14 were extracted for each of the courses they presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For F5–F14, we were not only interested in quantifying these characteristics into statistics but also in providing implica- tions of different course design choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Therefore, if the study stated, for example, that they had a two-semester cap- stone course because it provided students adequate time to learn, we recorded both of these information pieces: the quantifiable duration as well as any such insight relating to the characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This enabled us to analyse and discuss the course characteristics better in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' A data-driven the- matic analysis was applied to synthesise the qualitative data extracted as part of F5–F14 (Castleberry and Nolen, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' F5 and F6 were considered essential in assessing the gen- eral workload and duration of the course from the student’s perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Few sources have given the duration of their course (F5) as months or weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' These were rounded to the nearest amount of semesters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Courses lasting less than 4 months were categorised as “less than one semester“, 4 to 6 months as “one semester“ and anything more than 6 but less than 10 months as “two semesters“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Team sizes (F7) included the number of students per team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The courses were also examined on whether the projects in the course were done for a client (F8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The client could be ex- ternal to the course staff, the role of a client could be played by the course staff, and some projects did not have clients at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some sources present a mix of these categories, in which case, the source was labelled by the client category, which we thought was the most prevalent in the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We were also interested in how the project topics were generated (F9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Three main sources for projects were identi- fied during the data extraction: course staff, external clients and the students themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The projects were also found to vary regarding whether the students were working on the same project idea or whether each team had their own initial problem to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We extracted all the artefacts that students were expected to produce during the course (F10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This included both the deliverables used for grading the course as well as artefacts produced for project management reasons, as in most cases, it was hard to draw a distinction between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Evidence of project phases (F11) was extracted to find out which soft- ware life-cycle activities are gone through in these courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' F12 describes the technologies used in the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We found that most of the studies do not explicitly specify all the tech- nologies used for the projects in their courses, and more- over, these technologies could potentially include any soft- ware technologies available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We, therefore, categorised these into two categories, based on whether the main technology selections are made team-wise, or all use a common technol- ogy stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We extracted information on how the students learning process was assessed and improved throughout the course and how the student’s progress and achievement were as- sessed at the end of the course (F13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The key outcomes in Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 8 of 22 16 14 12 10 8 6 4 2 ■Conference papers ■ Journal articlesCapstone courses in software engineering Table 7 Duration of capstone courses Category Number of studies Percentage Study identifiers Less than one semester 10 8% S16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S46,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S61,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S78,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S94,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S99,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S113 One semester 87 66% S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S8b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S8d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S35,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3 for quality assessment and study filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Results and analysis This section represents quantitative statistics and qual- itative outcomes of capstone characteristics extracted from the primary studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The characterisation enables us to an- swer our research questions and ultimately helps educators when they are planning their capstone courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Duration (RQ1) Regarding the actual course characteristics, we first looked into the reported duration of these courses (F5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' A clear majority of institutions conduct capstone courses that last one semester (Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Interestingly, this is in conflict with the ACM/IEEE (2014) recommendations for undergraduate capstone courses, which propose having capstones lasting the entire academic year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' However, the unfortunate reality is that not all curricula can absorb a full-year implementa- tion [S117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The capstone courses often are very labour- intensive for the teaching staff, with many teams to man- age and evaluate throughout the projects [S39], [S112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Stu- dents might have full- or part-time work, which makes the longer courses harder to arrange [S49], [S58], [S73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Stu- dents also perceive two-semester capstone courses as labo- rious [S73], [S109], and some even the one-semester ones [S23], [S41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In order to provide an intensive and realis- tic experience, many of these courses take up at least half a work-week [S18], [S28], [S41], [S69], [S73], [S109], [S127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This again might make other courses taken simultaneously suffer [S41], which limits the possibilities for an intensive, year-long capstone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' However, the educators who had experiences with both, shorter and longer duration, had shifted to the longer du- ration since they felt it was impossible to reach the wanted depth in just a few months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S6 describes how they switched to a two-semester capstone as they found the one-semester projects inadequate in skill coverage and depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S70 is writ- ten by students of the course, and they strongly recommend that their course be lengthened into one academic year from the current duration of one semester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S33 have had experi- ences with one-period, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' a quarter of an academic year, and three-period courses, and stated that the change to a longer version received overwhelmingly positive feedback from all the participating parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Students were able to gain more hands-on experience in applying new and familiar tools and project management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Additionally, they learned to act when faced with unanticipated events as the teams experienced surprises – regarding both technologies and people – mul- tiple times during the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Industrial clients received more ambitious and polished products as a result of the course, and the course staff felt that the learning objectives for the course were finally truly met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Team sizes (RQ2) To find out how many students there generally are in a project group, we extracted the reported team sizes (F7) for each course in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' If a study refers to their course having teams of 4–5 students, this is thus reflected in both columns 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' By looking at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4, it is evident that capstone courses are almost always conducted as group projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Only three institutions in our research allow their capstone or se- nior project courses to be completed as single-student en- deavours [S11], [S89], [S111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Team sizes vary a great deal, ranging from 1 to 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Re- search has found that in very small groups, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', 2–3 stu- dents, the teams are unlikely to generate the dynamics and issues that are common in collaborative software develop- ment [S36], [S53], [S56], [S58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Such a small team size Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 9 of 22 Capstone courses in software engineering Figure 4: Team sizes in capstone courses does not present enough of a challenge [S36], and smaller groups are unable to complete substantial projects in a typ- ical one-semester course [S53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Having very small teams might also be unmaintainable in large programs with hun- dreds of, or even a thousand, students due to the extra or- ganisational overhead each team causes [S19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Going to the other extreme, larger groups with 7 or more students have often been found to be facing other kinds of problems, such as the inability to meet all together and other management and coordination issues [S30], [S36], [S39], [S53], [S56], [S78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' “Free-rider“ problem is also reportedly common in larger teams, where it is possible for few students to take the bigger responsibility for ensuring the overall success, and the small contribution of others might go unnoticed [S9], [S56], [S58], [S69], [S87], [S121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In larger teams, ensuring fair grading and an equal balance of work and responsibilities requires more attention from the course staff [S56], [S87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The course conducted in S106 had one of the largest team sizes found in our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The course had 15 students, all working on the same game project in one team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The idea was to simulate what large-scale game development in a diverse team feels like and what it takes to create production-quality games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The authors share that their approach was not en- tirely successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In the aftermath of the course, it came up that some students wanted explicit direction while others felt that they wanted more autonomy and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' According to the authors, for the latter group of students, it was clear that they were uncomfortable following the leadership of the vi- sion team and would have preferred to work on a project of their own design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' However, the authors also mention that getting to work with your own project vision is a very un- likely case for any recent graduate, which is why they did try to come up with such a real-world teamwork scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The majority of educators do seem to opt for the middle ground regarding team sizes and have 4 to 5 people working in a single group (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This size is perceived as the sweet spot, cancelling out the negatives of the two extremes [S36], [S52], [S53], [S56], [S58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Students themselves have also reported being satisfied with such a team size [S56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Addi- tional measures for combating any non-productive and op- portunistic group behaviour, such as social loafing and free riding, have also been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Conducting peer reviews has been proven to mitigate the risk of such behaviour [S72], [S98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some periodic monitoring should also be done by the course staff to ensure working team dynamics [S69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Both of these will be discussed further in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Clients and project ideas (RQ3) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Clients (RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1) We also looked at who is in the role of a client for these projects (F8) and how the project ideas are sourced (F9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Al- most half of the studies (42 %) report conducting their cap- stone courses without clients that are external to the course (Table 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In these courses, the course staff may act as clients or Product Owners for the projects or alternatively, the stu- dent teams work on their own and only report progress reg- ularly to the course staff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S6 explains that they have instruc- tors playing clients due to the difficulty of finding suitable clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Being a small program in a rural institution makes the businesses and organisations suited for such collabora- Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 10 of 22 70 62 60 56 50 40 Occurences of team sizes 34 33 30 20 19 18 15 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 6 3 2 2 1 1 1 1 1 1 1 0 8 1 06 9 30 3 V VotCapstone courses in software engineering Table 8 Clients of capstone courses Category Number of studies Percentage Study identifiers Clients external to the course staff 76 58% S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S7,' metadata={'source': 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would have suit- able clients available, there is always the upfront investment in time and effort that the course staff has to make to contact said clients, guiding them through creating project propos- als and assigning the students to these projects [S118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S42 aimed to create a course with students from five different technical and non-technical disciplines, such as computer science and business informatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S42 mentions that they need to be careful in how they organise the course so that it would suit the needs of all disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Bringing an ex- ternal client into the mix might not fulfil the learning goals for all students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some studies also explain how the course outcomes are less predictable with multiple external clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S114 have experienced several cases when the project spon- sors did not show up for the bi-weekly meetings with the stu- dents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Such client behaviour caused very low motivation in the student teams, and some capstone projects failed due to client unavailability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S3, S23, S78 and S122 have made sim- ilar observations and stress the importance of finding com- mitted clients to ensure a good experience for the students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Despite these risks having real, external clients other than the course supervisor for the projects is recommended for both undergraduate and graduate capstones (ACM/IEEE, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' ACM, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' These clients can be from other units within the university [S7], [S9], [S14], [S52], [S58], [S86], [S87], [S94], local businesses [S7], [S9], [S86], [S87], [S98], [S110], [S122] or various non-profit organisations [S7], [S9], [S11], [S16], [S58], [S110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Graduates of the program who already work in the industry are also a convenient way to find clients [S35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Working closely with real-world clients has also of- ten received highly positive feedback from students [S14], [S15], [S52], [S73], [S98], [S117] and organising staff alike [S14], [S35], [S84], [S98], [S117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' It has been found to in- crease the motivation and commitment of students, when there is an actual client with a real need behind the project [S9], [S14], [S15], [S35], [S66], [S73], [S84], [S121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' It has helped to keep the experience more realistic and credible in the students’ eyes [S19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Having industry clients improves the students’ technical and nontechnical skills and better pre- pares them for the challenges they will face in the work-life [S14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The collaboration has been reported to have benefits for the client too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S35 conducted a study to find out the rea- sons why clients participate in such project courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The rea- sons included getting a tailored software product, research- ing new technologies and, as a clear number one, recruit- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Recruiting students could happen directly from the team or more indirectly by adding visibility among the stu- dents as potential employers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Others have noticed this ben- efit, too;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' it is not uncommon for students to get hired by the industry partner who sponsored their capstone project [S15], [S28], [S35], [S73], [S84], [S114], [S121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S73 report hav- ing at least 60 out of a few hundred students gaining full- or part-time job offers based on the capstone project outcomes, in mere few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Keeping the experience positive also for the clients, might make them come back with further project ideas [S35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This helps to reduce the client acquisition over- head for years to come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some organising institutions have even managed to attract more external clients than there are student teams, which has enabled them to collect a small fee from the ones participating in the course [S35], [S121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Project sources (RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2) In addition to finding out the clients for these projects, we also looked into how the projects for these courses are sourced (F9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Three main ways for project sourcing were identified (Table 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' As the majority of courses have multiple external clients, the project ideas in these courses are mainly derived from the needs of the customer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In these cases, the organising staff often performs some pre-screening and scop- Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 11 of 22 Capstone courses in software engineering Table 9 Sources for projects in capstone courses Category Number of courses Percentage Study identifiers External stakeholders pro- pose project ideas 81 62% S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S8a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S8b,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S116,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S123 Students generate their own project ideas 22 17% S22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S36,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S44,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S47,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S51,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S53,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S56,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S70,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S76,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S83,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S89,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S95,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S101,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S102,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S103,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S107,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S111,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S118,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S119 Not specified 1 1% S32 ing in collaboration with the clients,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' to ensure that the ex- pectations for the projects are realistic and that the project scopes suit the intended learning outcomes [S9],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S35],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S52],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S61],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S87],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S90],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S94],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S98],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Capstone projects should generally not be on the critical path of any external organisation, as the course is intended to remain a safe learning place for the students [S35], [S52], [S87], [S90], [S121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some studies also emphasise that students are not supposed to be working for these clients, but be in collaboration with them [S9], [S52], [S87], [S121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Thus, the projects need to remain such that the students can have a say in how the project will be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For 17% of the courses, the students themselves are the main source for project ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' These studies state that stu- dents are more motivated if they get to choose the project idea rather than have teachers assigning the projects [S36], [S53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' According to S53, if the team selects and defines the projects, their level of commitment and excitement to the project rises as the software system grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' At the end of the semester, the students have a strong sense of owner- ship towards the project, rather than feeling that they have just done one additional assignment [S36], [S53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' However, there are some potential pitfalls with this approach that edu- cators should be aware of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S114 states that students should not be allowed to bring project ideas from the companies they work at or from their own businesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S114 have found that it causes a conflict of interests for the student with the proposal and creates an unfair situation for the rest of the team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S36 lets students form their own teams and generate their own project ideas, but states this might not accurately reflect the situation in the students’ future professional lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' If the project idea comes from the team itself, all complex- ities associated with requirements elicitation and analysis are eliminated [S73], making the experience less realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Real-life projects come with challenges relating to contra- dicting expectations coming from various external and in- ternal stakeholders [S73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For 21% of the courses, the course staff provides the project specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some educators have assigned the same project idea to all the student teams [S40], [S45], [S72], [S82] or even in some cases, all the students work on the ex- act same project in one team [S106], [S112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Having the same project has the benefit of giving the course staff a con- sistent basis for grading and teaching [S40], [S59], [S72] and providing technical assistance to the students [S40], [S53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In such cases, all teams will need to deal with the same complexities, project management issues and technology de- mands as in a typically constructed course, which makes the experience more predictable [S72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Having one project idea also opens up the possibility for competition amongst the teams, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', which team will create the best design and im- plementation [S5], [S30], [S37], [S42], [S54], [S72], [S106] potentially even for an external client [S30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' It also possi- bly allows the course to focus more on the quality of the developed software [S5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S5 has experience with both ap- proaches, having multiple project ideas and having only one project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' They have found it more productive and rewarding to focus on doing one project really well rather than juggling multiple projects and obtaining partial results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Project implementation (RQ4) We also aimed to uncover information on how capstone projects are implemented and what students are expected to do in these courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For this, we extracted various informa- tion on the project implementation (F10–F12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Produced artifacts (RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1) We were interested in finding out what kind of artefacts students are expected to produce on the course (F10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' It was often difficult to determine which artefacts were used for the final student assessment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' grading) and which were only produced to manage the project in some way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Therefore we Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 12 of 22 Capstone courses in software engineering Figure 5: Most commonly mentioned artefacts produced by students in capstone courses could not produce a list specifically of graded artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Ta- ble 5 provides a rough view of the explicitly mentioned arte- facts that students are expected to produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We were, however, able to determine that all but three courses [S16], [S17], [S103] expect some form of a software prototype, a software product or source code as the end de- liverable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Out of the courses which did not include produc- ing software, S16 conducted a course where the process and end deliverables are focused on students completing various research items, such as testing new team-based technolo- gies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' pair programming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For S17 the end deliverable was often a solution proposal for the client organisation’s IT department, and S103 delivered an inter-disciplinary course which might result in software deliverables or alternatively in reports describing a software-based solution to the defined problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' using 3D engines for art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Most studies mention requiring some documentation for the software project (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The actual number of courses that require documentation might be larger than reported here, as we only counted the times the study explicitly mentions that project documentation is done on the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Studies at the beginning of our time range are often following more “plan-first“ software development approaches, such as the waterfall model, and as a result, the quantity and detail of non-software artefacts are substantial [S2], [S11], [S22], [S70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The documentation usually starts heavily upfront by students producing project plans [S2], [S68], [S70], detailed designs [S2], [S22], [S52], [S68], architecture plans [S22], [S68] and test plans [S2], [S70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In later, more agile, courses, there is less evidence of extensive documentation and planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' With agile projects, the system documentation is largely de- veloped as the project evolves [S14], [S65], [S94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Students in these courses often also produce agile artefacts, such as product backlogs, sprint backlogs and burndown charts for project management and planning purposes [S6], [S65], [S94], [S124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' A large share of the studies (60%) explicitly mention that students must present or demonstrate their projects to wider audiences than just the immediate project team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This is a way to teach students how to present and explain their work also to a non-technical audience [S6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The most common time for presentations is at the end of the semester, and typ- ically these presentations are given at fairs or class sessions where all stakeholders of the course are invited [S2], [S4], [S6], [S33], [S95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The format of final presentations varies from poster sessions [S6], [S28], [S34], [S44] to live demon- stration sessions of the software [S89], [S95] to different kinds of demo videos [S1], [S73], [S118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Students are some- times also expected to draw up a project proposal or a pitch and then present it to the teachers or the class before start- ing to work on the implementation [S87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In these cases, the purpose is to offer the students a chance to practice their pitching skills [S100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Software requirements are something that students are often expected to detail during the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' These can be writ- ten down as a Software Requirements Specification created before the implementation phase begins [S30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For courses with agile methodologies, software requirements are often documented in an initial backlog with user stories, and the backlog is then updated as the project continues [S124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Stu- dents are also quite often expected to write some form of a report at the end of the experience, either individually or as a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The reports generally involve students reflecting on the learning done and development processes employed throughout the course [S6], [S19], [S21], [S24], [S27], [S41], [S63], [S65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 13 of 22 % of courses 0 10 20 30 40 50 60 70 80 90 100 Software or prototype 97% System documentation (technical/architecture/design) 70 % Presentations and demos 60 % Requirements specification 40 % Final and/or mid-term reports 36% 29% Agile artifacts Regular progress reports 14%Capstone courses in software engineering The balance between too little and too many other arte- facts is a delicate one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Having more documentation and de- liverables presents teachers with more opportunities to grade and assess students’ understanding of software development processes [S110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' On the other hand, more documentation means less product which might not be in the interests of project clients [S49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Indeed, some educators mandate only basic time tracking and reflective reporting from the students and have left the majority of the deliverables for the external client and team to decide [S24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Project phases (RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2) We sought evidence of the project phases and software life-cycle gone through on these courses (F11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Software life-cycle models include, with varying frequency and order, phases such as requirements gathering or solicitation, plan- ning and designing, developing, testing and maintaining the product (Mishra and Dubey, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Out of the studies that discuss the development process and end product quality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' the projects generally proceed from ideas to robust proof- of-concepts or products with few core requirements imple- mented [S1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S6],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S7],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S11],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S12] [S16],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S20],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S21],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S40],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S42],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S45],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S52],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S54],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S55],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S62],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S64],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S65],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S68],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S79],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S70],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S72],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S75],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S77],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S78],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S87],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S90],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S94],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S95],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S99],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S100],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S106],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S107],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S109],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S115],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S118],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S122],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S123],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S124],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Students thus get to get experience the phases of planning, designing, developing and testing the products in these projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In courses with clients, either external or internal, the stu- dents usually have to solicit the requirements from the clients (Table 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' However, sometimes the teachers provide stu- dents with ready-made feature or requirements lists [S12], [S21], [S45], [S79], [S82] and in some courses, students gen- erate their own project proposals (Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The experi- ence of requirements gathering is somewhat diminished in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Additionally, as the projects proceed from ideas to proofs- of-concept or simple, handed-off products, the projects gen- erally do not include developing existing products, especially ones that are in production-use during the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' There are some courses where some of the projects have been production- ready at the end of the course, but these too were then handed over to the customer [S5], [S9], [S117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This practice leaves students without the experience of working with existing prod- ucts or products in the true maintenance phase of their soft- ware life-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Assigning students to contribute to Free and Open Source Software (FOSS) projects is an emerging approach to remedy these shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The idea is to al- low students to deal with existing codebases, often large and complex, such as the one they will face when working in the industry [S8b], [S8c], [S112], [S113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some courses have also had a continuation of earlier projects in the course to ex- pose students to code generated by other people [S14], [S15], [S19], [S38], [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Both of these approaches allow students to maintain existing code, but they still present a minority in our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Project technologies (RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3) We also looked into the development technologies used in capstone courses and how they are selected (F12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Com- monly in multi-customer courses or in courses with other- wise very differing project ideas, the technology choices are made based on the project (Table 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In these cases, the course staff does not impose an entirely common technology stack for all the projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For some of these projects, the tech- nology stack is based on the client’s infrastructure [S35] and in some cases, the students get to make manager-like deci- sions on the suitable development technologies [S6], [S84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Having the teams decide on the tools and technologies makes the students explore available options and justify their se- lections [S6], [S56], [S84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S84 states that not only give them autonomy but also make them responsible for their own successes and failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' However, even though the majority of technologies would be selected based on the project and client, some studies recommend having some shared infras- tructural tools and technologies [S12], [S19], [S36], [S65], [S102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Version-control [S6], [S12], [S19], [S29], [S36], [S67], [S102], project management and communication tools [S19], [S29], [S65], [S67], [S80] and tools for continuous in- tegration and delivery are examples of these [S32], [S78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' It has been found to make the management and evaluation of projects easier [S19], [S29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Then again, having common development technologies for all projects is fairly common in cases, where the teachers provide students with the project requirements [S42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In some cases, the evaluation meth- ods focus heavily on the technical implementation, and the course graders might, for example, have sets of tests they like to run on each project to determine the quality [S109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some educators have the students compete on the same project proposal, which makes choosing a common stack justifiable [S37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Assessment of students (RQ5) We were also interested in finding out how the assess- ment is conducted in these courses (F13), and to which ex- tent students are given possibilities to reflect on their experi- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We looked at both the end-of-course student assess- ment (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1) and any continuous guidance and feed- back students are given during the course (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4.' metadata={'source': 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+page_content=' making the evaluation of these skills harder [S52],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S56],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' [S86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For these reasons, several studies report having ad- ditional sources for student assessment beyond the teachers’ evaluation of produced artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Different kinds of (anony- mous) peer evaluations are fairly common (31%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' They give course staff a look into the team dynamics during the course and help in detecting social loafing or free-rider behaviour [S12], [S86], [S112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Similarly, self-evaluation is often done either in combination with peer evaluations [S56] or as a part of the reflection done in a project’s final report [S65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some studies report utilising the client’s opinion in the course assessment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Clients can fill out a question- naire considering each student’s performance during the course [S86] or only evaluate the team’s deliverables or presenta- tions and their value from the client’s point-of-view [S35], [S52], [S89], [S127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' As with self- and peer-reviews, edu- cators use the client’s opinion as a complementary source of assessment when grading students (Table 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Continuous student assessment and guidance (RQ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2) Many studies specifically mention that the teams should not be left entirely on their own to complete the course project and should be guided along the way [S6], [S9], [S10], [S16], [S18], [S53], [S69], [S72], [S86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Three main ways for con- ducting such continuous assessment and guidance were found based on our research (Table 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Studies where there is no mention of the teacher, or anyone else, having an active role in how the teams work during the course, fall into the category “Not specified“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' If the course staff only passively receives reports of the student’s progress and evaluates the course outcomes after its completion, these are not the active guidance of the teams we were looking for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some courses have several types of guidance present, in which case the study has been listed under each corresponding category in Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Only 11% of the studies do not explicitly specify having any ongoing feedback and guidance system present during the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The most convenient way is to have the course staff, such as the responsible teacher or hired teaching assistants, act- ing in an advisory role (76%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The intensity of the guid- ance given by course staff varies a great deal between these courses, or even within these courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Sometimes course staff provides oversight in a more supervisory role and inter- venes in the team’s work if any conflicts arise or the team in- cludes clearly non-contributing students [S6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' On the other end, some instructors have weekly meetings with the stu- dents where the teachers actively propose solutions and guide the teams with technical and non-technical issues and team dynamics [S6], [S88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some teachers prefer even to prac- tically manage the team [S6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S38 explains that the most successful changes made on the course were those that al- lowed the course staff to take a more active role in each team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The grades of students improved, and the teams were able to complete more functionality of the software products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Another, often complementary, guidance form is to have industry experts occasionally participate in the course (16%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This can be seen as especially relevant when the course projects are focused around a common theme, for instance, the gam- ing industry [S25], [S106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' However, finding the correct bal- ance in this type of assessment, without a client relationship, has sometimes proven to be tricky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S106 had industry ex- perts from the gaming and software industries participating as advisers on their course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The advisers’ feedback on the students’ game product was mainly positive and encourag- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' While the staff took their remarks to mean that the game concept and development for the moment were commend- able, the students took the feedback to mean that the proto- type was, as presented, worthy of praise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This presented a dichotomy that never really resolved: students felt that the project was near-complete, whereas the instructors felt that the project was, at best, a rough sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Many authors have noticed the upsides of having more experienced students outside of the course staff, mentoring the students in the course (18%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' These can be, for instance, Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 15 of 22 Capstone courses in software engineering Table 11 End of course assessment Category Number of courses Percentage Study identifiers Course staff 93 71% S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S6,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S38,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S70,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S79,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S107,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S111,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S117 students who have completed the project course themselves in the past year [S122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This has been found to benefit both the project implementation and group dynamics: an active and knowledgeable coach can, for example, help students ask clarifying questions of the customer, overcoming fear of these being stupid and saving days or weeks [S122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Interestingly, forming a capstone team of the final year students with similar skill levels are in accordance with the ACM/IEEE Curriculum Guidelines for Undergraduate SE Degree Programmes (ACM/IEEE, 2014), but leaves out an Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 16 of 22 Capstone courses in software engineering integral part of the real software development team experi- ence: junior and senior positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This discrepancy has been noted in some studies [S19], [S35], [S58], [S98], where the course implementation has gone beyond having senior stu- dents just as advisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In these capstones, less-experienced students work as junior developers and more-experienced students as senior developers or team leaders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S19 organ- ised their capstone course in a way that students are required to work two course units on the same project, one unit as a junior member and one unit as a senior member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Each unit lasts one period (a quarter of an academic year), but the periods do not have to be consecutive to allow some flexibility for students in organising their studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In order for such an arrangement to work, the projects in the course are large, long-term products, which undergo enhancements over a number of semesters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S19 found that for junior stu- dents, this setup allowed a smooth transition to the project, up-skilling on relevant skills and acquiring the necessary ori- entation from senior students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Senior students, on the other hand, were enthusiastic about mentoring junior students and finding answers to their questions ranging from project re- quirements to the technology stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S98 have similarly split their capstone project into two parts with junior and senior positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' They also had faculty mentors with industrial ex- perience mentoring the student teams working with exter- nal clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' According to S98, having this course design en- abled them to create an effective industrial simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' S98 reports that students used tools and practices prevalent in the industry but frequently not taught in university and were able to develop professional and team working skills more inten- sively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Discussion In this research, the main objective was to understand how tertiary education institutions conduct their SE capstone courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This was done by looking at the characteristics of capstone courses through an extensive literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Firstly, we summarise the main findings of this study and compare them to the findings of previous systematic litera- ture reviews, whenever appropriate (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Secondly, we present suggestions for further research in this area in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Finally, we discuss the validity of the results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Main findings 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Duration (RQ1) Despite ACM/IEEE (2014) recommending that under- graduate SE capstones should span the whole academic year, most of courses identified here last only one semester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This is in line with findings regarding course models by Dugan Jr (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In our research, the studies presenting two-semester capstones often found one-semester courses inadequate in depth and breadth of skills they can provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' These stud- ies reported that a longer course better prepares students for the experiences they can expect in their working life in soft- ware engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' There are, however, some real-world con- straints to why the courses generally are shorter, such as cramped curricula and the time and effort capstones require from both, the staff and the students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Team sizes (RQ2) We found that capstone courses are generally conducted as large-scale group projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The team sizes varied greatly between courses, ranging from 1 to 35 students in one team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Dugan Jr (2011) found no agreement in the literature on the appropriate team sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Our results were somewhat contra- dictory to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In our research, several studies that reported experiences with different team sizes had found the optimal to be 4–6 students per group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This was also reflected in av- erage team sizes for capstones we found based on our re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For a group of 2–3 students, there is no communi- cation challenge to solve, and smaller groups often are not able to accomplish larger projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In contrast, larger teams often present too many issues for communication, coordina- tion and fair grading of students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Larger teams also require more effort from the teaching staff to ensure an even distri- bution of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Clients and project ideas (RQ3) In our research, 58% of the studies reported having ex- ternal clients for student projects (RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1) and projects are based on the real needs of external stakeholders in 62% of the courses (RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Having clients outside the immediate course staff presents more work for the teachers, but is often rewarding for students when they get to work for real clients on real projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The motivation boost in students, as well as the positive implications in their skills and employment after the course, were found to be among the top reasons for having real clients with real projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Using external clients is also recommended for both undergraduate and graduate degree programmes in software engineering (ACM/IEEE, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' ACM, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Despite these benefits, there still is a considerable number of capstone courses (42%), where the course staff acts as the client for these projects, or there is no client for students to interact with regularly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In addition, there are quite many courses where the teacher provides the students with project specifications (21%) or students them- selves generate project proposals (17%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Such courses gen- erally are less burdensome for teachers who do not have to get involved in sourcing multiple clients and projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Stu- dents are often also motivated when they get to choose a project topic of their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' However, in these cases, students do not get to experience requirement solicitation and plan- ning the project with an external stakeholder, who might not be technically knowledgeable at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The taxonomy used by Dugan Jr (2011) relates to project topics and not particularly project sources or clients making it difficult to assess whether there has been changes in this over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Moreover, our work demonstrates that external stakeholders can get in- volved in many ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In the future, it would be important to explore this in more detail, and evaluate the consequences as demonstrated by Steghöfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 17 of 22 Capstone courses in software engineering 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Project implementation (RQ4) To understand the project implementation, we first looked into the artefacts that students are expected to produce through- out the course (RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We found that in 97% of the courses, students were involved in developing some form of a soft- ware product as an end deliverable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Additionally, students were often required to produce agile development artefacts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' product and sprint backlogs), project plans and soft- ware documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The produced artefacts, therefore, sup- ported the idea of a well-rounded software development ex- perience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In our research, the role of documentation was slightly different from the role it had in the survey done by Dugan Jr (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In their classification, the core written doc- uments involve project proposals, requirements documents, project plans, designs, test plans and user manuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We, on the other hand, found that when courses had shifted towards more agile development approaches, the number of written assignments had reduced, and the documentation was being generated more throughout the course rather than as detailed plans up front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Secondly, we investigated the phases that these projects generally go through (RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We found that projects often proceeded from an idea or a ready-made list of requirements into project delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For a large number of courses, the stu- dents start from scratch and produce a prototype or a soft- ware product that is handed off to the clients or teachers at the end of the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Therefore, the maintenance of exist- ing software products and working with existing codebases is often not experienced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In contrast Dugan Jr (2011) states that regardless of the software process model, the common phases were requirements, design, implementation, testing, presentation and maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' However, they do make the same finding we made that, maintenance was frequently men- tioned in the literature with little supporting detail of its ac- tual implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Maintenance thus still remains an is- sue that is left with little attention in SE capstone litera- ture, despite its high relevance in the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some ed- ucators have solved this by involving large projects, which go through various incremental improvements over the years in their courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Others have students contributing to large Open Source projects, but both of these approaches were still found to present a minority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Dugan Jr (2011) makes no re- marks on Open Source projects being used as a solution to this problem like we did, which would indicate that it is an upcoming solution to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Indeed, the studies in our research covering large Open Source projects were writ- ten after the survey by Dugan Jr (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Finally, we investigated what technologies are used in these courses and how the selections are made (RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We found that studies mostly do not explicitly describe all the technologies used in these courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We also found that for the majority of courses, the technology selections are made based on the project specifications and needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This often en- tails students learning new technologies and having to justify their selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Assessment of students (RQ5) Regarding end-of-course student assessment (RQ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1) we aimed to find out what constitutes the course grade in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' A large number of studies gave inconclusive answers in this regard and did not describe grading rubrics in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Therefore, we were unable to draw any conclusions about what artefacts or assignments formed the final grades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' How- ever, we were able to determine fairly well, who does the final assessment and how well students are given a chance to reflect on their experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In all studies that discussed the student assessment, the teachers were involved in determin- ing the final grades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Any sort of concrete deliverables (pro- duced software, plans, agile artefacts, reports) were gener- ally graded by the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' These artefacts provided teachers with some understanding of how well students understood the phases of software development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' A minority of studies also mention including students in the assessment process, in the form of self- and peer-reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' These both have proven not only to hold the individual students more accountable during project work but also to give valuable insight for the teacher into the soft skill development of an individual stu- dent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Continuous assessment and guidance during the course (RQ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2) were explicitly addressed in most studies (89%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The survey by Trevisan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' (2006) focused on similar sort of assessment practices when they sought to find out, how of- ten engineering capstones implement classroom assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' They found only 32 articles in all engineering disciplines be- tween the years 1994 and 2006 which discussed classroom assessment schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' While their scope is tighter, it seems that the guidance of students during the course would have increased in the past 15 years or so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Trevisan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' (2006) also reflects on this, stating that the importance of classroom assessment has gained traction in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Our research showed that any sort of mentoring or coach- ing was found to be highly beneficial for the students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' It in- creased the success rate of projects and helped teachers to identify problems early on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Course staff were the ones that most often guided students during their projects, and sev- eral courses had hired teaching assistants for such positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some studies also found that having more experienced stu- dents advising the capstone participants was a rewarding ex- perience for both groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Mentoring activities are also com- mon in real-life companies where graduates generally join an existing team with various skill levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Implications for practitioners and researchers A large amount of research found on software engineer- ing capstones shows that capstones are a common way for educators to prepare students for varying aspects of working life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For an educator to find ways to implement a capstone course, it would be too a time-consuming task to go through all the published primary studies and distil the experience and evidence into concrete suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For such situations, we have provided an overview of the most common char- acteristics of capstone courses, and what kinds of choices regarding each of these can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 18 of 22 Capstone courses in software engineering The overriding guideline set by the ACM/IEEE (2014) for undergraduate SE capstone courses is that they should help to ensure that the curriculum has a significant real-world basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Capstones are expected to be the culminating expe- rience that ties everything learned so far together and pre- pares students for the working life in software engineering (ACM/IEEE, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' ACM, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Interestingly, our research revealed that primary studies on capstone courses only rarely included experiences, of course, alumni or industry clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This would be important to see how well these courses reflect what students are expected to do in real life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We would like to see more research done on how well these courses capture what students face later on in their careers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' It would also be worthwhile to see more controlled, comparative studies where one of the presented characteristics is changed and its impact on the course outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Most of the research iden- tified here does not provide controlled, comparative results on the capstone characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Threats to validity 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Deviations from the procedures for systematic reviews Although we aimed to use the guidelines provided in Kitchenham and Charters (2007) to perform our systematic review, we had deviations from their procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In our re- search, the study selection and data extraction were carried out by the first author rather than by a group of researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This means that some relevant papers might have been ex- cluded or that some of the collected data may be erroneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Inaccuracy and bias in selected papers for review One of the main limitations of any review is the possible bias in the study selection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In our case, we included only studies considering software-related capstone courses with a relatively tight scope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' for instance, we did not include any studies with courses on embedded systems or computer engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' However, we were clear in our goals of describ- ing only software capstones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' A similar kind of search has also been conducted in the earlier survey done by Dugan Jr (2011), whose search strings were “capstone“ and “software engineering course“ into a selected set of journals in SEE and CS education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' And to ensure that the selection pro- cess was as unbiased as possible, we described the employed search strategy and the inclusion and exclusion criteria in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This way, we aimed to make the selection process as visible to the reader as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The primary studies only represent the capstone courses with some aspects or outcomes worthy of publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' There- fore the study sample in our research might be skewed to- ward successful, well-planned courses or easy to research courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We believe this is not a problem in describing how versatile the projects can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' However, quantitative aspects of the data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', the portion of the courses having an ex- ternal client) should be addressed with caution as certain kinds of courses may be more likely in real life than in SE education research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Finally, we also acknowledge that there are similar courses organised under other related disciplines, such as data science and computer engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We, how- ever, knowingly chose to leave other disciplines out of the scope of this research as we wanted to provide a classifica- tion and insights specifically on software-related capstones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The ACM/IEEE (2014) recommendations we derived our re- search questions on, were also provided specifically for soft- ware engineering capstones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Inaccuracy and bias in data extraction As with any systematic review, one of the main limita- tions is the potential bias and inaccuracy of the data extrac- tion procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This is also the most likely step with inac- curacy in our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For example, the quality assessment was done by the first author, whose interpretation of quality might differ from that of another researcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The distinction between whether a study has explicitly discussed limitations (“Yes“) or they have only shortly referred to a limitation of the study (“Somewhat“) is something that another researcher might view differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' However, the two summed-up cate- gories presented, rigour and credibility, aimed to diminish the impact of a single quality assessment question and eval- uate the study rather as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' It is also worth noting that the primary studies presented in this review are not exclusively written to provide course descriptions or general course evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Some studies have a section dedicated to the course overview, which might have provided all the details of the course structure we needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Then again, some studies had the relevant details scattered across various sections and might not have been explicitly re- ferred to as our categories suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Especially regarding the produced artefacts and student assessment, the descriptions varied greatly in terms of detail and clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In situations like these, some interpretation was needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' To mitigate this problem, we tried to keep the categories generic and descrip- tive, so that it would be easy to grasp the general outline of each course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We also refrained from reading too much into the text itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For instance, if the study mentioned that the student teams were composed of “at most 5 students“, we left these courses in the category “not specified“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Lack of third party assessment of capstone courses Usually, at least one of the authors of the study was some- how involved in organising the course in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Addi- tionally, quite a large portion of these reports lacked an hon- est evaluation of the author bias, as can be seen in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Therefore there is an inherent lack of truly objective third-party assessment of these SE capstone courses in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' This is something that we were unable to affect but is worth noting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We would welcome more research on capstone courses, or on SE education in general, where the author is an unbiased third party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Evaluation of review For evaluating any SLR, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' present criteria based on four quality assessment (QA) questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We will briefly provide answers to each one of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' QA1 — Are the review’s inclusion and exclusion criteria Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 19 of 22 Capstone courses in software engineering described and appropriate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Yes, we have explicitly defined and described the inclusion and exclusion criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The foun- dation for the criteria stems from our research objectives and aims to ensure that the studies included in the review are of sufficient quality and help to answer our research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' QA2 – Is the literature search likely to have covered all relevant studies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' state that if the authors have searched 4 or more digital libraries and included additional search strate- gies, this criterion is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In this research, we did search 4 digital libraries and included a description of our search strategies, so this criterion is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' QA3 – Did the reviewers assess the quality/validity of the included studies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We did use a question set used by many similar SLRs to assess the quality and validity of the included studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Therefore this criterion is also met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' QA4 – Were basic data/studies adequately described?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We provided bibliographical references to each of the studies used, described from various viewpoints the target of their research (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' the capstone course presented in each of them), described how the data was collected in each of the studies and synthesised the reported outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Therefore it is safe to say, that this criterion was met as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Conclusions This research aimed to understand how software engi- neering capstone courses are organised in tertiary education institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' For this purpose, we conducted a systematic literature review, including 127 primary studies on SE cap- stone courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The characteristics were synthesised into a taxonomy consisting of duration, team sizes, clients and project sources, project implementation and student assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Based on the synthesised justifications and outcomes for these char- acteristics, we provided suggestions on how the courses can be organised and what the trade-offs are to be weighted re- garding each characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The main curriculum guideline that capstones should help to accomplish is “The curriculum should have a significant real-world basis“ (ACM/IEEE, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In our research, we focused on the concrete recommendations given to accom- plish this goal and formulated our research questions based on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' We found out that the courses have a software im- plementation as the main deliverable, the students are as- sessed based on various factors, not just the delivery of a working system, and the projects in these courses are almost always completed as group assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Students were also often given guidance and continuous assessment throughout the course via written and oral feedback on their progress and deliverables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' The area which educators should pay at- tention to is the duration of the course which in practice is one semester, whilst for instance, ACM/IEEE (2014) recom- mends having two-semester courses to reach adequate depth and breadth in skills and experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' A considerable num- ber of courses also did not have a client external to the course staff, despite external clients being recommended for under- graduate and graduate capstones (ACM/IEEE, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' ACM, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In these cases, the project specifications were gen- erated by the course staff or the students themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Such arrangements tend to leave students without the experience of having to solicit, negotiate and implement requirements set by a real client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' In addition, the projects usually progress from idea to product, and often do not include maintenance, especially that of pre-existing projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' These characteris- tics somewhat diminish the real-world compatibility of the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' References ACM, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Graduate software engineering 2009(gswe2009) curriculum guidelines for graduate degree programs in software engineering.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2014 Best practices for industry-sponsored CS cap- stone courses Journal of Computing Sciences in Colleges S32 Matthies, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Teusner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Hesse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2019 Beyond Surveys: Analyzing Software Develop- ment Artifacts to Assess Teaching Efforts IEEE Frontiers in Education Conference, FIE 2018 S33 Ziv, H.' metadata={'source': 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+page_content=' Black, Leonard 2009 Capstone projects as community connectors Journal of Computing Sciences in Colleges S35 Paasivaara, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Vanhanen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Lassenius, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2019 Collaborating with industrial customers in a cap- stone project course: The customers’ perspec- tive IEEE/ACM 41st International Conference on Software Engineering: Software En- gineering Education and Training, ICSE- SEET 2019 S36 Adams, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Kleiner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2016 Collaboration support in an international com- puter science capstone course International Conference on Social Com- puting and Social Media S37 Watkins, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Barnes, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2010 Competitive and agile software engineering ed- ucation IEEE SoutheastCon, SoutheastCon 2010 S38 Gustavsson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Brohede, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2019 Continuous assessment in software engineering project course using publicly available data from GitHub 15th International Symposium on Open Collaboration, OpenSym 2019 S39 Hadfield, Steven M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Jensen, Nathan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2007 Crafting a software engineering capstone project course Journal of Computing Sciences in Colleges S40 Rong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Shao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2012 Delivering software process-specific project courses in tertiary education environment: Chal- lenges and solution 25th IEEE Conference on Software Engi- neering Education and Training, CSEE&T 2012 S41 Nguyen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Truong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Le, N.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Lee, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Alejan- dre, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2009 Emphasizing soft skills and team development in an educational digital game design course 4th International Conference on the Foun- dations of Digital Games, FDG 2009 S47 Takala, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Malmi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Pugliese, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Takala, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2016 Empowering students to create better virtual re- ality applications: A longitudinal study of a VR capstone course Informatics in Education S48 Marques, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Ochoa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Bastarrica, M.' metadata={'source': 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+page_content=', Velasco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Aponte, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Marcus, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2017 Evolving a Project-Based Software Engineering Course: A Case Study 30th IEEE Conference on Software Engi- neering Education and Training, CSEE&T 2017 Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 23 of 22 Capstone courses in software engineering Table 13 Continued from previous page ID Author(s) Year Title Source title S55 Ras, Eric and Carbon, Ralf and Decker, Björn and Rech, Jörg 2007 Experience Management Wikis for Reflective Practice in Software Capstone Projects IEEE Transactions on Education S56 Schorr, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2020 Experience Report on Key Success Factors for Promoting Students’ Engagement in Software Development Group Projects 4th IEEE World Conference on Engineering Education, EDUNINE 2020 S57 Longstreet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Shaun;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' Cooper, Kendra 2013 Experience report: A sustainable serious educa- tional game capstone project CGAMES’2013 USA S58 Dupuis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Pieterse, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Thompson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Venter, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2016 Exploration of Participation in Student Software Engineering Teams ACM Transactions on Computing Educa- tion, TOCE S61 Ganci, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Ramnath, R.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Kärkkäinen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2008 The value of a real customer in a capstone project 21st Conference on Software Engineering Education and Training, CSEE&T 2008 S122 Mohan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Chenoweth, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Bohner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2012 Towards a better capstone experience 43rd ACM Technical Symposium on Com- puter Science Education, SIGCSE’12 S123 Rico, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Sayani, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2009 Use of agile methods in software engineering ed- ucation Agile Conference, AGILE 2009 S124 Tribelhorn, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Nuxoll, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2021 Using Agile and Active Learning in Software De- velopment Curriculum ASEE Virtual Annual Conference and Ex- position S125 McDonald, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Wolfe, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2008 Using computer graphics to foster interdisci- plinary collaboration in capstone courses Journal of Computing Sciences in Colleges S126 Ju, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Hemani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Dimitri- adis, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Fox, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2020 What agile processes should we use in software engineering course projects?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 51st ACM Technical Symposium on Com- puter Science Education, SIGCSE 2020 S127 Bastarrica, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Perovich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=', Samary, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 2017 What can students get from a software engi- neering capstone course?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' 39th IEEE/ACM International Conference on Software Engineering: Software Engi- neering and Education Track, ICSE-SEET 2017 Saara Tenhunen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 26 of 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E1T4oBgHgl3EQf9QZb/content/2301.03554v1.pdf'} diff --git a/B9E1T4oBgHgl3EQfpgVg/content/tmp_files/2301.03332v1.pdf.txt b/B9E1T4oBgHgl3EQfpgVg/content/tmp_files/2301.03332v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ce63f7f364301ff3bd4724826473a56346d3d5b --- /dev/null +++ b/B9E1T4oBgHgl3EQfpgVg/content/tmp_files/2301.03332v1.pdf.txt @@ -0,0 +1,2144 @@ +arXiv:2301.03332v1 [math.AP] 9 Jan 2023 +THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN +INEQUALITY ON THE H-TYPE GROUP +QIAOHUA YANG +Abstract. We determine the optimal constant in the L2 Folland-Stein in- +equality on the H-type group, which partially confirms the conjecture given +by Garofalo and Vassilev (Duke Math. J., 2001). The proof is inspired by the +work of Frank and Lieb (Ann. of Math., 2012) and Hang and Wang. +1. Introduction +Let G be a stratified, simply connected nilpotent Lie group (in short a Carnot +group) of step r. Denote by g the Lie algebra of G. It is known that g = �r +i=1 Vi +satisfying (see e.g. [10]) +[V1, Vj] = Vj+1, 1 ≤ j ≤ r − 1; [V1, Vr] = {0}. +As a simply connected nilpotent group, G is differential with RN, N = �r +i=1 dim Vi, +via the exponential map exp : g → G. There is a natural family of nonisotropic +dilations δλ : g → g for λ > 0 and we define it as follows: +δλ(X1 + · · · + Xr) = λX1 + · · · + λrXr, Xj ∈ Vj, 1 ≤ j ≤ r. +The homogeneous dimension of G, associated with δλ, is Q = �r +j=1 j dim Vj. Via +the exponential map exp : g → G, we define the group of dilations on G as follows: +δλ(g) = exp ◦δλ ◦ exp−1(g), g ∈ G. +Set nj = dim Vj, 1 ≤ j ≤ r. Let {X1, · · · , Xn1} be a basis of V1 and denote +by ∇G = (X1, · · · , Xn1) the horizontal gradient of G. The sub-Laplacian on G is +∆G = �n1 +i=1 X2 +i . The Sobolev space W 1,p +0 +(G) is the closure of C∞ +0 (G) with respect +to the norm +∥u∥W 1,p +0 +(G) = +�� +G +|∇Gu|pdg +� 1 +2 +, +where dg is the Haar measure on G. We remark that the Haar measure on G, +induced by the exponential mapping from the Lebesgue measure on g = RN, coin- +cides the Lebesgue measure on RN. The Folland-Stein inequality on G reads that +there exits some constant C > 0 such that for each u ∈ W 1,p +0 +(G) (see [8, 9]), +�� +G +|u| +pQ +Q−p dg +� Q−p +pQ +≤ C +�� +G +|∇Gu|pdg +� 1 +p +, 1 < p < Q. +(1.1) +2000 Mathematics Subject Classification. Primary: 43A80; 46E35; 22E25. +Key words and phrases. Folland-Stein inequality; Heisenberg group; H-type group; best +constant. +The +work +was +partially +supported +by +the +National +Natural +Science +Foundation +of +China(No.11201346). +1 + +2 +QIAOHUA YANG +For the existence and regularity of minimizers of the Folland-Stein inequality (1.1), +we refer to [24]. +The Heisenberg group is the simplest example of Carnot group of step 2. We +denote it by Hn = (Cn × R, ◦). The group law on Hn is given by +(z, t) ◦ (z′, t′) = (z + z′, t + t′ + 2Imz · z′), +where z · z′ = �n +j=1 zj¯z′ +j. The homogeneous norm on Hn is given by +|(z, t)| = (|z|4 + t2) +1 +4 . +In a series of papers [18, 19, 20], Jerison and Lee, among other results, determined +the explicit computation of the extremal functions in (1.1) in the case p = 2 and +G = Hn. In fact, the extremal functions are, up to group translations and dilations, +c((1 + |z|2)2 + t2)− Q−2 +4 , c ∈ R. +Such inequalities play an important role in the study of CR Yamabe problems. +Later, in a celebrated paper [11], Frank and Lieb established sharp Hardy-Littlewood- +Sobolev inequalities on Hn. We state the result as follows: +Theorem 1.1 (Frank-Lieb). Let 0 < λ < Q and p = +2Q +Q−λ. Then for any f, g ∈ +Lp(Hn), +����� +� � +Hn×Hn +f(z, t)g(z′, t′) +|(z, t)−1 ◦ (z′, t′)|λ dzdtdz′dt′ +����� ≤ +� πn+1 +2n−1n! +�λ/Q n!Γ( Q−λ +2 ) +Γ2( Q−2λ +4 +) +∥f∥p∥g∥p, +with equality if and only if, up to group translations and dilations, +f = c((1 + |z|2)2 + t2)− 2Q−λ +4 +, g = c′((1 + |z|2)2 + t2)− 2Q−λ +4 +for some c, c′ ∈ C. +In particular, choosing λ = Q−2 in Theorem 1.1 yields the Jerison and Lee’s in- +equality. Using the method in [11], Frank and Lieb [12] also gave a new, rearrangement- +free proof of sharp Hardy-Littlewood-Sobolev inequalities on Rn. Recently, Hang +and Wang [15] present a shorter proof of the Frank-Lieb inequality, in which they +bypasses the subtle proof of existence and the Hersch-type argument via subcritical +approximation. +Some of the results of Theorem 1.1 have been generalized to the cases of quater- +nionic Heisenberg group and octonionic Heisenberg group (see [4, 5, 16, 17]). We +note that Heisenberg group, quaternionic Heisenberg group and octonionic Heisen- +berg group are known as the groups of Iwasawa type, i.e., the nilpotent component +in the Iwasawa decomposition of simple groups of rank one (see e.g. [6]). +The aim of this paper is to look for the optimal constant of (1.1) when p = 2 and +G is a group of Heisenberg type (in short a H-type group). Recall that a H-type +group G is a Carnot group of step two with the following properties (see Kaplan +[21]): the Lie algebra g of G is endowed with an inner product ⟨, ⟩ such that, if z +is the center of g, then [z⊥, z⊥] = z and moreover, for every fixed z ∈ z, the map +Jz : z⊥ → z⊥ defined by +⟨Jz(v), ω⟩ = ⟨z, [v, ω]⟩, ∀ω ∈ z⊥ +(1.2) +is an orthogonal map whenever ⟨z, z⟩ = 1. It is known (see [6]) that a H-type group +G is the group of Iwasawa type if and only if its Lie algebra satisfies the following + +THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY +3 +J2-condition: for any v ∈ z⊥ and z, z′ ∈ z such that ⟨z, z′⟩ = 0, there exists z′′ ∈ z +such that +JzJz′v = Jz′′v. +Therefore, most of H-type groups are not groups of Iwasawa type. +Set m = dim z⊥ and n = dim z. Since G has step two, we can fix on G a system +of coordinates (x, t) such that the group law on G has the form (see [2]) +(x, t) ◦ (x′, t′) = +� +xi + x′ +i, i = 1, 2, · · · , m +tj + t′ +j + 1 +2⟨x, U (j)x′⟩, j = 1, 2, · · · , n +� +(1.3) +for suitable skew-symmetric matrices U (j)’s. Nextly, we set +U(ξ) = +�� +1 + |x|2 +4 +�2 ++ |t|2 +�− Q−2 +4 +, ξ = (x, t) ∈ G; +(1.4) +Uλ,η(ξ) =λ +Q−2 +2 U(δλ(η−1 ◦ ξ)), +η ∈ G. +(1.5) +It has been shown that [m(Q − 2)] +Q−2 +4 Uλ,η(ξ) satisfies the Yamabe-type equation +(see [13, 14]) +∆G[m(Q − 2)] +Q−2 +4 Uλ,η + {[m(Q − 2)] +Q−2 +4 Uλ,η} +Q+2 +Q−2 = 0, +or equivalently, +∆GUλ,η + m(Q − 2)U +Q+2 +Q−2 +λ,η += 0. +(1.6) +In the paper [13], Garofalo and Vassilev gave the following conjecture: +Conjecture (Garofalo-Vassilev). In a H-type group G, the functions [m(Q − +2)] +Q−2 +4 Uλ,η(ξ) are the only nontrivial entire solutions to +� +∆Gu + u +Q+2 +Q−2 = 0, +u ∈ W 1,2 +0 +(G), u ≥ 0. +If the conjecture is true, then one can obtain the optimal constant of L2 Folland- +Stein inequality on H-type groups. In this paper we shall use the method given by +Frank and Lieb [11, 12] and Hang and Wang [15] to determine the optimal constant, +instead of proving the conjecture directly. To this end, we have +Theorem 1.2. It holds that +� +G +|∇Gu|2dxdt ≥ Sm,n +�� +G +|u| +2Q +Q−2 dxdt +� Q−2 +Q +, u ∈ W 1,2 +0 +(G), +(1.7) +where +Sm,n = 4− 2n +Q m(Q − 2)π +m+n +Q +� Γ( m+n +2 +) +Γ(m + n) +�1/Q +. +The inequality is sharp and an extremal function is +U(x, t) = +�� +1 + |x|2 +4 +�2 ++ |t|2 +�− Q−2 +4 +. + +4 +QIAOHUA YANG +By Theorem 1.2, it is easy to see that the functions cUλ,η(ξ)(c ∈ R) are also +extremal functions of inequality (1.7). +As an application of Theorem 1.2, we study the eigenvalues of +−∆Gv = µU +4 +Q−2 +λ,η v, v ∈ W 1,2 +0 +(G). +(1.8) +We note that the eigenvalues of (1.8) play an important role in the study of stability +for the Folland-Stein inequality (see [1, 3, 7] for the case of Euclidean space). In +Lemma 3.2 we show that the embedding map W 1,2 +0 +(G) ֒→ L2(G, U(x, t) +4 +Q−2 dxdt) is +compact. So the spectrum of (1.8) is discrete. Furthermore, we have the following +theorem: +Theorem 1.3. Let µi, i = 1, 2, · · · be the eigenvalues of (1.8) given in increasing +order. Then +(1) µ1 = m(Q − 2) is simple with eigenfunction Uλ,η. +(2) µ2 = m(Q + 2) and {∂λUλ,η, ∇ηUλ,η} are eigenfunctions. +Furthermore, the eigenvalues do not depend on λ and η. +Remark 1.4. It seems that µ2 has multiplicity m + n + 1 with corresponding +eigenspace spanned by {∂λUλ,η, ∇ηUλ,η}. +However, we fail to prove it. +Once +it has been proven, it would provide a generalization of the results of Bianchi and +Egnell ([1], Lemma A1) to the setting of H-type groups. +2. preliminaries on H-type groups +In the rest of paper, we let G be a H-type group with group law given by (1.3). +The nonisotropic dilations δλ on G is +δλ(x, t) = (λx, λ2t). +For (x, t) ∈ G, the homogeneous norm of (x, t) is +ρ(x, t) = +�|x|4 +16 + |t|2 +� 1 +4 +. +With this norm ρ, we can define the ball centered at origin with radius R +BR(0) = {(x, t) ∈ G : ρ(x, t) < R} +and the unit sphere Σ = ∂B1(0) = {(x, t) ∈ G : ρ(x, t) = 1}. +Given any (x, t) ∈ G with ρ(x, t) ̸= 0, we set x∗ = +x +ρ(x,t) and t∗ = +t +ρ(x,t)2 . The +polar coordinates on G associated with ρ are the following (see [10]): +� +G +f(x, t)dxdt = +� ∞ +0 +� +Σ +f(ρx∗, ρ2t∗)ρQ−1dσdρ, f ∈ L1(G). +The following theorem was proved in [2], Theorem A.2.: +Theorem 2.1. G is a H-type group if and only if G is (isomorphic to) Rm+n +with the group law in (1.3) and the matrices U (1), U (2), · · · , U (n) have the following +properties: +(1) U (j) is a m × m skew symmetric and orthogonal matrix, for every j = +1, 2, · · · , n; +(2) U (i)U (j) + U (j)U (i) = 0 for every i, j ∈ {1, 2, · · · , n} with i ̸= j. + +THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY +5 +The vector field in the Lie algebra g that agrees at the origin with +∂ +∂xj (j = +1, · · · , m) is given by +Xj = +∂ +∂xj ++ 1 +2 +n +� +k=1 +� m +� +i=1 +U (k) +i,j xi +� +∂ +∂tk +and g is spanned by the left-invariant vector fields X1, · · · , Xm, T1 = +∂ +∂t1 , · · · , Tn = +∂ +∂tn . Furthermore (see [2], Page 200, (A.4) ), +[Xi, Xj] = +n +� +r=1 +U (r) +i,j Tr, i, j ∈ {1, 2, · · · , n}. +(2.1) +The exponential map exp : g → G is +exp : g → Rm+n, +m +� +i=1 +xiXi + +n +� +j=1 +tjTj �→ (x, t). +We note that by exponential mapping, the group law (1.3) is nothing but the +Baker-Campbell-Hausdorff formula (see [2], the proof of Theorem A.2) +exp X ◦ exp Y = exp(X + Y + 1 +2[X, Y ]), X, Y ∈ g. +Using (2.1), we have that for t = (t1, · · · , tn) = t1T1 + · · · + tnTn and x = +(x1, · · · , xm) = x1X1 + · · · + xmXm, the map Jt, defined by (1.2), is (see also +[2], Page 201) +Jtx = +n +� +r=1 +m +� +i=1 +trxiJTr(Xi) = +n +� +r=1 +m +� +i=1 +trxi + + +m +� +j=1 +U (r) +i,j Xj + + += +m +� +j=1 +� n +� +r=1 +m +� +i=1 +trxiU (r) +i,j +� +Xj. +Since Jt is an orthogonal map whenever |t| = 1, we obtain +|Jtx|2 = |t|2|x|2 = +m +� +j=1 +� n +� +r=1 +m +� +i=1 +trxiU (r) +i,j +�2 +. +(2.2) +The horizontal gradient on G is ∇G = (X1, · · · , Xm). The sub-Laplacian on G +is given by (see [2], Remark A.6.) +∆G = +m +� +j=1 +X2 +j = +m +� +j=1 +� +∂ +∂xj ++ 1 +2 +n +� +k=1 +� m +� +i=1 +U (k) +i,j xi +� +∂ +∂tk +�2 += ∆x + 1 +4|x|2∆t + +n +� +k=1 +⟨x, U (k)∇x⟩ ∂ +∂tk +, +where +∆x = +m +� +j=1 +� ∂ +∂xj +�2 +, ∆t = +n +� +k=1 +� ∂ +∂tk +�2 +. +We remark that ∆G is homogeneous of degree two with respect to δλ. +By using (1.6), we have the following Hardy inequality (see [22], Corollary 1.4 +for Hardy inequality of fractional powers of the sublaplacian on G). + +6 +QIAOHUA YANG +Lemma 2.2. It holds that, for u ∈ W 1,2 +0 +(G), +� +G +|∇Gu|2dxdt ≥ m(Q − 2) +� +G +u2 +(1 + |x|2 +4 )2 + |t|2 dxdt, +with equality if and only if u = cU(x, t), where c ∈ R and U(x, t) is given by (1.4). +Proof. We have, through integration by parts, +0 ≤ +� +G +U 2|∇G(U(x, t)−1u)|2dxdt += +� +G +���∇Gu − u +U ∇GU +��� +2 +dxdt += +� +G +|∇Gu|2dxdt + +� +G +|∇GU|2 +U 2 +u2dxdt − +� +G +1 +U ⟨∇Gu2, ∇GU⟩dxdt += +� +G +|∇Gu|2dxdt + +� +G +u2 1 +U ∆GUdxdt += +� +G +|∇Gu|2dxdt − m(Q − 2) +� +G +u2 +(1 + |x|2 +4 )2 + |t|2 dxdt. +(2.3) +To get the last equality, we use (1.6). The desired result follows. +□ +Set η = (y1, · · · , ym, w1, · · · , wn) ∈ G. By (1.6), we have +∆G +∂Uλ,η +∂yj ++ m(Q + 2)U +4 +Q−2 +λ,η +∂Uλ,η +∂yj += 0, +j = 1, , · · · , m, +∆G +∂Uλ,η +∂wr ++ m(Q + 2)U +4 +Q−2 +λ,η +∂Uλ,η +∂wr += 0, +r = 1, , · · · , n, +∆G +∂Uλ,η +∂λ ++ m(Q + 2)U +4 +Q−2 +λ,η +∂Uλ,η +∂λ += 0. +(2.4) +Furthermore, we have the following lemma: +Lemma 2.3. It holds that +m +� +j=1 +���� +∂Uλ,η +∂yj +|λ=1,η=0 +���� +2 ++ +n +� +r=1 +���� +∂Uλ,η +∂wr +|λ=1,η=0 +���� +2 ++ 1 +4 +���� +∂Uλ,η +∂λ |λ=1,η=0 +���� +2 += (Q − 2)2 +16 +U(ξ)2. +Proof. It is easy to see η−1 = −η. Therefore, by (1.3) and (1.5), we have +Uλ,η(x, t) =λ +Q−2 +2 + + +� +1 + λ2 +4 +m +� +i=1 +(xi − yi)2 +�2 ++ λ4 +n +� +r=1 +� +tr − wr − ⟨y, U (r)x⟩ +2 +�2 + +− Q−2 +4 +. + +THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY +7 +We compute +∂Uλ,η +∂yj +|λ=1,η=0 = − Q − 2 +4 +U(ξ) +Q+2 +Q−2 +� +2 +� +1 + |x|2 +4 +� +· +� +−xj +2 +� ++ +n +� +r=1 +tr +� +− +m +� +i=1 +U (r) +j,i xi +�� +=Q − 2 +4 +U(ξ) +Q+2 +Q−2 +�� +1 + |x|2 +4 +� +xj + +n +� +r=1 +m +� +i=1 +trxiU (r) +j,i +� +, j = 1, · · · , m; +∂Uλ,η +∂wr +|λ=1,η=0 = − Q − 2 +4 +U(ξ) +Q+2 +Q−2 (−2tr) +=Q − 2 +2 +U(ξ) +Q+2 +Q−2 tr, +r = 1, · · · , n; +∂Uλ,η +∂λ |λ=1,η=0 =Q − 2 +2 +U(ξ) − Q − 2 +4 +U(ξ) +Q+2 +Q−2 +� +2 +� +1 + |x|2 +4 +� +· |x|2 +2 ++ 4|t|2 +� += − Q − 2 +2 +U(ξ) +Q+2 +Q−2 +� +−1 + |x|4 +16 + |t|2 +� +. +Since each U (j)(1 ≤ j ≤ n) is a m × m skew symmetric matrix, we have, by using +(2.2), +m +� +j=1 +���� +∂Uλ,η +∂yj +|λ=1,η=0 +���� +2 +=(Q − 2)2 +16 +U(ξ)2 Q+2 +Q−2 +m +� +j=1 +�� +1 + |x|2 +4 +� +xj − +n +� +r=1 +m +� +i=1 +trxiU (r) +i,j +�2 +=(Q − 2)2 +16 +U(ξ)2 Q+2 +Q−2 +�� +1 + |x|2 +4 +�2 +|x|2 + |t|2|x|2− +2 +� +1 + |x|2 +4 +� +n +� +r=1 +tr + + +m +� +i=1 +m +� +j=1 +U (r) +i,j xixj + + + + +=(Q − 2)2 +16 +U(ξ)2 Q+2 +Q−2 +�� +1 + |x|2 +4 +�2 +|x|2 + |t|2|x|2 +� +. +To get the last equality, we use the fact +m +� +i=1 +m +� +j=1 +U (r) +i,j xixj = 0 + +8 +QIAOHUA YANG +since U (r)(1 ≤ r ≤ n) is a m × m skew symmetric matrix. Therefore, we have +m +� +j=1 +���� +∂Uλ,η +∂yj +|λ=1,η=0 +���� +2 ++ +n +� +r=1 +���� +∂Uλ,η +∂wr +|λ=1,η=0 +���� +2 ++ 1 +4 +���� +∂Uλ,η +∂λ |λ=1,η=0 +���� +2 +=(Q − 2)2 +16 +U(ξ)2 Q+2 +Q−2 +�� +1 + |x|2 +4 +�2 +|x|2 + |t|2|x|2 +� ++ (Q − 2)2 +4 +U(ξ)2 Q+2 +Q−2 |t|2+ +(Q − 2)2 +16 +U(ξ)2 Q+2 +Q−2 +� +−1 + |x|4 +16 + |t|2 +�2 +=(Q − 2)2 +16 +U(ξ)2 Q+2 +Q−2 +�� +1 + |x|2 +4 +�2 +|x|2 + |t|2|x|2 + 4|t|2 + +� +−1 + |x|4 +16 + |t|2 +�2� +=(Q − 2)2 +16 +U(ξ)2 Q+2 +Q−2 +�� +1 + |x|2 +4 +�2 ++ |t|2 +�2 +=(Q − 2)2 +16 +U(ξ)2. +To get the third equality, we use the fact +� +−1 + |x|4 +16 + |t|2 +�2 += +�� +1 + |x|2 +4 +�2 ++ |t|2 − 2 +� +1 + |x|2 +4 +��2 += +�� +1 + |x|2 +4 +�2 ++ |t|2 +�2 ++ 4 +� +1 + |x|2 +4 +�2 +− 4 +� +1 + |x|2 +4 +� �� +1 + |x|2 +4 +�2 ++ |t|2 +� += +�� +1 + |x|2 +4 +�2 ++ |t|2 +�2 +− +� +1 + |x|2 +4 +�2 +|x|2 − |t|2|x|2 − 4|t|2. +This completes the proof of Lemma 2.3. +□ +For simplicity, we set +ωj = +4 +Q − 2U(ξ)−1 ∂Uλ,η +∂yj +|λ=1,η=0, j = 1, · · · , m; +ωj+r = +4 +Q − 2U(ξ)−1 ∂Uλ,η +∂wr +|λ=1,η=0, r = 1, · · · , n; +ωm+n+1 = +2 +Q − 2U(ξ)−1 ∂Uλ,η +∂λ |λ=1,η=0. +(2.5) +By Lemma 2.3 and (2.4), we have +m+n+1 +� +j=1 +ω2 +j =1; +(2.6) +∆G(U(ξ)ωj) + m(Q + 2)U(ξ) +Q+2 +Q−2 ωj =0, 1 ≤ j ≤ m + n + 1. +(2.7) + +THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY +9 +3. Proof of Theorem 1.2 and 1.3 +In this section, we shall prove Theorem 1.2 and 1.3. The proof depends on a +scheme of subcritical approximation due to Hang and Wang [15]. We first establish +the following subcritical Sobolev inequality on G. +Lemma 3.1. Let 2 ≤ p < +2Q +Q−2. There exists C > 0 such that for each u ∈ W 1,2 +0 +(G), +� +G +|∇Gu|2dxdt ≥ C +�� +G +|u|pU(x, t) +2Q +Q−2 −pdxdt +� 2 +p +. +Proof. By H¨older’s inequality, we have +� +G +|u|pU(x, t) +2Q +Q−2 −pdxdt += +� +G +� +|u|U(x, t) +2 +Q−2 +�Q− Q−2 +2 +p +|u| +Q +2 (p−2)dxdt +≤ +�� +G +|u|2U(x, t) +4 +Q−2 dxdt +� 2Q−(Q−2)p +4 +�� +G +|u| +2Q +Q−2 dxdt +� (Q−2)(p−2) +4 += +�� +G +u2 +(1 + |x|2 +4 )2 + |t|2 dxdt +� 2Q−(Q−2)p +4 +�� +G +|u| +2Q +Q−2 dxdt +� (Q−2)(p−2) +4 +≤C +� +G +|∇Gu|2dxdt, +where C is a positive constant independent of u. To get the last inequality above, +we use Folland-Stein inequality (1.1) and Lemma 2.2. This completes the proof of +Lemma 3.1. +□ +By Lemma 3.1, we have +W 1,2 +0 +(G) ֒→ Lp(G, U(x, t) +2Q +Q−2 −pdxdt). +Furthermore, the embedding map is compact. +Lemma 3.2. Let 2 ≤ p < +2Q +Q−2. The embedding map W 1,2 +0 +(G) ֒→ Lp(G, U(x, t) +2Q +Q−2 −pdxdt) +is compact. +Proof. The proof is similar to that given by Schneider (see [23], section 2.2). +Let φ : G → [0, 1] be a cut-off function that is equal to one in B1(0) and zero +outside of B2(0). Consider the operator +IR : W 1,2 +0 +(G) ֒→ Lp(G, U(x, t) +2Q +Q−2 −pdxdt) + +10 +QIAOHUA YANG +defined by IR(u) = u(x, t)φ +� x +R, +t +R2 +� +. Since the imbedding map W 1,2 +0 +(B2(0)) ֒→ +Lp(B2(0)) is compact, so is IR. Moreover, by H¨older’s inequality, +� +G +|u − IR(u)|pU(x, t) +2Q +Q−2 −pdxdt +≤ +� +G\BR(0) +|u|pU(x, t) +2Q +Q−2 −pdxdt +≤ +�� +G\BR(0) +|u| +2Q +Q−2 dxdt +� Q−2 +2Q p �� +G\BR(0) +U(x, t) +2Q +Q−2 dxdt +�1− Q−2 +2Q p +≤C +�� +G +|∇Gu|2dxdt +� p +2 �� +G\BR(0) +U(x, t) +2Q +Q−2 dxdt +�1− Q−2 +2Q p +. +To get the last inequality above, we use the Folland-Stein inequality (1.1). By polar +coordinates, we have +� +G\BR(0) +U(x, t) +2Q +Q−2 dxdt = +� +G\BR(0) +�� +1 + |x|2 +4 +�2 ++ |t|2 +�− Q +2 +dxdt +≤ +� +G\BR(0) +1 +( |x|2 +4 + |t|2) +Q +2 dxdt += +� ∞ +R +� +Σ +1 +ρ2Q ρQ−1dρdσ +=|Σ| +1 +QRQ → 0, R → ∞, +where |Σ| is the volume of Σ. Therefore, the embedding map +W 1,2 +0 +(G) ֒→ Lp(G, U(x, t) +2Q +Q−2 −pdxdt) +is a limit of compact operators and thus it is compact. The proof of Lemma 3.2 is +thereby completed. +□ +By Lemma 3.2, the minimization problem +Λp = inf +�� +G +|∇Gu|2dxdt : +� +G +|u|pU(x, t) +2Q +Q−2 −pdxdt = 1 +� +, 2 ≤ p < +2Q +Q − 2, +(3.1) +has a positive solution u. +We shall show that such u satisfies a moment zero +condition. The main result is the following lemma: +Lemma 3.3. Let 2 ≤ p < +2Q +Q−2 and u be a positive solution of (3.1). Then we have +� +G +upU(x, t) +2Q +Q−2 −pωidxdt = 0, i = 1, 2, · · · , m + n + 1, +(3.2) +where ωi(1 ≤ i ≤ m + n + 1) is given by (2.5). +Proof. For simplicity, we set +Fp(u) = +� +G |∇Gu|2dxdt +�� +G |u|pU(x, t) +2Q +Q−2 −pdxdt +� 2 +p + +THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY +11 +and +uλ−1,η−1(ξ) =λ− Q−2 +2 u(δλ−1(η ◦ ξ)), λ > 0, η = (y1, · · · , ym, w1, · · · , wn) ∈ G. +A simple calculation shows +� +G +|∇Guλ−1,η−1|2dxdt = +� +G +|∇Gu|2dxdt; +� +G +up +λ−1,η−1U(x, t) +2Q +Q−2 −pdxdt = +� +G +upUλ,η(x, t) +2Q +Q−2 −pdxdt, +where Uλ,η is given by (1.5). Therefore, +Fp(uλ−1,η−1(ξ)) = +� +G |∇Gu|2dxdt +�� +G |u|pUλ,η(x, t) +2Q +Q−2 −pdxdt +� 2 +p . +(3.3) +Since u is a positive solution of (3.1), we have +∂ +∂yj +Fp(uλ−1,η−1(ξ))|λ=1,η=0 =0, j = 1, · · · , m; +∂ +∂wr +Fp(uλ−1,η−1(ξ))|λ=1,η=0 =0, r = 1, · · · , n; +∂ +∂λFp(uλ−1,η−1(ξ))|λ=1,η=0 =0. +(3.4) +Combining (3.3) and (3.4) yields (3.2). This completes the proof of Lemma 3.3. +□ +Remark 3.4. We remark that Lemma 3.3 is also valid for u > 0 satisfying the +Yamabe-type equation +∆Gu + ΛpupU(x, t) +2Q +Q−2 −p = 0. +The proof is same and we omit it (see [15], Corollary 1 for the case of CR sphere). +Lemma 3.5. It holds that, for any u ∈ W 1,2 +0 +(G), +m+n+1 +� +i=1 +� +G +|∇G(uωi)|2dxdt = +� +G +|∇Gu|2dxdt + 4m +� +G +u2 +(1 + |x|2 +4 )2 + |t|2 dxdt. + +12 +QIAOHUA YANG +Proof. Let u = U(ξ)v. We compute, through integration by parts, +m+n+1 +� +i=1 +� +G +|∇G(uωi)|2dxdt = +m+n+1 +� +i=1 +� +G +|∇G(vUωi)|2dxdt += +m+n+1 +� +i=1 +� +G +|Uωi∇Gv + v∇G(Uωi)|2dxdt += +m+n+1 +� +i=1 +�� +G +|∇Gv|2U 2ω2 +i dxdt + +� +G +|∇G(Uωi)|2v2dxdt+ +1 +2 +� +G +⟨∇G(Uωi)2, ∇Gv2⟩dxdt +� += +m+n+1 +� +i=1 +�� +G +|∇Gv|2U 2ω2 +i dxdt − +� +G +v2Uωi∆G(Uωi)dxdt +� += +� +G +|∇Gv|2U 2dxdt + m(Q + 2) +� +G +v2U +2Q +Q−2 dxdt. +(3.5) +To get the last equality, we use (2.7). On the other hand, by (2.3), we have +� +G +|∇Gv|2U 2dxdt = +� +G +���∇G +u +U +��� +2 +U 2dxdt += +� +G +|∇Gu|2dxdt − m(Q − 2) +� +G +u2 +(1 + |x|2 +4 )2 + |t|2 dxdt. +(3.6) +Substituting (3.6) into (3.5), we obtain +m+n+1 +� +i=1 +� +G +|∇G(uωi)|2dxdt = +� +G +|∇Gu|2dxdt + 4m +� +G +u2 +(1 + |x|2 +4 )2 + |t|2 dxdt. +This completes the proof of Lemma 3.5. +□ +Now we can give the proof of Theorem 1.2. The idea is due to Frank and Lieb +[11, 12] and Hang and Wang [15]. +Proof of Theorem 1.2. Let 2 ≤ p < +2Q +Q−2 and up be a positive solution of +(3.1). The 2nd variation of the functional Fp around up shows that +� +G +|∇Gf|2dxdt +� +G +up +pU(x, t) +2Q +Q−2 −pdxdt− +(p − 1) +� +G +|∇Gup|2dxdt +� +G +up−2 +p +Uλ,η(x, t) +2Q +Q−2 −pf 2dxdt ≥ 0 +for any f with +� +G +up +pUλ,η(x, t) +2Q +Q−2 −pfdxdt = 0. + +THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY +13 +By Lemma 3.3, we may choose f = upωi, i = 1, 2, · · · , m + n + 1. Summing the +corresponding inequalities for all such f’s yields, in view of (2.6) and Lemma 3.5, +0 ≤ +m+n+1 +� +i=1 +� +G +|∇G(upωi)|2dxdt − (p − 1) +� +G +|∇Gup|2dxdt +=4m +� +G +u2 +p +(1 + |x|2 +4 )2 + |t|2 dxdt − (p − 2) +� +G +|∇Gup|2dxdt, +i.e. +(p − 2) +�� +G +|∇Gup|2dxdt − m(Q − 2) +� +G +u2 +p +(1 + |x|2 +4 )2 + |t|2 dxdt +� +≤m(Q − 2) +� 2Q +Q − 2 − p +� � +G +u2 +p +(1 + |x|2 +4 )2 + |t|2 dxdt +≤m(Q − 2) +� 2Q +Q − 2 − p +� �� +G +up +pU(x, t) +2Q +Q−2 −pdxdt +� 2 +p �� +G +U(x, t) +2Q +Q−2 dxdt +�1− 2 +p +=m(Q − 2) +� 2Q +Q − 2 − p +� �� +G +U(x, t) +2Q +Q−2 dxdt +�1− 2 +p +→ 0, p ր +2Q +Q − 2. +To get the last equality, we use the fact +� +G up +pU(x, t) +2Q +Q−2 −pdxdt = 1. Therefore, by +Lemma 2.2, we obtain +� +G +|∇Gup|2dxdt − m(Q − 2) +� +G +u2 +p +(1 + |x|2 +4 )2 + |t|2 dxdt → 0, p ր +2Q +Q − 2, +or equivalently, +� +G +|∇G(U −1up)|2U 2dxdt → 0, p ր +2Q +Q − 2. +So we can choose a sequence {pk : k = 1, 2, · · ·} such that pk ր +2Q +Q−2 and upk +converges to a nonzero function c0U (for reader’s convenience, we prove it in Lemma +3.6). Thus c0U is an extremal function of +Λ = inf +�� +G +|∇Gu|2dxdt : +� +G +|u| +2Q +Q−2 dxdt = 1 +� +. +The value Sm,n has been calculated in [13], Theorem 1.6. The proof of Theorem +1.2 is thereby completed. +Lemma 3.6. Let up(2 ≤ p < +2Q +Q−2) be a positive solution of (3.1). If +� +G +|∇Gup|2dxdt − m(Q − 2) +� +G +u2 +p +(1 + |x|2 +4 )2 + |t|2 dxdt → 0, p ր +2Q +Q − 2, +then there exists c0 > 0 and a sequence {pk : k = 1, 2, · · ·} such that pk ր +2Q +Q−2 and +� +G +|∇G(upk − c0U)|2dxdt → 0, k → ∞. + +14 +QIAOHUA YANG +Proof. By Lemma 2.2, µ1 = m(Q − 2) is simple with eigenfunction U of (1.8) with +λ = 1 and η = 0. Decompose up as +up = λpU + vp +(3.7) +with +λp = +� +G U +Q+2 +Q−2 updxdt +� +G U +2Q +Q−2 dxdt +> 0. +Then vp ⊥ U, i.e. +� +G +U +4 +Q−2 · Uvpdx = +� +G +U +Q+2 +Q−2 vpdxdt = 0, +� +G +⟨∇GU, ∇Gvp⟩dx = 0. +(3.8) +Therefore, we have +� +G +|∇Gvp|2dxdt ≥ µ2 +� +G +U +4 +Q−2 · v2 +pdx = µ2 +� +G +v2 +p +(1 + |x|2 +4 )2 + |t|2 dxdt, +(3.9) +where µ2 is the second eigenvalue of (1.8) with with λ = 1 and η = 0. We compute, +by using (3.8) and (3.9), +� +G +|∇Gup|2dxdt − m(Q − 2) +� +G +u2 +p +(1 + |x|2 +4 )2 + |t|2 dxdt += +� +G +� +λ2 +p|∇GU|2 + |∇Gvp|2� +dxdt − µ1 +� +G +λ2 +pU 2 + v2 +p +(1 + |x|2 +4 )2 + |t|2 dxdt += +� +G +|∇Gvp|2dxdt − µ1 +� +G +v2 +p +(1 + |x|2 +4 )2 + |t|2 dxdt +=µ1 +µ2 +�� +G +|∇Gvp|2dxdt − µ2 +� +G +u2 +p +(1 + |x|2 +4 )2 + |t|2 dxdt +� ++ +µ2 − µ1 +µ2 +� +G +|∇Gvp|2dxdt +≥µ2 − µ1 +µ2 +� +G +|∇Gvp|2dxdt. +Therefore, +� +G +|∇Gvp|2dxdt +≤ +µ2 +µ2 − µ1 +�� +G +|∇Gup|2dxdt − m(Q − 2) +� +G +u2 +p +(1 + |x|2 +4 )2 + |t|2 dxdt +� +→ 0, p ր +2Q +Q − 2. +(3.10) + +THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY +15 +On the other hand, by (3.7), Minkowski’s inequalities, H¨older’s inequality and +(1.1), we have +λp +�� +G +U(x, t) +2Q +Q−2 dxdt +� 1 +p += +�� +G +(up − vp)pU(x, t) +2Q +Q−2 −pdxdt +� 1 +p +≤ +�� +G +up +pU(x, t) +2Q +Q−2 −pdxdt +� 1 +p ++ +�� +G +|vp|pU(x, t) +2Q +Q−2 −pdxdt +� 1 +p +=1 + +�� +G +|vp|pU(x, t) +2Q +Q−2 −pdxdt +� 1 +p +≤1 + +�� +G +|vp| +2Q +Q−2 dxdt +� Q−2 +2Q �� +G +U +2Q +Q−2 dxdt +� 1 +p − Q−2 +2Q +≤1 + C +�� +G +|∇Gvp|2dxdt +� 1 +2 �� +G +U +2Q +Q−2 dxdt +� 1 +p − Q−2 +2Q +. +(3.11) +Substituting (3.10) into (3.11), we obtain +lim sup +pր 2Q +Q−2 +λp ≤ +�� +G +U(x, t) +2Q +Q−2 dxdt +�− Q−2 +2Q +. +Therefore, there exists c0 ≥ 0 and a sequence {pk : k = 1, 2, · · · } such that pk ր +2Q +Q−2 and +λpk → c0, k → ∞. +(3.12) +We claim that +� +G +|∇G(upk − c0U)|2dxdt → 0, k → ∞. +(3.13) +In fact, by using (3.10) and (3.12), we obtain +� +G +|∇G(upk − c0U)|2dxdt += +� +G +|∇G(vpk + (λpk − c0)U)|2dxdt += +� +G +|∇Gvpk|2dxdt + (λpk − c0)2 +� +G +|∇GU|2dxdt → 0, k → ∞. +This proves the claim. +Finally, we show that c0 > 0. In fact, if c0 = 0, then by (3.13), +� +G +|∇Gupk|2dxdt → 0, k → ∞. + +16 +QIAOHUA YANG +On the other hand, by H¨older’s inequality and (1.1), we obtain +1 = +�� +G +upk +pkU(x, t) +2Q +Q−2 −pkdxdt +� 1 +pk +≤ +�� +G +|upk| +2Q +Q−2 dxdt +� Q−2 +2Q �� +G +U +2Q +Q−2 dxdt +� 1 +pk − Q−2 +2Q +≤C +�� +G +|∇Gupk|2dxdt +� 1 +2 �� +G +U +2Q +Q−2 dxdt +� 1 +pk − Q−2 +2Q +→ 0, k → ∞, +which is a contradiction. So c0 > 0. The proof of Lemma 3.6 is thereby completed. +□ +Finally, we give the proof of Theorem 1.3. +Proof of Theorem 1.3 A simple scaling argument shows that the eigenvalues +do not depend on λ and η. So we may assume λ = 1 and η = 0. +From Lemma 2.2 we know that µ1 = m(Q − 2) is simple with eigenfunction U. +Nextly, we show µ2 ≥ m(Q + 2). Let V ̸= 0 be a eigenfunction of µ2. Then +(3.14) +µ2 = +� +G |∇GV |2dxdt +� +G U +4 +Q−2 V 2dxdt +. +Furthermore, since V ⊥ U, we have +(3.15) +� +G +⟨∇GU, ∇GV ⟩dxdt = 0, +� +G +U +4 +Q−2 · UV dxdt = +� +G +U +Q+2 +Q−2 V dxdt = 0. +Set +Φ(ǫ) = +� +G |∇G(U + ǫV )|2dxdt +�� +G |U + ǫV | +2Q +Q−2 dxdt +� Q−2 +Q , ǫ ∈ R. +By Theorem 1.2, U is an extremal function of Folland-Stein inequality (1.7). So we +have Φ′(0) = 0 and Φ′′(0) ≥ 0. We compute +Φ′(ǫ) =2 +� +G⟨∇G(U + ǫV ), ∇GV ⟩dxdt +�� +G |U + ǫV | +2Q +Q−2 dxdt +� Q−2 +Q +− +2 +� +G |∇G(U + ǫV )|2dxdt +�� +G |U + ǫV | +2Q +Q−2 dxdt +� 2Q−2 +Q +� +G +|U + ǫV | +4 +Q−2 (U + ǫV )V dxdt +=Φ1(ǫ) − Φ2(ǫ), +where +Φ1(ǫ) =2 +� +G⟨∇G(U + ǫV ), ∇GV ⟩dxdt +�� +G |U + ǫV | +2Q +Q−2 dxdt +� Q−2 +Q +; +Φ2(ǫ) =2 +� +G |∇G(U + ǫV )|2dxdt +�� +G |U + ǫV | +2Q +Q−2 dxdt +� 2Q−2 +Q +� +G +|U + ǫV | +4 +Q−2 (U + ǫV )V dxdt. + +THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY +17 +By using (3.15), we have +Φ′ +1(0) =2 +� +G |∇GV |2dxdt +�� +G |U| +2Q +Q−2 dxdt +� Q−2 +Q +− 4 +� +G⟨∇GU, ∇GV ⟩dxdt +�� +G |U| +2Q +Q−2 dxdt +� 2Q−2 +Q +� +G +U +Q+2 +Q−2 V dxdt +=2 +� +G |∇GV |2dxdt +�� +G |U| +2Q +Q−2 dxdt +� Q−2 +Q ; +Φ′ +2(0) =4 +� +G⟨∇GU, ∇GV ⟩dxdt +�� +G |U| +2Q +Q−2 dxdt +� 2Q−2 +Q +� +G +U +Q+2 +Q−2 V dxdt +− 8(Q − 1) +Q − 2 +� +G |∇GU|2dxdt +�� +G |U| +2Q +Q−2 dxdt +� 3Q−2 +Q +�� +G +U +4 +Q−2 V 2dxdt +�2 ++ 2(Q + 2) +Q − 2 +� +G |∇GU|2dxdt +�� +G U +2Q +Q−2 dxdt +� 2Q−2 +Q +� +G +U +4 +Q−2 V 2dxdt +=2(Q + 2) +Q − 2 +� +G |∇GU|2dxdt +�� +G U +2Q +Q−2 dxdt +� 2Q−2 +Q +� +G +U +4 +Q−2 V 2dxdt. +Therefore, +0 ≤ Φ′′(0) =Φ′ +1(0) − Φ′ +2(0) +=2 +� +G |∇GV |2dxdt +�� +G |U| +2Q +Q−2 dxdt +� Q−2 +Q +− 2(Q + 2) +Q − 2 +� +G |∇GU|2dxdt +�� +G U +2Q +Q−2 dxdt +� 2Q−2 +Q +� +G +U +4 +Q−2 V 2dxdt, +i.e. +� +G |∇GV |2dxdt +� +G |U| +4 +Q−2 V 2dxdt +≥ Q + 2 +Q − 2 +� +G |∇GU|2dxdt +� +G |U| +2Q +Q−2 dxdt +. +(3.16) +Combing (3.14) and (3.16) yields +µ2 ≥ Q + 2 +Q − 2 +� +G |∇GU|2dxdt +� +G |U| +2Q +Q−2 dxdt += Q + 2 +Q − 2µ1 = m(Q + 2). +On the other hand, by (2.4), {∂λUλ,η|λ=1,η=0, ∇ηUλ,η|λ=1,η=0} are eigenfunctions +of m(Q + 2). So µ2 = m(Q + 2). This completes the proof of Theorem 1.3. +References +[1] G. Bianchi, H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24. +[2] A. Bonfiglioli, F. Uguzzoni, Nonlinear Liouville theorems for some critical problems on H-type +groups, J. Funct. Anal., 207(2004), 161-215. +[3] H. Brezis, E. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal., 62(1985), +73-86. +[4] M. Christ, H. Liu, A. Zhang, Sharp Hardy-Littlewood-Sobolev inequalities on quaternionic +Heisenberg groups, Nonlinear Analysis, 130(2016), 361-395. +[5] M. Christ, H. Liu, A. Zhang, Sharp Hardy-Littlewood-Sobolev inequalities on octonionic +Heisenberg group, Calc. Var. 55, Article number: 11 (2016). + +18 +QIAOHUA YANG +[6] M. Cowling, A. H. Dooley, A. Kor´anyi, F. Ricci, H-type groups and Iwasawa decompositions, +Adv. Math., 87 (1991), 1-41. +[7] J. Dolbeault, M. J. Esteban, A. Figalli, R. L. Frank, M. Loss, Stability for the Sobolev +inequality with explicit constants, arXiv:2209.08651v2 [math.AP]. +[8] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. +13 (1975), 161-207. +[9] G. B. Folland, E. M. Stein, Estimates for the ¯∂b complex and analysis on the Heisenberg +group, Comm. Pure Appl. Math. 27 (1974), 429-522. +[10] G.B. Folland, E.M. Stein, Hardy spaces on homogeneous groups, Princeton University Press, +Princeton, NJ, 1982. +[11] R. L. Frank, E. Lieb, Sharp constants in several inequalities on the Heisenberg group. Ann. +of Math. (2) 176 (2012), no. 1, 349-381. +[12] R. L. Frank, E. Lieb, A new, rearrangement-free proof of the sharp Hardy-Littlewood- Sobolev +inequality. (English summary) Spectral theory, function spaces and inequalities, 55-67, Oper. +Theory Adv. Appl., 219, Birkh¨auser/Springer Basel AG, Basel, 2012. +[13] N. Garofalo, D. Vassilev, Symmetry properties of positive entire solutions of Yamabe type +equations on groups of Heisenberg type, Duke Math. J., 106 (2001), no. 3, 411-449. +[14] N. Garofalo, D. Vassilev, Regularity near the characteristic set in the non-linear Dirich- +let problem and conformal geometry of sub-Laplacians on Carnot groups, Math. Ann., 318 +(2000), no. 3, 453-516. +[15] F. Hang, X. Wang, A simpler proof of Frank and Lieb’s sharp inequality on the Heisenberg +Group, arXiv:2211.10301v2 [math.AP]. +[16] S. Ivanov, I. Minchev, D. Vassilev, Extremals for the Sobolev inequality on the seven- +dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem, +J. Euro. Math. Soc., 12 (2010), no. 4, 1041-1067. +[17] S. Ivanov, I. Minchev, D. Vassilev, The optimal constant in the L2 Folland-Stein inequality +on the quaternionic Heisenberg group, Ann. Sc. Norm. Super. PisaCl. Sci. XI (2012), no. (5), +635-652. +[18] D. Jerison, J. M. Lee, The Yamabe problem on CR manifolds. J. Diff. Geom. 25 (1987), no. +2, 167-197. +[19] D. Jerison, J. M. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the +CR Yamabe problem. J. Amer. Math. Soc. 1 (1988), no. 1, 1-13. +[20] D. Jerison, J. M. Lee, Intrinsic CR normal coordinates and the CR Yamabe problem, J. Diff. +Geom. 29 (1989), 303-343. +[21] A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition +of quadratic forms, Trans. Amer. Math. Soc., 258 (1) (1980) 147-153. +[22] L. Roncal, S. Thangavelu, An extension problem and trace Hardy inequality for the sublapla- +cian on H-type groups, arXiv:1708.09258 [math.AP]. +[23] M. Schneider, Entire solutions of semilinear elliptic problems with indefinite nonlinearities, +Shaker Verlag GmbH, Germany, 2001. +[24] D. Vassilev, Regularity near the characteristic boundary for sub-laplacian operators, Pacific +J. Math., 227 (2006), no. 2, 361-397. +School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s +Republic of China +Email address: qhyang.math@whu.edu.cn + diff --git a/B9E1T4oBgHgl3EQfpgVg/content/tmp_files/load_file.txt b/B9E1T4oBgHgl3EQfpgVg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..94e20131c1596add60677c5801379cc7957bb5b0 --- /dev/null +++ b/B9E1T4oBgHgl3EQfpgVg/content/tmp_files/load_file.txt @@ -0,0 +1,561 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf,len=560 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='03332v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='AP] 9 Jan 2023 THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY ON THE H-TYPE GROUP QIAOHUA YANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We determine the optimal constant in the L2 Folland-Stein in- equality on the H-type group, which partially confirms the conjecture given by Garofalo and Vassilev (Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The proof is inspired by the work of Frank and Lieb (Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=', 2012) and Hang and Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Introduction Let G be a stratified, simply connected nilpotent Lie group (in short a Carnot group) of step r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Denote by g the Lie algebra of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It is known that g = �r i=1 Vi satisfying (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' [10]) [V1, Vj] = Vj+1, 1 ≤ j ≤ r − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' [V1, Vr] = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' As a simply connected nilpotent group, G is differential with RN, N = �r i=1 dim Vi, via the exponential map exp : g → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' There is a natural family of nonisotropic dilations δλ : g → g for λ > 0 and we define it as follows: δλ(X1 + · · · + Xr) = λX1 + · · · + λrXr, Xj ∈ Vj, 1 ≤ j ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The homogeneous dimension of G, associated with δλ, is Q = �r j=1 j dim Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Via the exponential map exp : g → G, we define the group of dilations on G as follows: δλ(g) = exp ◦δλ ◦ exp−1(g), g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Set nj = dim Vj, 1 ≤ j ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let {X1, · · · , Xn1} be a basis of V1 and denote by ∇G = (X1, · · · , Xn1) the horizontal gradient of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The sub-Laplacian on G is ∆G = �n1 i=1 X2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The Sobolev space W 1,p 0 (G) is the closure of C∞ 0 (G) with respect to the norm ∥u∥W 1,p 0 (G) = �� G |∇Gu|pdg � 1 2 , where dg is the Haar measure on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We remark that the Haar measure on G, induced by the exponential mapping from the Lebesgue measure on g = RN, coin- cides the Lebesgue measure on RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The Folland-Stein inequality on G reads that there exits some constant C > 0 such that for each u ∈ W 1,p 0 (G) (see [8, 9]), �� G |u| pQ Q−p dg � Q−p pQ ≤ C �� G |∇Gu|pdg � 1 p , 1 < p < Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1) 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Primary: 43A80;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 46E35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 22E25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Folland-Stein inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Heisenberg group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' H-type group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' best constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The work was partially supported by the National Natural Science Foundation of China(No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='11201346).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 1 2 QIAOHUA YANG For the existence and regularity of minimizers of the Folland-Stein inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1), we refer to [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The Heisenberg group is the simplest example of Carnot group of step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We denote it by Hn = (Cn × R, ◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The group law on Hn is given by (z, t) ◦ (z′, t′) = (z + z′, t + t′ + 2Imz · z′), where z · z′ = �n j=1 zj¯z′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The homogeneous norm on Hn is given by |(z, t)| = (|z|4 + t2) 1 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' In a series of papers [18, 19, 20], Jerison and Lee, among other results, determined the explicit computation of the extremal functions in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1) in the case p = 2 and G = Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' In fact, the extremal functions are, up to group translations and dilations, c((1 + |z|2)2 + t2)− Q−2 4 , c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Such inequalities play an important role in the study of CR Yamabe problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Later, in a celebrated paper [11], Frank and Lieb established sharp Hardy-Littlewood- Sobolev inequalities on Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We state the result as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1 (Frank-Lieb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let 0 < λ < Q and p = 2Q Q−λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Then for any f, g ∈ Lp(Hn), ����� � � Hn×Hn f(z, t)g(z′, t′) |(z, t)−1 ◦ (z′, t′)|λ dzdtdz′dt′ ����� ≤ � πn+1 2n−1n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' �λ/Q n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='Γ( Q−λ 2 ) Γ2( Q−2λ 4 ) ∥f∥p∥g∥p, with equality if and only if, up to group translations and dilations, f = c((1 + |z|2)2 + t2)− 2Q−λ 4 , g = c′((1 + |z|2)2 + t2)− 2Q−λ 4 for some c, c′ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' In particular, choosing λ = Q−2 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1 yields the Jerison and Lee’s in- equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Using the method in [11], Frank and Lieb [12] also gave a new, rearrangement- free proof of sharp Hardy-Littlewood-Sobolev inequalities on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Recently, Hang and Wang [15] present a shorter proof of the Frank-Lieb inequality, in which they bypasses the subtle proof of existence and the Hersch-type argument via subcritical approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Some of the results of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1 have been generalized to the cases of quater- nionic Heisenberg group and octonionic Heisenberg group (see [4, 5, 16, 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We note that Heisenberg group, quaternionic Heisenberg group and octonionic Heisen- berg group are known as the groups of Iwasawa type, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=', the nilpotent component in the Iwasawa decomposition of simple groups of rank one (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The aim of this paper is to look for the optimal constant of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1) when p = 2 and G is a group of Heisenberg type (in short a H-type group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Recall that a H-type group G is a Carnot group of step two with the following properties (see Kaplan [21]): the Lie algebra g of G is endowed with an inner product ⟨, ⟩ such that, if z is the center of g, then [z⊥, z⊥] = z and moreover, for every fixed z ∈ z, the map Jz : z⊥ → z⊥ defined by ⟨Jz(v), ω⟩ = ⟨z, [v, ω]⟩, ∀ω ∈ z⊥ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2) is an orthogonal map whenever ⟨z, z⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It is known (see [6]) that a H-type group G is the group of Iwasawa type if and only if its Lie algebra satisfies the following THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 3 J2-condition: for any v ∈ z⊥ and z, z′ ∈ z such that ⟨z, z′⟩ = 0, there exists z′′ ∈ z such that JzJz′v = Jz′′v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, most of H-type groups are not groups of Iwasawa type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Set m = dim z⊥ and n = dim z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Since G has step two, we can fix on G a system of coordinates (x, t) such that the group law on G has the form (see [2]) (x, t) ◦ (x′, t′) = � xi + x′ i, i = 1, 2, · · · , m tj + t′ j + 1 2⟨x, U (j)x′⟩, j = 1, 2, · · · , n � (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3) for suitable skew-symmetric matrices U (j)’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Nextly, we set U(ξ) = �� 1 + |x|2 4 �2 + |t|2 �− Q−2 4 , ξ = (x, t) ∈ G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4) Uλ,η(ξ) =λ Q−2 2 U(δλ(η−1 ◦ ξ)), η ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5) It has been shown that [m(Q − 2)] Q−2 4 Uλ,η(ξ) satisfies the Yamabe-type equation (see [13, 14]) ∆G[m(Q − 2)] Q−2 4 Uλ,η + {[m(Q − 2)] Q−2 4 Uλ,η} Q+2 Q−2 = 0, or equivalently, ∆GUλ,η + m(Q − 2)U Q+2 Q−2 λ,η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6) In the paper [13], Garofalo and Vassilev gave the following conjecture: Conjecture (Garofalo-Vassilev).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' In a H-type group G, the functions [m(Q − 2)] Q−2 4 Uλ,η(ξ) are the only nontrivial entire solutions to � ∆Gu + u Q+2 Q−2 = 0, u ∈ W 1,2 0 (G), u ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' If the conjecture is true, then one can obtain the optimal constant of L2 Folland- Stein inequality on H-type groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' In this paper we shall use the method given by Frank and Lieb [11, 12] and Hang and Wang [15] to determine the optimal constant, instead of proving the conjecture directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' To this end, we have Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It holds that � G |∇Gu|2dxdt ≥ Sm,n �� G |u| 2Q Q−2 dxdt � Q−2 Q , u ∈ W 1,2 0 (G), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='7) where Sm,n = 4− 2n Q m(Q − 2)π m+n Q � Γ( m+n 2 ) Γ(m + n) �1/Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The inequality is sharp and an extremal function is U(x, t) = �� 1 + |x|2 4 �2 + |t|2 �− Q−2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 4 QIAOHUA YANG By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2, it is easy to see that the functions cUλ,η(ξ)(c ∈ R) are also extremal functions of inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' As an application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2, we study the eigenvalues of −∆Gv = µU 4 Q−2 λ,η v, v ∈ W 1,2 0 (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) We note that the eigenvalues of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) play an important role in the study of stability for the Folland-Stein inequality (see [1, 3, 7] for the case of Euclidean space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' In Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2 we show that the embedding map W 1,2 0 (G) ֒→ L2(G, U(x, t) 4 Q−2 dxdt) is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' So the spectrum of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Furthermore, we have the following theorem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let µi, i = 1, 2, · · · be the eigenvalues of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) given in increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Then (1) µ1 = m(Q − 2) is simple with eigenfunction Uλ,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2) µ2 = m(Q + 2) and {∂λUλ,η, ∇ηUλ,η} are eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Furthermore, the eigenvalues do not depend on λ and η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It seems that µ2 has multiplicity m + n + 1 with corresponding eigenspace spanned by {∂λUλ,η, ∇ηUλ,η}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' However, we fail to prove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Once it has been proven, it would provide a generalization of the results of Bianchi and Egnell ([1], Lemma A1) to the setting of H-type groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' preliminaries on H-type groups In the rest of paper, we let G be a H-type group with group law given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The nonisotropic dilations δλ on G is δλ(x, t) = (λx, λ2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' For (x, t) ∈ G, the homogeneous norm of (x, t) is ρ(x, t) = �|x|4 16 + |t|2 � 1 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' With this norm ρ, we can define the ball centered at origin with radius R BR(0) = {(x, t) ∈ G : ρ(x, t) < R} and the unit sphere Σ = ∂B1(0) = {(x, t) ∈ G : ρ(x, t) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Given any (x, t) ∈ G with ρ(x, t) ̸= 0, we set x∗ = x ρ(x,t) and t∗ = t ρ(x,t)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The polar coordinates on G associated with ρ are the following (see [10]): � G f(x, t)dxdt = � ∞ 0 � Σ f(ρx∗, ρ2t∗)ρQ−1dσdρ, f ∈ L1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The following theorem was proved in [2], Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' : Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' G is a H-type group if and only if G is (isomorphic to) Rm+n with the group law in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3) and the matrices U (1), U (2), · · · , U (n) have the following properties: (1) U (j) is a m × m skew symmetric and orthogonal matrix, for every j = 1, 2, · · · , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2) U (i)U (j) + U (j)U (i) = 0 for every i, j ∈ {1, 2, · · · , n} with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 5 The vector field in the Lie algebra g that agrees at the origin with ∂ ∂xj (j = 1, · · · , m) is given by Xj = ∂ ∂xj + 1 2 n � k=1 � m � i=1 U (k) i,j xi � ∂ ∂tk and g is spanned by the left-invariant vector fields X1, · · · , Xm, T1 = ∂ ∂t1 , · · · , Tn = ∂ ∂tn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Furthermore (see [2], Page 200, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4) ), [Xi, Xj] = n � r=1 U (r) i,j Tr, i, j ∈ {1, 2, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1) The exponential map exp : g → G is exp : g → Rm+n, m � i=1 xiXi + n � j=1 tjTj �→ (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We note that by exponential mapping, the group law (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3) is nothing but the Baker-Campbell-Hausdorff formula (see [2], the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2) exp X ◦ exp Y = exp(X + Y + 1 2[X, Y ]), X, Y ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1), we have that for t = (t1, · · · , tn) = t1T1 + · · · + tnTn and x = (x1, · · · , xm) = x1X1 + · · · + xmXm, the map Jt, defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2), is (see also [2], Page 201) Jtx = n � r=1 m � i=1 trxiJTr(Xi) = n � r=1 m � i=1 trxi \uf8eb \uf8ed m � j=1 U (r) i,j Xj \uf8f6 \uf8f8 = m � j=1 � n � r=1 m � i=1 trxiU (r) i,j � Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Since Jt is an orthogonal map whenever |t| = 1, we obtain |Jtx|2 = |t|2|x|2 = m � j=1 � n � r=1 m � i=1 trxiU (r) i,j �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2) The horizontal gradient on G is ∇G = (X1, · · · , Xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The sub-Laplacian on G is given by (see [2], Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=') ∆G = m � j=1 X2 j = m � j=1 � ∂ ∂xj + 1 2 n � k=1 � m � i=1 U (k) i,j xi � ∂ ∂tk �2 = ∆x + 1 4|x|2∆t + n � k=1 ⟨x, U (k)∇x⟩ ∂ ∂tk , where ∆x = m � j=1 � ∂ ∂xj �2 , ∆t = n � k=1 � ∂ ∂tk �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We remark that ∆G is homogeneous of degree two with respect to δλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' By using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6), we have the following Hardy inequality (see [22], Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4 for Hardy inequality of fractional powers of the sublaplacian on G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 6 QIAOHUA YANG Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It holds that, for u ∈ W 1,2 0 (G), � G |∇Gu|2dxdt ≥ m(Q − 2) � G u2 (1 + |x|2 4 )2 + |t|2 dxdt, with equality if and only if u = cU(x, t), where c ∈ R and U(x, t) is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We have, through integration by parts, 0 ≤ � G U 2|∇G(U(x, t)−1u)|2dxdt = � G ���∇Gu − u U ∇GU ��� 2 dxdt = � G |∇Gu|2dxdt + � G |∇GU|2 U 2 u2dxdt − � G 1 U ⟨∇Gu2, ∇GU⟩dxdt = � G |∇Gu|2dxdt + � G u2 1 U ∆GUdxdt = � G |∇Gu|2dxdt − m(Q − 2) � G u2 (1 + |x|2 4 )2 + |t|2 dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3) To get the last equality, we use (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The desired result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' □ Set η = (y1, · · · , ym, w1, · · · , wn) ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6), we have ∆G ∂Uλ,η ∂yj + m(Q + 2)U 4 Q−2 λ,η ∂Uλ,η ∂yj = 0, j = 1, , · · · , m, ∆G ∂Uλ,η ∂wr + m(Q + 2)U 4 Q−2 λ,η ∂Uλ,η ∂wr = 0, r = 1, , · · · , n, ∆G ∂Uλ,η ∂λ + m(Q + 2)U 4 Q−2 λ,η ∂Uλ,η ∂λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4) Furthermore, we have the following lemma: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It holds that m � j=1 ���� ∂Uλ,η ∂yj |λ=1,η=0 ���� 2 + n � r=1 ���� ∂Uλ,η ∂wr |λ=1,η=0 ���� 2 + 1 4 ���� ∂Uλ,η ∂λ |λ=1,η=0 ���� 2 = (Q − 2)2 16 U(ξ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It is easy to see η−1 = −η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5), we have Uλ,η(x, t) =λ Q−2 2 \uf8ee \uf8f0 � 1 + λ2 4 m � i=1 (xi − yi)2 �2 + λ4 n � r=1 � tr − wr − ⟨y, U (r)x⟩ 2 �2\uf8f9 \uf8fb − Q−2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 7 We compute ∂Uλ,η ∂yj |λ=1,η=0 = − Q − 2 4 U(ξ) Q+2 Q−2 � 2 � 1 + |x|2 4 � � −xj 2 � + n � r=1 tr � − m � i=1 U (r) j,i xi �� =Q − 2 4 U(ξ) Q+2 Q−2 �� 1 + |x|2 4 � xj + n � r=1 m � i=1 trxiU (r) j,i � , j = 1, · · · , m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' ∂Uλ,η ∂wr |λ=1,η=0 = − Q − 2 4 U(ξ) Q+2 Q−2 (−2tr) =Q − 2 2 U(ξ) Q+2 Q−2 tr, r = 1, · · · , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' ∂Uλ,η ∂λ |λ=1,η=0 =Q − 2 2 U(ξ) − Q − 2 4 U(ξ) Q+2 Q−2 � 2 � 1 + |x|2 4 � |x|2 2 + 4|t|2 � = − Q − 2 2 U(ξ) Q+2 Q−2 � −1 + |x|4 16 + |t|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Since each U (j)(1 ≤ j ≤ n) is a m × m skew symmetric matrix, we have, by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2), m � j=1 ���� ∂Uλ,η ∂yj |λ=1,η=0 ���� 2 =(Q − 2)2 16 U(ξ)2 Q+2 Q−2 m � j=1 �� 1 + |x|2 4 � xj − n � r=1 m � i=1 trxiU (r) i,j �2 =(Q − 2)2 16 U(ξ)2 Q+2 Q−2 �� 1 + |x|2 4 �2 |x|2 + |t|2|x|2− 2 � 1 + |x|2 4 � n � r=1 tr \uf8eb \uf8ed m � i=1 m � j=1 U (r) i,j xixj \uf8f6 \uf8f8 \uf8f9 \uf8fb =(Q − 2)2 16 U(ξ)2 Q+2 Q−2 �� 1 + |x|2 4 �2 |x|2 + |t|2|x|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' To get the last equality, we use the fact m � i=1 m � j=1 U (r) i,j xixj = 0 8 QIAOHUA YANG since U (r)(1 ≤ r ≤ n) is a m × m skew symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' we have m � j=1 ���� ∂Uλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='η ∂yj |λ=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='η=0 ���� 2 + n � r=1 ���� ∂Uλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='η ∂wr |λ=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='η=0 ���� 2 + 1 4 ���� ∂Uλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='η ∂λ |λ=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='η=0 ���� 2 =(Q − 2)2 16 U(ξ)2 Q+2 Q−2 �� 1 + |x|2 4 �2 |x|2 + |t|2|x|2 � + (Q − 2)2 4 U(ξ)2 Q+2 Q−2 |t|2+ (Q − 2)2 16 U(ξ)2 Q+2 Q−2 � −1 + |x|4 16 + |t|2 �2 =(Q − 2)2 16 U(ξ)2 Q+2 Q−2 �� 1 + |x|2 4 �2 |x|2 + |t|2|x|2 + 4|t|2 + � −1 + |x|4 16 + |t|2 �2� =(Q − 2)2 16 U(ξ)2 Q+2 Q−2 �� 1 + |x|2 4 �2 + |t|2 �2 =(Q − 2)2 16 U(ξ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' To get the third equality, we use the fact � −1 + |x|4 16 + |t|2 �2 = �� 1 + |x|2 4 �2 + |t|2 − 2 � 1 + |x|2 4 ��2 = �� 1 + |x|2 4 �2 + |t|2 �2 + 4 � 1 + |x|2 4 �2 − 4 � 1 + |x|2 4 � �� 1 + |x|2 4 �2 + |t|2 � = �� 1 + |x|2 4 �2 + |t|2 �2 − � 1 + |x|2 4 �2 |x|2 − |t|2|x|2 − 4|t|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' This completes the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' □ For simplicity, we set ωj = 4 Q − 2U(ξ)−1 ∂Uλ,η ∂yj |λ=1,η=0, j = 1, · · · , m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' ωj+r = 4 Q − 2U(ξ)−1 ∂Uλ,η ∂wr |λ=1,η=0, r = 1, · · · , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' ωm+n+1 = 2 Q − 2U(ξ)−1 ∂Uλ,η ∂λ |λ=1,η=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4), we have m+n+1 � j=1 ω2 j =1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6) ∆G(U(ξ)ωj) + m(Q + 2)U(ξ) Q+2 Q−2 ωj =0, 1 ≤ j ≤ m + n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='7) THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3 In this section, we shall prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The proof depends on a scheme of subcritical approximation due to Hang and Wang [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We first establish the following subcritical Sobolev inequality on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let 2 ≤ p < 2Q Q−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' There exists C > 0 such that for each u ∈ W 1,2 0 (G), � G |∇Gu|2dxdt ≥ C �� G |u|pU(x, t) 2Q Q−2 −pdxdt � 2 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' By H¨older’s inequality, we have � G |u|pU(x, t) 2Q Q−2 −pdxdt = � G � |u|U(x, t) 2 Q−2 �Q− Q−2 2 p |u| Q 2 (p−2)dxdt ≤ �� G |u|2U(x, t) 4 Q−2 dxdt � 2Q−(Q−2)p 4 �� G |u| 2Q Q−2 dxdt � (Q−2)(p−2) 4 = �� G u2 (1 + |x|2 4 )2 + |t|2 dxdt � 2Q−(Q−2)p 4 �� G |u| 2Q Q−2 dxdt � (Q−2)(p−2) 4 ≤C � G |∇Gu|2dxdt, where C is a positive constant independent of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' To get the last inequality above, we use Folland-Stein inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' □ By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1, we have W 1,2 0 (G) ֒→ Lp(G, U(x, t) 2Q Q−2 −pdxdt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Furthermore, the embedding map is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let 2 ≤ p < 2Q Q−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The embedding map W 1,2 0 (G) ֒→ Lp(G, U(x, t) 2Q Q−2 −pdxdt) is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The proof is similar to that given by Schneider (see [23], section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let φ : G → [0, 1] be a cut-off function that is equal to one in B1(0) and zero outside of B2(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Consider the operator IR : W 1,2 0 (G) ֒→ Lp(G, U(x, t) 2Q Q−2 −pdxdt) 10 QIAOHUA YANG defined by IR(u) = u(x, t)φ � x R, t R2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Since the imbedding map W 1,2 0 (B2(0)) ֒→ Lp(B2(0)) is compact, so is IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Moreover, by H¨older’s inequality, � G |u − IR(u)|pU(x, t) 2Q Q−2 −pdxdt ≤ � G\\BR(0) |u|pU(x, t) 2Q Q−2 −pdxdt ≤ �� G\\BR(0) |u| 2Q Q−2 dxdt � Q−2 2Q p �� G\\BR(0) U(x, t) 2Q Q−2 dxdt �1− Q−2 2Q p ≤C �� G |∇Gu|2dxdt � p 2 �� G\\BR(0) U(x, t) 2Q Q−2 dxdt �1− Q−2 2Q p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' To get the last inequality above, we use the Folland-Stein inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' By polar coordinates, we have � G\\BR(0) U(x, t) 2Q Q−2 dxdt = � G\\BR(0) �� 1 + |x|2 4 �2 + |t|2 �− Q 2 dxdt ≤ � G\\BR(0) 1 ( |x|2 4 + |t|2) Q 2 dxdt = � ∞ R � Σ 1 ρ2Q ρQ−1dρdσ =|Σ| 1 QRQ → 0, R → ∞, where |Σ| is the volume of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, the embedding map W 1,2 0 (G) ֒→ Lp(G, U(x, t) 2Q Q−2 −pdxdt) is a limit of compact operators and thus it is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2 is thereby completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' □ By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2, the minimization problem Λp = inf �� G |∇Gu|2dxdt : � G |u|pU(x, t) 2Q Q−2 −pdxdt = 1 � , 2 ≤ p < 2Q Q − 2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1) has a positive solution u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We shall show that such u satisfies a moment zero condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The main result is the following lemma: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let 2 ≤ p < 2Q Q−2 and u be a positive solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Then we have � G upU(x, t) 2Q Q−2 −pωidxdt = 0, i = 1, 2, · · · , m + n + 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2) where ωi(1 ≤ i ≤ m + n + 1) is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' For simplicity, we set Fp(u) = � G |∇Gu|2dxdt �� G |u|pU(x, t) 2Q Q−2 −pdxdt � 2 p THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 11 and uλ−1,η−1(ξ) =λ− Q−2 2 u(δλ−1(η ◦ ξ)), λ > 0, η = (y1, · · · , ym, w1, · · · , wn) ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' A simple calculation shows � G |∇Guλ−1,η−1|2dxdt = � G |∇Gu|2dxdt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' � G up λ−1,η−1U(x, t) 2Q Q−2 −pdxdt = � G upUλ,η(x, t) 2Q Q−2 −pdxdt, where Uλ,η is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, Fp(uλ−1,η−1(ξ)) = � G |∇Gu|2dxdt �� G |u|pUλ,η(x, t) 2Q Q−2 −pdxdt � 2 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3) Since u is a positive solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1), we have ∂ ∂yj Fp(uλ−1,η−1(ξ))|λ=1,η=0 =0, j = 1, · · · , m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' ∂ ∂wr Fp(uλ−1,η−1(ξ))|λ=1,η=0 =0, r = 1, · · · , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' ∂ ∂λFp(uλ−1,η−1(ξ))|λ=1,η=0 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4) yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We remark that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3 is also valid for u > 0 satisfying the Yamabe-type equation ∆Gu + ΛpupU(x, t) 2Q Q−2 −p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The proof is same and we omit it (see [15], Corollary 1 for the case of CR sphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' It holds that, for any u ∈ W 1,2 0 (G), m+n+1 � i=1 � G |∇G(uωi)|2dxdt = � G |∇Gu|2dxdt + 4m � G u2 (1 + |x|2 4 )2 + |t|2 dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 12 QIAOHUA YANG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let u = U(ξ)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We compute, through integration by parts, m+n+1 � i=1 � G |∇G(uωi)|2dxdt = m+n+1 � i=1 � G |∇G(vUωi)|2dxdt = m+n+1 � i=1 � G |Uωi∇Gv + v∇G(Uωi)|2dxdt = m+n+1 � i=1 �� G |∇Gv|2U 2ω2 i dxdt + � G |∇G(Uωi)|2v2dxdt+ 1 2 � G ⟨∇G(Uωi)2, ∇Gv2⟩dxdt � = m+n+1 � i=1 �� G |∇Gv|2U 2ω2 i dxdt − � G v2Uωi∆G(Uωi)dxdt � = � G |∇Gv|2U 2dxdt + m(Q + 2) � G v2U 2Q Q−2 dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5) To get the last equality, we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' On the other hand, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3), we have � G |∇Gv|2U 2dxdt = � G ���∇G u U ��� 2 U 2dxdt = � G |∇Gu|2dxdt − m(Q − 2) � G u2 (1 + |x|2 4 )2 + |t|2 dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6) Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5), we obtain m+n+1 � i=1 � G |∇G(uωi)|2dxdt = � G |∇Gu|2dxdt + 4m � G u2 (1 + |x|2 4 )2 + |t|2 dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' □ Now we can give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The idea is due to Frank and Lieb [11, 12] and Hang and Wang [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let 2 ≤ p < 2Q Q−2 and up be a positive solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The 2nd variation of the functional Fp around up shows that � G |∇Gf|2dxdt � G up pU(x, t) 2Q Q−2 −pdxdt− (p − 1) � G |∇Gup|2dxdt � G up−2 p Uλ,η(x, t) 2Q Q−2 −pf 2dxdt ≥ 0 for any f with � G up pUλ,η(x, t) 2Q Q−2 −pfdxdt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 13 By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3, we may choose f = upωi, i = 1, 2, · · · , m + n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Summing the corresponding inequalities for all such f’s yields, in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='5, 0 ≤ m+n+1 � i=1 � G |∇G(upωi)|2dxdt − (p − 1) � G |∇Gup|2dxdt =4m � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt − (p − 2) � G |∇Gup|2dxdt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (p − 2) �� G |∇Gup|2dxdt − m(Q − 2) � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt � ≤m(Q − 2) � 2Q Q − 2 − p � � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt ≤m(Q − 2) � 2Q Q − 2 − p � �� G up pU(x, t) 2Q Q−2 −pdxdt � 2 p �� G U(x, t) 2Q Q−2 dxdt �1− 2 p =m(Q − 2) � 2Q Q − 2 − p � �� G U(x, t) 2Q Q−2 dxdt �1− 2 p → 0, p ր 2Q Q − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' To get the last equality, we use the fact � G up pU(x, t) 2Q Q−2 −pdxdt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2, we obtain � G |∇Gup|2dxdt − m(Q − 2) � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt → 0, p ր 2Q Q − 2, or equivalently, � G |∇G(U −1up)|2U 2dxdt → 0, p ր 2Q Q − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' So we can choose a sequence {pk : k = 1, 2, · · ·} such that pk ր 2Q Q−2 and upk converges to a nonzero function c0U (for reader’s convenience, we prove it in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Thus c0U is an extremal function of Λ = inf �� G |∇Gu|2dxdt : � G |u| 2Q Q−2 dxdt = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The value Sm,n has been calculated in [13], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2 is thereby completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let up(2 ≤ p < 2Q Q−2) be a positive solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' If � G |∇Gup|2dxdt − m(Q − 2) � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt → 0, p ր 2Q Q − 2, then there exists c0 > 0 and a sequence {pk : k = 1, 2, · · ·} such that pk ր 2Q Q−2 and � G |∇G(upk − c0U)|2dxdt → 0, k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 14 QIAOHUA YANG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2, µ1 = m(Q − 2) is simple with eigenfunction U of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) with λ = 1 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Decompose up as up = λpU + vp (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='7) with λp = � G U Q+2 Q−2 updxdt � G U 2Q Q−2 dxdt > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Then vp ⊥ U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' � G U 4 Q−2 · Uvpdx = � G U Q+2 Q−2 vpdxdt = 0, � G ⟨∇GU, ∇Gvp⟩dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) Therefore, we have � G |∇Gvp|2dxdt ≥ µ2 � G U 4 Q−2 · v2 pdx = µ2 � G v2 p (1 + |x|2 4 )2 + |t|2 dxdt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='9) where µ2 is the second eigenvalue of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) with with λ = 1 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We compute, by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='9), � G |∇Gup|2dxdt − m(Q − 2) � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt = � G � λ2 p|∇GU|2 + |∇Gvp|2� dxdt − µ1 � G λ2 pU 2 + v2 p (1 + |x|2 4 )2 + |t|2 dxdt = � G |∇Gvp|2dxdt − µ1 � G v2 p (1 + |x|2 4 )2 + |t|2 dxdt =µ1 µ2 �� G |∇Gvp|2dxdt − µ2 � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt � + µ2 − µ1 µ2 � G |∇Gvp|2dxdt ≥µ2 − µ1 µ2 � G |∇Gvp|2dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, � G |∇Gvp|2dxdt ≤ µ2 µ2 − µ1 �� G |∇Gup|2dxdt − m(Q − 2) � G u2 p (1 + |x|2 4 )2 + |t|2 dxdt � → 0, p ր 2Q Q − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='10) THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 15 On the other hand, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='7), Minkowski’s inequalities, H¨older’s inequality and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1), we have λp �� G U(x, t) 2Q Q−2 dxdt � 1 p = �� G (up − vp)pU(x, t) 2Q Q−2 −pdxdt � 1 p ≤ �� G up pU(x, t) 2Q Q−2 −pdxdt � 1 p + �� G |vp|pU(x, t) 2Q Q−2 −pdxdt � 1 p =1 + �� G |vp|pU(x, t) 2Q Q−2 −pdxdt � 1 p ≤1 + �� G |vp| 2Q Q−2 dxdt � Q−2 2Q �� G U 2Q Q−2 dxdt � 1 p − Q−2 2Q ≤1 + C �� G |∇Gvp|2dxdt � 1 2 �� G U 2Q Q−2 dxdt � 1 p − Q−2 2Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='11) Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='10) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='11), we obtain lim sup pր 2Q Q−2 λp ≤ �� G U(x, t) 2Q Q−2 dxdt �− Q−2 2Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, there exists c0 ≥ 0 and a sequence {pk : k = 1, 2, · · · } such that pk ր 2Q Q−2 and λpk → c0, k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='12) We claim that � G |∇G(upk − c0U)|2dxdt → 0, k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='13) In fact, by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='12), we obtain � G |∇G(upk − c0U)|2dxdt = � G |∇G(vpk + (λpk − c0)U)|2dxdt = � G |∇Gvpk|2dxdt + (λpk − c0)2 � G |∇GU|2dxdt → 0, k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Finally, we show that c0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' In fact, if c0 = 0, then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='13), � G |∇Gupk|2dxdt → 0, k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' 16 QIAOHUA YANG On the other hand, by H¨older’s inequality and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='1), we obtain 1 = �� G upk pkU(x, t) 2Q Q−2 −pkdxdt � 1 pk ≤ �� G |upk| 2Q Q−2 dxdt � Q−2 2Q �� G U 2Q Q−2 dxdt � 1 pk − Q−2 2Q ≤C �� G |∇Gupk|2dxdt � 1 2 �� G U 2Q Q−2 dxdt � 1 pk − Q−2 2Q → 0, k → ∞, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' So c0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='6 is thereby completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' □ Finally, we give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3 A simple scaling argument shows that the eigenvalues do not depend on λ and η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' So we may assume λ = 1 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2 we know that µ1 = m(Q − 2) is simple with eigenfunction U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Nextly, we show µ2 ≥ m(Q + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Let V ̸= 0 be a eigenfunction of µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='14) µ2 = � G |∇GV |2dxdt � G U 4 Q−2 V 2dxdt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Furthermore, since V ⊥ U, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='15) � G ⟨∇GU, ∇GV ⟩dxdt = 0, � G U 4 Q−2 · UV dxdt = � G U Q+2 Q−2 V dxdt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Set Φ(ǫ) = � G |∇G(U + ǫV )|2dxdt �� G |U + ǫV | 2Q Q−2 dxdt � Q−2 Q , ǫ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='2, U is an extremal function of Folland-Stein inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' So we have Φ′(0) = 0 and Φ′′(0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' We compute Φ′(ǫ) =2 � G⟨∇G(U + ǫV ), ∇GV ⟩dxdt �� G |U + ǫV | 2Q Q−2 dxdt � Q−2 Q − 2 � G |∇G(U + ǫV )|2dxdt �� G |U + ǫV | 2Q Q−2 dxdt � 2Q−2 Q � G |U + ǫV | 4 Q−2 (U + ǫV )V dxdt =Φ1(ǫ) − Φ2(ǫ), where Φ1(ǫ) =2 � G⟨∇G(U + ǫV ), ∇GV ⟩dxdt �� G |U + ǫV | 2Q Q−2 dxdt � Q−2 Q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Φ2(ǫ) =2 � G |∇G(U + ǫV )|2dxdt �� G |U + ǫV | 2Q Q−2 dxdt � 2Q−2 Q � G |U + ǫV | 4 Q−2 (U + ǫV )V dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY 17 By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='15), we have Φ′ 1(0) =2 � G |∇GV |2dxdt �� G |U| 2Q Q−2 dxdt � Q−2 Q − 4 � G⟨∇GU, ∇GV ⟩dxdt �� G |U| 2Q Q−2 dxdt � 2Q−2 Q � G U Q+2 Q−2 V dxdt =2 � G |∇GV |2dxdt �� G |U| 2Q Q−2 dxdt � Q−2 Q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Φ′ 2(0) =4 � G⟨∇GU, ∇GV ⟩dxdt �� G |U| 2Q Q−2 dxdt � 2Q−2 Q � G U Q+2 Q−2 V dxdt − 8(Q − 1) Q − 2 � G |∇GU|2dxdt �� G |U| 2Q Q−2 dxdt � 3Q−2 Q �� G U 4 Q−2 V 2dxdt �2 + 2(Q + 2) Q − 2 � G |∇GU|2dxdt �� G U 2Q Q−2 dxdt � 2Q−2 Q � G U 4 Q−2 V 2dxdt =2(Q + 2) Q − 2 � G |∇GU|2dxdt �� G U 2Q Q−2 dxdt � 2Q−2 Q � G U 4 Q−2 V 2dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Therefore, 0 ≤ Φ′′(0) =Φ′ 1(0) − Φ′ 2(0) =2 � G |∇GV |2dxdt �� G |U| 2Q Q−2 dxdt � Q−2 Q − 2(Q + 2) Q − 2 � G |∇GU|2dxdt �� G U 2Q Q−2 dxdt � 2Q−2 Q � G U 4 Q−2 V 2dxdt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' � G |∇GV |2dxdt � G |U| 4 Q−2 V 2dxdt ≥ Q + 2 Q − 2 � G |∇GU|2dxdt � G |U| 2Q Q−2 dxdt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='16) Combing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='16) yields µ2 ≥ Q + 2 Q − 2 � G |∇GU|2dxdt � G |U| 2Q Q−2 dxdt = Q + 2 Q − 2µ1 = m(Q + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' On the other hand, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='4), {∂λUλ,η|λ=1,η=0, ∇ηUλ,η|λ=1,η=0} are eigenfunctions of m(Q + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' So µ2 = m(Q + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Bianchi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content=' Egnell, A note on the Sobolev inequality, J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='math@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfpgVg/content/2301.03332v1.pdf'} diff --git a/C9E0T4oBgHgl3EQfyQLe/content/tmp_files/2301.02658v1.pdf.txt b/C9E0T4oBgHgl3EQfyQLe/content/tmp_files/2301.02658v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..555c29eed6519e4a9af9bd5742493178d03a2182 --- /dev/null +++ b/C9E0T4oBgHgl3EQfyQLe/content/tmp_files/2301.02658v1.pdf.txt @@ -0,0 +1,313 @@ +Measuring Power with a Saturated Photodiode +Shiekh Zia Uddin1,* +1Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA +*suddin@mit.edu +ABSTRACT +Accurate measurement of optical power is pivotal in many applications and scientific research. However, traditional power +meters are unable to measure power levels beyond a certain saturation point, limiting their usefulness in high-power applications. +In this technical note, I discuss how optical power can be measured using a saturated photodiode. I demonstrate that by +monitoring both the dc photocurrent and ac noise, it is possible to accurately measure power levels beyond its saturation point. +Keywords: Power meter, Photodiode, Saturation, Noise. +Introduction +Optical power measurement is a critical aspect of many applications. It is the conventional wisdom that a saturated photodiode +can not be used to measure power. Saturation power of photodiodes can be pushed to higher levels by applying a reverse bias +voltage, however there is a limit to the amount of bias voltage due to the reverse breakdown which can be catastrophic to the +diode. This limitation can be problematic in high-power applications, where it is important to be able to accurately measure +power levels at high speed. In this technical note, I discuss a method for measuring optical power using a saturated photodiode. +I demonstrate that the photocurrent noise decreases with power beyond saturation which can be used to accurately measure +power at levels beyond its saturation point. This information might be useful in photon noise measurements. +Background +If a photoevent generated at t = 0 produces an electric pulse h(t), of area e, in the external circuit. A photoevent generated +at time t1 then produces a displaced pulse, h(t −t1). Dividing the time axis into incremental time intervals ∆t so that the +probability p that a photoevent occurs within an interval is P = ηΦ∆t. The electric current i at time t is written as +i(t) = ∑ +l +Xlh(t −l∆t), +(1) +where Xl assumes the value 1 with probability p, and 0 with probability 1− p. The variables Xl are independent. The mean +value of Xl is E[Xl] = 0×(1− p)+1× p = p. Its mean-square value is E[X2 +l ] = 02 ×(1− p)+12 × p = p. The mean of the +product XlXk is p2 if l ̸= k, and p if l = k. The mean and mean-square values of i(t) are now determined via +i = E[i(t)] = ∑ +l +ph(t −l∆t), +(2) +E[i2(t)] = ∑ +l ∑ +k +E[XlXk]h(t −l∆t)h(t −k∆t) +(3) += ∑∑ +l̸=k +p2h(t −l∆t)h(t −k∆t)+∑ +l +ph2(t −l∆t) +(4) +Substituting p = ηΦ∆t, and taking the limit ∆t → 0 so that the summations become integrals, previous equations yield, +respectively, +E[i(t)] = ηΦ +� +h(t)dt, +(5) +E[i2(t)] = +� +ηΦ +� +h(t)dt +�2 ++ηΦ +� +h2(t)dt +(6) +The limits of the integration is zero to infinity. It follows that +σ2 +i = E[i2]−E[i]2 = ηΦ +� +h2(t)dt +(7) +arXiv:2301.02658v1 [physics.ins-det] 7 Jan 2023 + +Definition of the bandwidth B as +B = 1 +2e2 +� ∞ +0 h2(t)dt = +� ∞ +0 h2(t)dt +2( +� ∞ +0 h(t)dt)2 , +(8) +can be readily verified by noting that the Fourier transform of h(t) is its transfer function H(v). The area under h(t) is simply +H(0) = e. In accordance with Parseval’s theorem, the area under h2(t) is equal to the area under the symmetric function |H(v)|2, +so that +B = +� ∞ +0 +���� +H(v) +H(0) +���� +2 +dv +(9) +The quantity B is therefore the power-equivalent spectral width of the function H(v) (i.e., the bandwidth of the device/circuit +combination). As an example, if H(v) = 1 for −Vc < v < Vc and 0 elsewhere, we get B = Vc. Using this definition of bandwidth, +we get back our familiar expression for the noise in photocurrent +σ2 +i = 2eE[i]B. +(10) +So far this is a standard derivation of the shot noise1. Note that this expression hinges on the assumption that the area under h(t) +is simply e, basically one photoevent cause one electrons worth of charge to flow as current. However in saturation its definitely +not the case and the photodiode response becomes a function of intensity. In the first order approximation, one can make the +assumption that h(t) = f(Φ)g(t), where f(Φ) is a power dependent function that is 1 at low power and decreases at high power +and area under g(t) is e. Then we get +E[i] = ηΦ f(Φ) +� +g(t)dt +(11) +σ2 +i = ηΦ f 2(Φ) +� +g2(t)dt, +(12) +which gives us the key insight that the average value of current and noise power scales differently with incident optical power. +If we take a ratio +E[i]2 +σ2 +i += η2Φ2 f 2(Φ)e2 +ηΦ f 2(Φ)2e2B ∝ Φ +(13) +we find that the signal to noise ratio (SNR) in principle is proportional to incident power despite the nonlinearity in the response. +Even if this proportionality does not hold exactly, with proper calibration therefore it should be possible to measure the power +with a saturated photodiode by measuring both the average photocurrent and the photocurrent noise. +Results +Figure 1A shows a schematic diagram of a standard photodiode driving circuit. The photodiode can be modelled as a current +source in parallel with a diode2. A reverse bias voltage is applied to set the operating point, but there is a limit to how much +voltage can be applied defined by junction breakdown3. Such a circuit can be simulated using conventional electrical circuit +theory4 (MATLAB code below), the resulting operating current with realistic circuit parameters is shown in Fig. 1B. We can see +that the photodiode saturates at some power and the saturation knee increases with reverse bias voltage. The highest measurable +power is determined by the highest reverse bias voltage that can be applied across a junction. Below saturation the current is +linear with incident power, which is expected. +Experimental data of a reverse biased photodiode is shown in Fig. 2 where reverse bias is seen to increase the saturation +power. Before saturation the voltage is proportional to power. After saturation no measurement of power is possible, which is +the current paradigm. +We can now attempt to measure power beyond saturation. In Fig. 3A we show the photovoltage and the photocurrent noise +as a function of incident power. Photocurrent noise is measured around 1.5 MHz with a spectrum analyzer (10 kHz resolution +bandwidth, 1 MHz span, with preamplifier on and no attenuation, electronic noise floor is −165 dBm/Hz). Below saturation +photovoltage and noise increases simultaneously. As the photovoltage is saturated, the photocurrent noise suddenly decreases. +This qualitatively follows our theoretical voltage and noise shown in Fig. 1. In Fig. 3B we show the photovoltage and SNR +calculated from the experimental data. Beyond saturation SNR changes with incident power, which can be used to measure +power after proper calibration. Such behaviour also holds at other noise frequencies as long as they are lower than the badngap +of the photodiode and away from 1/ f noise. +. +2/4 + +Figure 1. Schematic diagram of a photodiode driving circuit and simulated operating current I0. +Figure 2. Output voltage across the 25Ω load resistance from a Thorlabs FDGA055 InGaAs photodiode at different reverse +bias excited by a 1070 nm CW laser. It has a 0.95 A/W responsivity, 2.5 ns rise time and 0.5 mm active area diameter. +Figure 3. (A) Output voltage and noise from a Excelitas C30641GH6 InGaAs photodiode at 30 V reverse bias excited by a +1550 nm femtosecond pulsed laser. It saturates around 25 mW. (B) Even though photovoltage has saturated, the SNR shows +response beyond the saturation power. +3/4 + +A +B +100 +C +-140 +Current +Photon +Vr (V) + (mA) +80 +Noise (dBm/Hz) +0 +g Current ( +10 +60 +VR (V) +Load +20 +-160 +0 +R +30 +40 +10 +Photocurrent +20 +Ip +Operating +30 +20 +Bias +0 +VR +-180 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +Incident Power (mW) +Incident Power (mW)1000 +Voltage (mV) +100 +Vr (V) +0 +5 +10 +15 +18 +20 +25 +30 +SL +10 +1 +10 +100 +Incident Power (mWPhotovoltage (mV) +B +30 +1.0 +-135 +Photovoltage (mV) +25 +10 +Noise (dBm/Hz) +0.8 +20 +(norm.) +-145 +50 +0.6 +I +0.4 +SNR +-155 +工 +5 +0.2 +-165 +0.1 +0.0 +0 +10 +20 +30 +40 +50 +0 +10 +20 +30 +40 +50 +Power (mW) +Power (mw)Discussion +Even when a photodiode is saturated, the information about the photon flux intensity are not completely lost and in a way +encoded in the photocurrent noise, which can be practically used to measure power at high speed after proper calibration. +Codes +1 +%% Matlab code to solve for photocurrent and noise in a circuit +2 +clc;clear all;close all; +3 +P=linspace(0,100,1000)*1e-3; % Incident power in W +4 +Responsivity=1; +5 +Ip=P*Responsivity; % Expected photocurrent +6 +R=500; +7 +V=[linspace(-50,0,1e5),linspace(0,.7,1e5)]; +8 +VR=30; % Reverse Bias Voltage +9 +Iop=zeros(size(P)); % Operating Photocurrent +10 +11 +for indx=1:length(P) +12 +I1=-Ip(indx)+0.1e-9*exp(V/.0259); +13 +I2=-(V+VR)/R; +14 +[¬,pos]=min(abs(I1-I2)); +15 +Iop(indx)=-I2(pos); +16 +end +17 +18 +figure(1);subplot(121), plot(P/1e-3,Iop); % Operating Current vs Power +19 +f=Iop./P; %nonlinear response function +20 +S=10*log10(2*1.6e-19*R*1*(P/1e-3).*(f.^2)); % Noise power vs optical power +21 +subplot(122), plot(P/1e-3,S); +References +1. Saleh, B. E. & Teich, M. C. Fundamentals of Photonics (John Wiley & Sons, 2019). +2. Bhattacharya, P. Semiconductor Optoelectronic Devices (Prentice-Hall, Inc., 1997). +3. Neamen, D. A. Semiconductor Physics and Devices: Basic Principles (McGraw-hill, 2003). +4. Sedra, A. S., Smith, K. C., Carusone, T. C. & Gaudet, V. Microelectronic Circuits, vol. 4 (Oxford University Press New +York, 2004). +5. Thorlabs FDGA05. https://www.thorlabs.com/thorproduct.cfm?partnumber=FDGA05 (2022). Accessed: 2022-12-10. +6. Excelitas C30641GH. https://www.excelitas.com/product/c30641gh-ingaas-pin-1mm-18 (2022). Accessed: 2022-12-10. +Acknowledgements +The author acknowledges Nicholas Rivera, Jamison Sloan, Yannick Salamin, chatGPT for their discussions. All equipment +used in the experiments are properties of MIT. +4/4 + diff --git a/C9E0T4oBgHgl3EQfyQLe/content/tmp_files/load_file.txt b/C9E0T4oBgHgl3EQfyQLe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e4ff2c735f29ff321c253db5a7f0baf5a88659c --- /dev/null +++ b/C9E0T4oBgHgl3EQfyQLe/content/tmp_files/load_file.txt @@ -0,0 +1,153 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf,len=152 +page_content='Measuring Power with a Saturated Photodiode Shiekh Zia Uddin1,* 1Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA suddin@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='edu ABSTRACT Accurate measurement of optical power is pivotal in many applications and scientific research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' However, traditional power meters are unable to measure power levels beyond a certain saturation point, limiting their usefulness in high-power applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' In this technical note, I discuss how optical power can be measured using a saturated photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' I demonstrate that by monitoring both the dc photocurrent and ac noise, it is possible to accurately measure power levels beyond its saturation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Keywords: Power meter, Photodiode, Saturation, Noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Introduction Optical power measurement is a critical aspect of many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' It is the conventional wisdom that a saturated photodiode can not be used to measure power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Saturation power of photodiodes can be pushed to higher levels by applying a reverse bias voltage, however there is a limit to the amount of bias voltage due to the reverse breakdown which can be catastrophic to the diode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' This limitation can be problematic in high-power applications, where it is important to be able to accurately measure power levels at high speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' In this technical note, I discuss a method for measuring optical power using a saturated photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' I demonstrate that the photocurrent noise decreases with power beyond saturation which can be used to accurately measure power at levels beyond its saturation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' This information might be useful in photon noise measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Background If a photoevent generated at t = 0 produces an electric pulse h(t), of area e, in the external circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' A photoevent generated at time t1 then produces a displaced pulse, h(t −t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Dividing the time axis into incremental time intervals ∆t so that the probability p that a photoevent occurs within an interval is P = ηΦ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' The electric current i at time t is written as i(t) = ∑ l Xlh(t −l∆t), (1) where Xl assumes the value 1 with probability p, and 0 with probability 1− p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' The variables Xl are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' The mean value of Xl is E[Xl] = 0×(1− p)+1× p = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Its mean-square value is E[X2 l ] = 02 ×(1− p)+12 × p = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' The mean of the product XlXk is p2 if l ̸= k, and p if l = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' The mean and mean-square values of i(t) are now determined via i = E[i(t)] = ∑ l ph(t −l∆t), (2) E[i2(t)] = ∑ l ∑ k E[XlXk]h(t −l∆t)h(t −k∆t) (3) = ∑∑ l̸=k p2h(t −l∆t)h(t −k∆t)+∑ l ph2(t −l∆t) (4) Substituting p = ηΦ∆t, and taking the limit ∆t → 0 so that the summations become integrals, previous equations yield, respectively, E[i(t)] = ηΦ � h(t)dt, (5) E[i2(t)] = � ηΦ � h(t)dt �2 +ηΦ � h2(t)dt (6) The limits of the integration is zero to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' It follows that σ2 i = E[i2]−E[i]2 = ηΦ � h2(t)dt (7) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='02658v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='ins-det] 7 Jan 2023 Definition of the bandwidth B as B = 1 2e2 � ∞ 0 h2(t)dt = � ∞ 0 h2(t)dt 2( � ∞ 0 h(t)dt)2 , (8) can be readily verified by noting that the Fourier transform of h(t) is its transfer function H(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' The area under h(t) is simply H(0) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' In accordance with Parseval’s theorem, the area under h2(t) is equal to the area under the symmetric function |H(v)|2, so that B = � ∞ 0 ���� H(v) H(0) ���� 2 dv (9) The quantity B is therefore the power-equivalent spectral width of the function H(v) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=', the bandwidth of the device/circuit combination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' As an example, if H(v) = 1 for −Vc < v < Vc and 0 elsewhere, we get B = Vc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Using this definition of bandwidth, we get back our familiar expression for the noise in photocurrent σ2 i = 2eE[i]B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' (10) So far this is a standard derivation of the shot noise1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Note that this expression hinges on the assumption that the area under h(t) is simply e, basically one photoevent cause one electrons worth of charge to flow as current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' However in saturation its definitely not the case and the photodiode response becomes a function of intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' In the first order approximation, one can make the assumption that h(t) = f(Φ)g(t), where f(Φ) is a power dependent function that is 1 at low power and decreases at high power and area under g(t) is e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Then we get E[i] = ηΦ f(Φ) � g(t)dt (11) σ2 i = ηΦ f 2(Φ) � g2(t)dt, (12) which gives us the key insight that the average value of current and noise power scales differently with incident optical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' If we take a ratio E[i]2 σ2 i = η2Φ2 f 2(Φ)e2 ηΦ f 2(Φ)2e2B ∝ Φ (13) we find that the signal to noise ratio (SNR) in principle is proportional to incident power despite the nonlinearity in the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Even if this proportionality does not hold exactly, with proper calibration therefore it should be possible to measure the power with a saturated photodiode by measuring both the average photocurrent and the photocurrent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Results Figure 1A shows a schematic diagram of a standard photodiode driving circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' The photodiode can be modelled as a current source in parallel with a diode2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' A reverse bias voltage is applied to set the operating point, but there is a limit to how much voltage can be applied defined by junction breakdown3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Such a circuit can be simulated using conventional electrical circuit theory4 (MATLAB code below), the resulting operating current with realistic circuit parameters is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' We can see that the photodiode saturates at some power and the saturation knee increases with reverse bias voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' The highest measurable power is determined by the highest reverse bias voltage that can be applied across a junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Below saturation the current is linear with incident power, which is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Experimental data of a reverse biased photodiode is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 2 where reverse bias is seen to increase the saturation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Before saturation the voltage is proportional to power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' After saturation no measurement of power is possible, which is the current paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' We can now attempt to measure power beyond saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 3A we show the photovoltage and the photocurrent noise as a function of incident power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Photocurrent noise is measured around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='5 MHz with a spectrum analyzer (10 kHz resolution bandwidth, 1 MHz span, with preamplifier on and no attenuation, electronic noise floor is −165 dBm/Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Below saturation photovoltage and noise increases simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' As the photovoltage is saturated, the photocurrent noise suddenly decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' This qualitatively follows our theoretical voltage and noise shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 3B we show the photovoltage and SNR calculated from the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Beyond saturation SNR changes with incident power, which can be used to measure power after proper calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Such behaviour also holds at other noise frequencies as long as they are lower than the badngap of the photodiode and away from 1/ f noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 2/4 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Schematic diagram of a photodiode driving circuit and simulated operating current I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Output voltage across the 25Ω load resistance from a Thorlabs FDGA055 InGaAs photodiode at different reverse bias excited by a 1070 nm CW laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' It has a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='95 A/W responsivity, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='5 ns rise time and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='5 mm active area diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' (A) Output voltage and noise from a Excelitas C30641GH6 InGaAs photodiode at 30 V reverse bias excited by a 1550 nm femtosecond pulsed laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' It saturates around 25 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' (B) Even though photovoltage has saturated, the SNR shows response beyond the saturation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 3/4 A B 100 C 140 Current Photon Vr (V) (mA) 80 Noise (dBm/Hz) 0 g Current ( 10 60 VR (V) Load 20 160 0 R 30 40 10 Photocurrent 20 Ip Operating 30 20 Bias 0 VR 180 0 20 40 60 80 100 0 20 40 60 80 100 Incident Power (mW) Incident Power (mW)1000 Voltage (mV) 100 Vr (V) 0 5 10 15 18 20 25 30 SL 10 1 10 100 Incident Power (mWPhotovoltage (mV) B 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='0 135 Photovoltage (mV) 25 10 Noise (dBm/Hz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='8 20 (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=') 145 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='6 I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='4 SNR 155 工 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='2 165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='0 0 10 20 30 40 50 0 10 20 30 40 50 Power (mW) Power (mw)Discussion Even when a photodiode is saturated, the information about the photon flux intensity are not completely lost and in a way encoded in the photocurrent noise, which can be practically used to measure power at high speed after proper calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Codes 1 %% Matlab code to solve for photocurrent and noise in a circuit 2 clc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='clear all;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='close all;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 3 P=linspace(0,100,1000)*1e-3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' % Incident power in W 4 Responsivity=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 5 Ip=P*Responsivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' % Expected photocurrent 6 R=500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 7 V=[linspace(-50,0,1e5),linspace(0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='7,1e5)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 8 VR=30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' % Reverse Bias Voltage 9 Iop=zeros(size(P));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' % Operating Photocurrent 10 11 for indx=1:length(P) 12 I1=-Ip(indx)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='1e-9*exp(V/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='0259);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 13 I2=-(V+VR)/R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 14 [¬,pos]=min(abs(I1-I2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 15 Iop(indx)=-I2(pos);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 16 end 17 18 figure(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='subplot(121), plot(P/1e-3,Iop);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' % Operating Current vs Power 19 f=Iop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='/P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' %nonlinear response function 20 S=10*log10(2*1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='6e-19*R*1*(P/1e-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' *(f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='^2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' % Noise power vs optical power 21 subplot(122), plot(P/1e-3,S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Saleh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' & Teich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Fundamentals of Photonics (John Wiley & Sons, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Bhattacharya, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Semiconductor Optoelectronic Devices (Prentice-Hall, Inc.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Sedra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=', Smith, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=', Carusone, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' & Gaudet, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Microelectronic Circuits, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 4 (Oxford University Press New York, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Thorlabs FDGA05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='thorlabs.' metadata={'source': 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+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='excelitas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content='com/product/c30641gh-ingaas-pin-1mm-18 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Accessed: 2022-12-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' Acknowledgements The author acknowledges Nicholas Rivera, Jamison Sloan, Yannick Salamin, chatGPT for their discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' All equipment used in the experiments are properties of MIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} +page_content=' 4/4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'} diff --git a/D9E0T4oBgHgl3EQfywIT/content/tmp_files/2301.02662v1.pdf.txt b/D9E0T4oBgHgl3EQfywIT/content/tmp_files/2301.02662v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a296624ee8b093d08ab0da249d24146c847b8c04 --- /dev/null +++ b/D9E0T4oBgHgl3EQfywIT/content/tmp_files/2301.02662v1.pdf.txt @@ -0,0 +1,2540 @@ +Robust knapsack ordering for a partially-informed +newsvendor with budget constraint +Guus Boonstra +Retail Consulting Department, IG&H Consultants, guus.boonstra@igh.com +Wouter J.E.C. van Eekelen +Department of Econometrics and Operations Research, Tilburg University, w.j.e.c.vaneekelen@tilburguniversity.edu +Johan S.H. van Leeuwaarden +Department of Econometrics and Operations Research, Tilburg University, j.s.h.vanleeuwaarden@tilburguniversity.edu +This paper studies the multi-item newsvendor problem with a constrained budget and information about +demand limited to its range, mean and mean absolute deviation. We consider a minimax model that deter- +mines order quantities by minimizing the expected overage and underage costs for the worst-case demand +distributions. The resulting optimization problem turns out to be solvable by a method reminiscent of the +greedy algorithm that solves the continuous knapsack problem, purchasing items in order of marginal value. +This method has lower computational complexity compared to directly solving the model and leads to a +simple policy that (i) sorts items based on their marginal effect on the total cost and (ii) determines order +quantities according to this ranking until the budget is spent. +Key words : distributionally robust optimization, multi-item newsvendor model, knapsack problem, +minimax analysis, inventory management +History : This paper was first submitted on March 8, 2022. +1. +Introduction +The newsvendor model is one of the cornerstones of inventory management, introduced by +Arrow et al. (1951) for finding the order quantity that minimizes expected costs in view +of unknown demand and the trade-off between leftovers and lost sales. The newsvendor +model finds many applications in e.g. perishable food, fashion and high-tech industries, +particularly when the total time span of production and lead times exceeds the market +lifetime of a product; see Nahmias (1982) and Fisher and Raman (1996). +Manufacturers and retailers need to decide how to employ the available budget or re- +sources when determining the optimal order quantities of different products. A budget +constraint makes the problem multidimensional—as ordering more of one item leaves less +budget for other items—and gives rise to a challenging optimization problem. Hadley and +Whitin (1963) solve this problem with Lagrangian optimization. Abdel-Malek et al. (2004) +and Lau and Lau (1996) provide alternative solution methods, Erlebacher (2000) estab- +lishes closed-form solutions for special demand distributions and Nahmias and Schmidt +1 +arXiv:2301.02662v1 [math.OC] 5 Jan 2023 + +2 +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +(1984) develop heuristic solutions. All these works are for the full information setting, +where the demand distributions for all items are fully specified. In this paper we perform +a distribution-free analysis of the multi-item newsvendor problem with budget constraint. +This analysis does not rely on full specification of the demand distributions, but only re- +quires for each item knowledge of the mean, mean absolute deviation (MAD) and range. +Given this partial demand information, we obtain a robust ordering policy by employing +distributionally robust optimization (DRO) methods. +The newsvendor model in this paper seeks to minimize the expected costs as function +of the order quantity. The cost function depends on the order quantity, but also on the +demand, which is a random variable with some distribution. Given the demand distribu- +tion, the single-item newsvendor model finds the optimal order quantity that minimizes +the expected costs. In traditional approaches, the demand distribution is fully specified, +so that the expected costs can be calculated, and the optimal order quantity can be deter- +mined. A robust version of this problem assumes partial information, and only knows that +the demand distribution belongs to some ambiguity set that contains all distributions that +comply with this partial information. We adopt a minimax strategy that can be viewed as +a game between the newsvendor and nature: the newsvendor first picks the order quantity +after which nature chooses a demand distribution that maximizes the expected costs. The +goal then becomes to solve this minimax problem. +The way we solve this minimax problem in this paper fits in a much richer class of DRO +approaches that first calculate worst-case model performance, over the set of distributions +satisfying some partial information, and then optimize against these worst-case circum- +stances. Such DRO techniques found applications in many domains including scheduling +(Kong et al., 2013; Mak et al., 2014), portfolio optimization (Popescu, 2007; Delage and +Ye, 2010), pricing (Elmachtoub et al., 2021; Chen et al., 2022; Kleer and van Leeuwaarden, +2022), complex networks (van Leeuwaarden and Stegehuis, 2021), and inventory manage- +ment (Scarf, 1958; Gallego, 1992; Perakis and Roels, 2008; Ben-Tal et al., 2013). A classic +distributionally robust approach is due to Scarf (1958), who considered the single-item +newsvendor problem with mean-variance demand information. Scarf was able to derive +explicit expressions for the worst-case distribution, and solved the minimax problem to +obtain the optimal order quantity. Whether a minimax problem is solvable depends on +both the function to be optimized and the choice of ambiguity set. There are many ways + +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +3 +to characterize a set of distributions. In DRO, one can define ambiguity by using distance- +based metrics, such as total variation or Kullback-Leibler distance. Another popular class +of ambiguity uses summary statistics. The ambiguity set studied in this paper contains all +distributions with known mean and MAD. The maximization part of the minimax problem +can then be viewed as a semi-infinite linear optimization problem with three constraints, +and an infinite number of variables (all distributions in the ambiguity set). In fact, such +minimax problems are related to generalized moment bound problems, for which general +theory says there exists an extremal distribution solving the maximization part with at +most a number of support points equal to the number of moment constraints (Rogosinski, +1958). See Rahimian and Mehrotra (2019) for overviews of many more DRO applications +and techniques. +For the multi-item newsvendor model in this paper, we solve the multi-dimensional mini- +max problem with a random vector that describes the demand for all items. Compared with +tractable one-dimensional problems such as the single-item newsvendor model, applying +DRO techniques to such problems with multiple random variables might present consider- +able challenges in terms of computational complexity. For example, given information on +the mean and covariance of the demands, the distributionally robust multi-item newsvendor +is significantly harder to solve than its single-item counterpart (Hanasusanto et al., 2015). +However, for the multi-item newsvendor model in conjunction with mean-MAD ambiguity, +solving the minimax problem becomes tractable, and in fact has an elegant algorithmic +solution. The key insight will prove to be that the worst-case demand distribution—the +solution to the maximization part of the minimax problem—is identical for any order +quantity. As a result, the minimax problem reduces to a known-distribution optimization +problem. This known distribution is in fact, for each item, a unique three-point distribu- +tion. In turn, the minimization problem with this known (discrete) distribution can be +solved using a reduction to a knapsack problem. +The main contributions of this paper are as follows: +(i) Solution of minimax problem. We solve the minimax problem for mean-MAD ambi- +guity and a budget constraint. We first show that the worst-case scenarios arise when +item demands follow specific three-point distributions that comply with the partial +demand information. We minimize the associated worst-case costs to obtain a robust + +4 +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +ordering policy as the solution to a knapsack problem. As opposed to existing meth- +ods for the newsvendor model under full demand information, the knapsack problem +leads to an effective closed-form ordering policy, also for scenarios with many items. +As such, the present paper further develops DRO theory that uses MAD information +to formulate tractable minimax problems. +(ii) Budget consistency. The robust ordering policy only depends on the minimal, mean +and maximal demand for each item. Hence, the worst-case distributions are indepen- +dent of all other model parameters, which makes the robust ordering policy ‘budget +consistent’. When the budget is increased, the orders for the original budget remain +unaltered, while only the additional budget is further divided over the items. Such +budget consistency is useful because the optimization model needs to be solved only +once. That is, for the initial budget value the decision maker can generate an ordered +list of items as the solution to the knapsack problem, using only standard spreadsheet +software, and this solution is valid for all budget levels. In contrast, most other exact +and robust methods for the multi-item newsvendor model do not have this feature, +which means that the decision maker has to recompute the optimal policy for each +budget level. +(iii) Performance of ordering policy. Through a range of numerical examples we demon- +strate the performance of the knapsack ordering. We draw comparisons with full infor- +mation settings and other robust approaches that require partial demand information +by assessing the so-called expected value of additional information (EVAI). Overall, +the performance of the robust policy only deviates a few percent from the optimal +performance with full information availability. We also quantify the value of MAD +information by comparing the performance with the situations when only the mean +and range of demand is known, and show that MAD indeed provides crucial infor- +mation for providing good performance. In addition, we construct an ordering policy +that attains the optimal value of a matching minimin problem which, in conjunction +with the optimal value of the minimax problem, yields tight performance guarantees. +We next discuss some related literature on the newsvendor model. Gallego and Moon +(1993) consider the multi-item newsvendor model with budget constraint when the mean +and variance of demand is known. Gallego and Moon (1993) extend the ideas in Scarf +(1958) to obtain an optimization problem that can be solved with Lagrange multiplier + +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +5 +techniques, similar to the full information setting with a known distribution. In contrast, +our minimax analysis with mean-MAD-range information yields a knapsack ordering pol- +icy that generates a sorted list and prescribes to sort items successively according to that +list, with order sizes equal to the minimal, mean or maximum demand. Other related +works that consider the multi-item newsvendor model under partial information include +Vairaktarakis (2000), who assumes only the support of demand is known, and Ardestani- +Jaafari and Delage (2016) who assume knowledge of partial moments and rephrase the +robust optimization problem as a tractable linear program. Natarajan et al. (2018) assume +knowledge of mean, variance and semivariance, for which the newsvendor model is solvable +in the single-item setting using a semi-infinite linear program, but largely intractable in +the multi-item setting. Natarajan et al. (2018) therefore consider a relaxation that gives +a semidefinite program (SDP) to find a lower bound (which is not tight). Hanasusanto +et al. (2015) consider mean and covariance knowledge. They prove that the distributionally +robust problem is NP-hard but admits a semidefinite programming formulation with an ex- +ponential number of inequalities (that grows in the number of items). Xu et al. (2018) and +Natarajan and Teo (2017) present more tractable bounds for mean-covariance information. +In the present paper we assume only marginal information is available, since covariance +information and other dependency structures are difficult to estimate, and fixing covari- +ance information often leads to difficult optimization problems with non-intuitive solutions +(policies). The knapsack ordering policy that we obtain in this paper deals with the worst- +case demand distributions among all demand distributions with a given mean, MAD and +range, not conditioning on a specific dependency structure. This approach makes the knap- +sack ordering policy robust, but also suitable for scarce-data settings, as the mean, MAD +and range are relatively easy to estimate. +Section 2 introduces the single-item model and the multi-item model with budget, under +the traditional assumption of full information about the demand distributions. In Section 3 +we present our main results for the distributionally robust setting with partial information. +Section 4 presents a detailed numerical study that demonstrates the robust policies. We +present conclusions and several directions for future work in Section 5. Supplementary +material appears in the Electronic Companion (EC), including several proofs, additional +numerical experiments, and model extensions. + +6 +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +2. +Classical newsvendor analysis +We introduce the newsvendor model and several well-known results in Section 2.1 for the +single-item setting, and in Section 2.2 for the multi-item setting with budget constraint. +2.1. +Classical single-item setting +Consider an item with purchase price c and selling pricing p. The decision maker places +an order of size q. The demand for items is assumed to be the random variable D with +distribution function FD(·). Unsold items will be salvaged at the end of the period for +salvage value s per item. The mark-up m > 0 represents the profit per sold item and +satisfies p = c(1 + m) and the discount factor d > 0 captures the loss through s = (1 − d)c. +The expected costs consist of two terms: opportunity costs of lost sales and overage costs +in case of overstocking. This gives the cost function +G(q,D) = +� +� +� +� +� +(p − c)(D − q) +if q ⩽ D, +(c − s)(q − D) +if q > D. +(1) +The case q ⩽ D amounts to lost sales and q > D results in overstocking. The objective is to +order the quantity q of items that minimizes the expected costs. Let E denote expectation, +and define µ = E[D] and x+ = max(x,0). Write the expected costs as +C(q) := E[G(q,D)] = (c−s)q+(p−s)E(D−q)+−(c−s)µ = c +� +d(q − µ) + (m + d)E(D − q)+� +. +(2) +To keep notation simple (and without loss of generality) set c = 1. Then, the optimal order +quantity +q∗ = argmin +q⩾0 +C(q) ≡ argmin +q⩾0 +dq + (m + d)E(D − q)+, +(3) +is given by +q∗ = inf +� +q : F(q) ⩾ +m +m + d +� +. +(4) +A proof of (4) is provided in most standard textbooks on inventory management; see e.g. +Hadley and Whitin (1963); Silver et al. (1998); Nahmias (2009). +2.2. +Multi-item setting +Consider n different items and order qi units for item i for a given period where i = 1,...,n. +For item i, the unit purchasing and selling price are ci and pi respectively. Possible leftovers +will be salvaged at the end of the period for unit salvage value si. We define the model + +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +7 +in terms of the mark-up mi > 0 and discount factor di > 0. The mark-up represents the +profit per sold unit and the discount factor the loss, i.e. pi = ci(1 + mi) and si = (1 − di)ci. +The random demand for item i in one period is represented by the nonnegative random +variable Di, distributed according to Fi(·). +As in the single-item setting, we minimize the expected costs. Define the multi-item cost +function as +G(q,D) := +n +� +i=1 +ci +� +di(qi − Di) + (mi + di)(Di − qi)+� +. +(5) +We also introduce the budget constraint �n +i=1 ciqi ⩽ B with B the available budget. The +multi-item newsvendor model, with decision vector q = (q1,...,qn), is then given by +min +q +C(q) := E[G(q,D)] = +n +� +i=1 +ci +� +di(qi − µi) + (mi + di)E(Di − qi)+� +s.t. +n +� +i=1 +ciqi ⩽ B, +qi ⩾ 0, +i = 1,...,n. +(6) +Its solution, referred to as the optimal ordering policy, will be denoted by q∗. In the +single-item setting the purchase costs had no influence on the objective function, but in +the multi-item setting the optimal order quantity is affected by ci. It is well known that +model (3) is a convex optimization problem. In (6) we take the summation over n convex +functions, which preserves convexity. Moreover, the constraints form a convex set, so that +(6) is a convex optimization problem (Boyd and Vandenberghe, 2004). +3. +Proposed robust approach +Section 3.1 presents the robust ordering policy for the single-item setting. This result serves +as building block for the robust analysis of the multi-item setting in Section 3.2, which +describes the optimal policy as the solution of a linear program (LP). In Section 3.3 we +show that this LP can be viewed as a knapsack problem. All these results are based on a +tight upper bound for the cost function. In Section 3.4 we derive a matching tight lower +bound for the cost function. +3.1. +Distribution-free ordering policy for single item +Let P denote a probability distribution, and write EP for E to emphasize that the expec- +tation is taken with respect to the distribution P of D. The MAD for random demand D + +8 +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +is defined as δ := EP|D − µ|, where µ is the expected value of D. Similar to the variance, +the MAD is a measure of dispersion or variability. We mention several properties of MAD +in EC.2. For the random variable D with mean µ, MAD δ, and (bounded) support [a,b], +where 0 ⩽ a ⩽ b < ∞, the mean-MAD ambiguity set is defined as +P(µ,δ) := {P|EP[D] = µ, EP|D − µ| = δ, supp(D) ⊆ [a,b]}. +We thus assume that the ‘true’ distribution ˜P of the random demand D is contained in +this ambiguity set, that is, ˜P ∈ P(µ,δ). +To obtain the robust order quantity, we solve +min +q +max +P∈P(µ,δ) dq + (m + d)EP(D − q)+, +for which we first consider maxP∈P(µ,δ) EP(D − q)+. To characterize this tight bound, we +apply a general upper bound for convex functions of a random variable by Ben-Tal and +Hochman (1972). To make this paper self-contained, we provide a proof of the following +result in EC.1. +Lemma 1. The extremal distribution that solves +max +P∈P(µ,δ) EP(D − q)+ is a three-point dis- +tribution on the values a, µ and b that does not depend on q. +From the proof of Lemma 1, it follows that the worst-case probability distribution of D, +the extremal distribution that solves maxP∈P(µ,δ) EP(D − q)+, is a three-point distribution +defined as +P(D = x) = +� +� +� +� +� +� +� +� +� +� +� +� +� +δ +2(µ − a), +for x = a, +1 − +δ +2(µ − a) − +δ +2(b − µ), for x = µ, +δ +2(b − µ), +for x = b. +(7) +Applying this worst-case distribution, the robust order quantity follows from solving +qU = argminq CU(q) with +CU(q) := d(q − µ) + δ(m + d) +2(µ − a) (a − q)+ + (m + d) +� +1 − +δ +2(µ − a) − +δ +2(b − µ) +� +(µ − q)+ ++ δ(m + d) +2(b − µ) (b − q)+. +(8) +To illustrate the mean-MAD bound and robust order quantity qU, consider an example in +which D is distributed according to a beta distribution with both shape parameters set + +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +9 +to 1. For a general beta distribution, a = 0 and b = 1. In Figure 1a, we have m = 1 and +d = 0.8. This leads to qU = µ. In Figure 1b, the mark-up increases to m = 3. In this case +the mean-MAD order quantity increases to qU = b. When computing this upper bound, +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Order quantity +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +Expected costs +Beta +Mean-MAD bound +Mean-variance bound +(a) m = 1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Order quantity +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Expected costs +Beta +Mean-MAD bound +Mean-variance bound +(b) m = 3 +Figure 1 +Mean-MAD and mean-variance bounds and corresponding ordering policies. The upper curve corre- +sponds to the mean-variance upper bound that follows from P(1/2,1/12). The middle curve depicts the +mean-MAD upper bound. The ‘true’ cost function assumes that D follows a beta distribution with +both shape parameters equal to 1 (the lower curve). +observe that the mean-MAD bound touches the ‘true’ cost function in the points a,µ and b. +This property actually holds in general. Clearly, for q = a or b, it holds that CU(q) = C(q). +When q = µ, the cost function equals +C(µ) = d(µ − µ) + (m + d)E(D − µ)+ = δ(m + d) +2 += CU(µ), +since E(D − µ)+ = E|D − µ|/2. +By analyzing (8) one can obtain an explicit ordering rule for qU. The objective func- +tion of (8) is composed of piecewise linear functions. By exploiting this structure, we +can construct an explicit ordering policy. For scalars α1,...,αm,ν1,...,νm ∈ R, f(x) = +maxi=1,...,m{αix+νi} denotes a convex, piecewise linear function. The function CU(q) in (8) +admits a representation of the form +CU(q) = d(q − µ) + (m + d)E(D − q) = m(µ − q) =: f0(q), +for q ∈ [0,a) and +CU(q) = d(q − µ) + (m + d) +� +1 − +δ +2(µ − a) − +δ +2(b − µ) +� +(µ − q) + δ(m + d) +2(b − µ) (b − q) += q(δ(m + d) +2(µ − a) − m) + ν1 =: f1(q), + +10 +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +for q ∈ [a,µ), where ν1 is some constant value. For q ∈ [a,µ), the mean-MAD objective +function is defined by the linear function f1(q). For the interval q ∈ [µ,b], we obtain +CU(q) = d(q − µ) + δ(m + d) +2(b − µ) (b − q) = q +� +d − δ(m + d) +2(b − µ) +� ++ ν2 =: f2(q) +for some constant ν2. The cost function is thus the pointwise maximum of the three linear +functions f0(q), f1(q) and f2(q): +CU(q) = max{f0(q), f1(q), f2(q)}. +Since CU(q) = maxj=0,1,2{αjq + νj} is a convex function, it holds that α0 ⩽ α1 ⩽ α2. Since +we assume that m > 0, we know that α0 < 0. Therefore, from the derivatives α1, α2 of +CU(q), we can derive an explicit order quantity by examining for which linear piece the +slope turns positive. This allows us to state Theorem 1. +Theorem 1 (Mean-MAD order quantity). The +robust +order +quantity +qU +∈ +argminq CU(q) is given by +(a) If m < +δd +2(µ − a) − δ, then qU = a. +(b) If +δd +2(µ − a) − δ < m < d(2(b − µ) − δ) +δ +, then qU = µ. +(c) If d(2(b − µ) − δ) +δ +< m, then qU = b. +(d) If m = +δd +2(µ − a) − δ and m = d(2(b − µ) − δ) +δ +, then qU ∈ [a,µ] and qU ∈ [µ,b], respec- +tively. +According to Theorem 1, the robust order quantity qU for mean-MAD-range information +consists of three predictable values (minimal, mean, maximum demand) that do not depend +on the mark-up m and discount factor d, whereas the conditions that dictate how much +to order do depend on them (in addition to the demand mean, MAD and range). +3.2. +Multiple items and budget constraint +A distribution-free analysis of the multi-item model requires a multivariate ambiguity set. +As in the single-item case, the partial information is the mean µi, MAD δi and support +supp(Di) = [ai,bi] for each random variable Di, i = 1,...,n. The mean-MAD ambiguity set +is defined as +P(µ,δ) := {P|EP (Di) = µi, EP |Di − µi| = δi, supp(Di) ⊆ [ai,bi], ∀i}. +(9) + +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +11 +We henceforth assume that the distribution of the vector of random variables D = +(D1,...,Dn) belongs to this ambiguity set, i.e., P ∈ P(µ,δ). Since the objective function in +(6) is separable, one can apply the single-item bound to each term E(Di − qi)+ in the +summation individually. The following result, for the multi-item problem, is then a direct +consequence of Lemma 1. +Lemma 2. The extremal distribution that solves max +P∈P(µ,δ) EP[G(q,D)] consists for each Di +of a three-point distribution with values ξ(i) +1 = ai, ξ(i) +2 = µi, ξ(i) +3 = bi and probabilities +p(i) +1 = +δi +2(µi − ai), +p(i) +2 = 1 − +δi +2(µi − ai) − +δi +2(bi − µi), +p(i) +3 = +δi +2(bi − µi). +(10) +For the multi-item newsvendor model based on mean-MAD ambiguity, we use Lemma 2 +to solve the maximization part of +min +q:� +i ciqi⩽B,qi⩾0 max +P∈P(µ,δ) EP +� +n +� +i=1 +cidi(qi − µi) + ci(mi + di)(Di − qi)+ � +, +(11) +and obtain +min +q +n +� +i=1 +ci +� +di(qi − µi) + (mi + di) +� +p(i) +1 (ai − qi)+ + p(i) +2 (µi − qi)+ + p(i) +3 (bi − qi)+�� +s.t. +n +� +i=1 +ciqi ⩽ B, +qi ⩾ 0, +i = 1,...,n. +(12) +The objective function of (12) has a piecewise linear structure. Moreover, because of this +result and since the constraints are linear, (12) can be cast as a linear program (LP). In +particular, as explained below, the robust ordering policy qU can be found by solving +min +q +n +� +i=1 +max +j=0,1,2{αi,jqi + νi,j} +s.t. +n +� +i=1 +ciqi ⩽ B, +qi ⩾ 0, +i = 1,...,n, +(13) +where +αi,0 = −cimi, +νi,0 = cimiµi, +αi,1 = ci +�δi(mi + di) +2(µi − ai) − mi +� +, +νi,1 = ci(mi + di) +� +µi − +δiai +2(µi − ai) +� +− cidiµi, +αi,2 = ci +� +di − δi(mi + di) +2(bi − µi) +� +, +νi,2 = ciδi(mi + di)bi +2(bi − µi) +− cidiµi, +for i = 1,...,n. + +12 +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +Let fi,j(x) = αi,jx + νi,j for i = 1,...,n and j = 0,1,2. From the single-item case, we know +that the objective, for each item i, can be written as maxj=0,1,2{fi,j(qi)} with αi,0 ⩽ αi,1 ⩽ +αi,2, and thus the objective functions of (12) and (13) are equal, which makes the two +models equivalent. Since we know from linear programming theory that convex, piecewise +linear objective functions can be written as linear constraints, problem (13) admits an LP +representation (Boyd and Vandenberghe, 2004). +3.3. +Knapsack algorithm +It turns out that problem (13) is intimately related to the continuous knapsack problem, +thus making available efficient sorting-based algorithms to solve (13). We next describe an +efficient algorithm that determines the robust ordering policy. +Define the linear funtion fi,j for each item i, and let αi,j represent its derivative with +respect to qi, for items i = 1,...,n and linear pieces j = 0,1,2. That is, +dfi,j(qi) +dqi += αi,j. +For each item i, fi,0, fi,1 and fi,2 represent the marginal effect on the value of (13) when +we increase qi to ai,µi and bi respectively. The parameter αi,j represents the slope of these +linear functions and an order quantity is increased only when αi,j < 0, because otherwise +it will not reduce the expected costs. We consecutively allocate budget to the item that +causes the largest relative decrease in expected costs; that is, item k with the smallest +negative derivative αk,i relative to its cost ck. Define the set of all items as N = {1,...,n}. +Since only order quantities that decrease the expected costs are considered, define the +ordered set: +G := {(i,j) | αi,j < 0,i ∈ N,j ∈ {0,1,2}}, +(14) +where the ordering is determined according to the value of αi,j/ci. For m = |G|, this +ordering is represented by the sequence (i1,j1),...,(im,jm) for which it holds that +αi1,j1/ci1 ⩽ ··· ⩽ αim,jm/cim. Here G contains tuples (i,j) for which i represents an item in +the newsvendor model and j a linear piece of the piecewise function. As these functions +are convex, the linear pieces appear for each item i in increasing order in the set G. We +can now state the knapsack algorithm for the distribution-free multi-item newsvendor +model. + +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +13 +Algorithm 1 (Knapsack algorithm). For a budget level B ⩾ 0, the ordering policy +qU is found by the following procedure: +(i) Initialize by setting q = (0,...,0), and construct G. Continue to (ii). +(ii) Select the first element (i,j) ∈ G. If the set G is empty, the optimal solution is qU = q. +Otherwise, continue to (iii). +(iii) If j = 0, set qi = ai. If j = 1, set qi = µi. If j = 2, set qi = bi. Continue to (iv). +(iv) Determine whether the budget constraint �n +i=1 ciqi ⩽ B is violated. If so, set qi such +that ciqi = B − � +k∈N|k̸=i ckqk, and the optimal solution is qU = q. Otherwise, remove +element (i,j) from G and return to step (ii). +This algorithm yields an optimal solution to (13), as asserted in the following theorem. +Theorem 2 (Knapsack ordering policy). The robust ordering policy qU that solves +the multi-item newsvendor model (13) is determined by Algorithm 1. +Proof. +To prove that this algorithm produces an optimal solution, we construct a con- +tinuous knapsack problem that solves (13). In the following, (ik,jk) corresponds to the kth +entry of the ordered sequence of items in G. Define the following auxiliary model: +min +x +m +� +k=1 +pkxk +s.t. +m +� +k=1 +ckxk ⩽ B, +0 ⩽ xk ⩽ uk +∀k = 1,...,m, +(15) +where +uk = +� +� +� +� +� +� +� +aik, +for jk = 0 +µik − aik, for jk = 1 +bik − µik, for jk = 2 +and pk = αik,jk and ck = cik. From the order of the sequence, it follows that p1/c1 ⩽ ... ⩽ +pm/cm. Assume that (x∗ +1,...,x∗ +m) is an optimal solution to optimization problem (15). +For i ∈ N, let qU +i = � +k=1,...,m|i=ik x∗ +k. Since αi,0 ⩽ αi,1 ⩽ αi,2, the pieces jk appear in G in +increasing order for each item i. Thus, in an optimal solution, uik,jk will only be attained if +its predecessor uik,jl is also attained. By construction, qU is feasible for (13). Moreover, the +objective values of problems (13) and (15) only differ by a constant term, so both problems +have the same optimal solution. For the continuous knapsack problem, a greedy allocation +produces an optimal solution (see EC.3). Hence, qU = (qU +1 ,...,qU +n ) is optimal for (13). +□ + +14 +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +Theorem 2 shows that there exists a ranking for the selection of items. Take an initial +budget B = 0. If we increase the budget B by some small value, we first increase item i +to ai for the item that has the highest mark-up mi. This makes sense intuitively because +the product with the highest mark-up is most profitable and, since qi < ai, we have no +risk of overstocking. We successively select the items with the greatest marginal benefit +αi,j/ci, and increase the order quantity consecutively to either ai, µi or bi. This procedure +continues until we have spent the entire budget, or reached the uncapacitated optimum. +Items that are ordered in the beginning of this procedure have the largest impact on the +decrease in costs for the multi-item newsvendor model. +As the main complexity of the knapsack algorithm in Theorem 2 stems from sorting +the set G, the greedy approach is of computational complexity O(nlog n). Moreover, the +solution can be found in O(n) time by first identifying the critical element (is,js) that will +violate the budget constraint, as proposed by Balas and Zemel (1980) for the continuous +knapsack problem. One then compares each αi,j/ci with the ratio of the critical element to +determine the optimal allocation of budget to the items. The optimal solution can also be +found through the LP (13), which we solve with the simplex method. We remark that a +single iteration of the simplex method takes O(n2) arithmetic operations (Ill´es and Terlaky, +2002), which exceeds the time requirement of the knapsack algorithm. +3.4. +A matching lower bound +The robust analysis so far was based on finding a tight upper bound on the cost function +when we know the mean, MAD and range of the demand distributions. When additional +information is available, we can also construct a matching lower bound. We include the +skewness information βi = P(Di ⩾ µi) in the mean-MAD ambiguity set to obtain the tight +lower bound. For the random variables D = (D1,...,Dn), define the ambiguity set as +P(µ,δ,β) := {P|P ∈ P(µ,δ), P(Di ⩾ µi) = βi, i = 1,...,n} +with P(µ,δ,β) ⊆ P(µ,δ). The proof of the following result is identical to that of Lemma 2, +but now uses the tight lower bound for a convex function of random variables discussed in +Ben-Tal and Hochman (1972). To make this paper self-contained, a proof for the univariate +case is provided in EC.1. This is sufficient since the univariate result can be applied to +each term of the summation in G(q,D) separately, as with Lemma 2. + +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +15 +Lemma 3. The extremal distribution that solves +min +P∈P(µ,δ,β) EP[G(q,D)] consists for each +Di of a two-point distribution with values µi + δi +2βi, µi − +δi +2(1−βi) and probabilities βi, 1 − βi, +respectively. +Using this result, we obtain +min +q +CL(q) := +n +� +i=1 +ci +� +di(qi − µi) + (mi + di) +� +βi(µi + δi +2βi +− qi)+ + (1 − βi)(µi − +δi +2(1 − βi) − qi)+ +�� +s.t +n +� +i=1 +ciqi ⩽ B, +qi ⩾ 0, +for i = 1,...,n, +(16) +as a model to provide a lower bound for the multi-item newsvendor. As the objective +function in problem (16) also consists of piecewise linear functions, there exists an LP +representation and knapsack algorithm for (16) similar to the results for problem (12). +We can now solve (13) and (16) to obtain tight performance intervals for the multi-item +newsvendor model, using recent DRO results (see EC.4 and Postek et al., 2018). For all +feasible ordering policies q and P ∈ P(µ,δ,β), it holds that +C(q) ∈ +� +CL(q),CU(q) +� +. +In addition, for the optimal solutions to the newsvendor problem and its distributionally +robust counterparts, +C(q∗) ∈ +� +CL(qL),CU(qU) +� +. +One can find the tightest upper and lower bounds, based on mean-MAD ambiguity, for +the multi-item newsvendor model by calculating the optimal solutions to models (12) and +(16), respectively. +4. +Numerical examples of robust ordering +We will now illustrate and visualize the robust ordering policies. To demonstrate the +‘budget-consistency’ property, Section 4.1 applies the knapsack algorithm for a setting +where the budget is increased. In Section 4.2 we contrast the performance of the knapsack +policy for partial demand information against that of the optimal solution for the full +information setting. Our code is made available in the form of an online supplement. + +16 +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +4.1. +Numerical illustration of the ‘budget-consistency’ property +We illustrate the knapsack algorithm and the process of allocating budget to different order +quantities for items in the newsvendor model. Consider n = 5 identically distributed items +with support a = 10, b = 50 and mean µ = 30. From Figure 2, we can infer that item 1 +is the most profitable. Low budget levels are allocated to this item such that we obtain +q1 = µ. Item number 3 is the last item to which the budget is allocated. Hence, it is the +least profitable item. Table 1 displays the ordered set G. From this table, we can indeed +infer that item 1 has the smallest value for αi,0/ci and therefore is increased first. +0 +50 +100 +150 +200 +250 +300 +Budget +0 +10 +20 +30 +40 +50 +Order quantity +Item 1 +Item 2 +Item 3 +Item 4 +Item 5 +Figure 2 +Development of the order quantities when the budget increases according to the knapsack algorithm +Table 1 +Table containing αi,j/ci and corresponding information of the ordered set G +G +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +αi,j/ci +-0.92 -0.75 -0.72 -0.49 -0.3 -0.15 -0.1 -0.08 -0.03 -0.01 0.14 0.42 0.45 0.7 0.7 +Function piece +0 +1 +0 +1 +0 +0 +1 +2 +1 +0 +1 +2 +2 +2 +2 +Item +1 +1 +2 +2 +4 +5 +4 +1 +5 +3 +3 +5 +2 +4 +3 +Figure 2 nicely illustrates that when the budget is increased, the orders for the original +budget remain unaltered, while only the additional budget is further divided over the items. +To further illustrate the ‘budget-consistency’ property, consider the multi-item newsvendor +model for which n = 2, m2 = 2, the remaining cost parameters equal 1, and demand is +identically distributed according to a symmetric triangle distribution supported on [10,50]. +In Figure 3 we plot the expected costs and order quantities for various budget levels. + +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +17 +Figure 3a contains the allocation between both order quantities. For low budget values, one +first increases the order quantity of item one, the most profitable item. Figure 3b shows the +upper bound (12) and lower bound (16) that together lead to a tight performance interval +for the expected costs. +For the sake of comparison, we also show results for the partial demand information +setting considered in Gallego and Moon (1993), assuming that the mean and variance of +demands are known; see EC.5 for more details. The results of Gallego and Moon (1993) de- +pend (non-trivially) on all model parameters, including the budget B. This lack of budget- +consistency forces the decision maker to solve an optimization problem, see (EC.13), for +each budget level separately, and explains the smooth curve in Figure 3a. In contrast, +our knapsack algorithm generates a sorted ordering list that does not depend on B, and +prescribes to sort items successively according to that list, with order sizes equal to the +minimal, mean or maximum demand. +0 +5 +10 +15 +20 +25 +30 +Order quantity of item 1 +0 +5 +10 +15 +20 +25 +30 +35 +Order quantity of item 2 +Optimal order quantity +Mean-MAD policy +Mean-variance policy +(a) Ordering policy +0 +10 +20 +30 +40 +50 +60 +Budget +10 +20 +30 +40 +50 +60 +70 +80 +90 +Expected costs +Triangular +Mean-MAD lower bound +Mean-MAD upper bound +Mean-variance bound +(b) Newsvendor costs +Figure 3 +Mean-variance and mean-MAD bounds and ordering policies for the newsvendor model. The mean- +variance curves are obtained through solving (EC.13). The mean-MAD policy corresponds to the optimal +solution of (12). The mean-MAD upper and lower bounds correspond to the extremal three- and two- +point distributions, respectively. The ‘true’ cost function assumes that D follows a symmetric triangular +distribution on [10,50]. +We emphasize that these results are not meant to numerically compare the mean-MAD +and mean-variance policies, because the displayed differences merely express different ways +of dealing with ambiguity. Indeed, it is hard to compare both policies as the respective +ambiguity sets can contain vastly different distributions. For instance, a finite variance +excludes distributions with an infinite second moment, while finite MAD does not. For + +18 +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +our purposes, MAD and variance are equally adequate descriptors of dispersion, and both +are easily calibrated on data using basic statistical estimators. The crucial difference in +the DRO context of this paper is that MAD leads to a simple, budget-consistent ordering +policy. +4.2. +Expected value of additional information +We introduce as performance measure the expected value of additional information (EVAI), +defined as +EVAI(qU +B) = C(qU +B) − C(q∗ +B) +C(q∗ +B) +, +where qU +B is the robust ordering policy and q∗ +B is the optimal ordering policy when the +joint demand distribution is known. We let B run from 0 to �n +i=1 q∗ +i =: Bopt, and consider +nine different demand distributions, listed in Table 2. +Table 2 +Nine distributions used for multi-item performance analysis +Case +Case +Case +1 +Uniform[10,50] +4 +Beta(1,3) on [0,50] +7 +Triangular(10,50,18) +2 +Uniform[10,100] +5 +Beta(2,2) on [0,50] +8 +Triangular(10,50,30) +3 +Uniform[10,200] +6 +Beta(3,1) on [0,50] +9 +Triangular(10,50,42) +We consider n = 25 items. For each item i, let ci = di = 1 and assume identically dis- +tributed demand. For example, in Case 2 the demand Di for each item i follows the uniform +distribution with parameters ai = 10 and bi = 100. Table 3 provides an overview for the +mark-up, representing low, average and high margins. +For the low margin regime, Figure 4 shows results for each of the nine cases, for both the +robust ordering policy with mean-MAD-range information, and for the policy that uses +the additional information βi = P(Di ⩾ µi). For the former, the worst performance over all +nine cases has a maximum deviation of approximately 23% compared to the optimal order +quantity q∗ +B. Overall, the performance of the robust policy only deviates a few percent from +the optimal performance with full information availability. For the uniformly distributed +cases (Cases 1-3), the performance decreases when the range increases. For beta distributed +demand (Cases 4-6), right-tailed distributions perform worse than left-tailed distributions. +This effect is also observed for the triangular distributions (Cases 7-9). The policy with +additional information βi = P(Di ⩾ µi) performs somewhat better in most cases. + +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +19 +Table 3 +Mark-up values for all 25 items in the newsvendor model +Mark-up +m1 +m2 +m3 +m4 +m5 +m6 +m7 +m8 +m9 +m10 +m11 +m12 +m13 +Low margin +0.1 +0.14 0.18 0.21 0.25 0.29 0.33 0.36 +0.4 +0.44 0.48 0.51 0.55 +Average margin +1 +1.13 1.25 1.38 +1.5 +1.63 1.75 1.88 +2 +2.13 2.25 2.38 +2.5 +High margin +4 +4.21 4.42 4.63 4.83 5.04 5.25 5.46 5.67 5.88 6.08 6.29 +6.5 +Mark-up +m14 +m15 +m16 +m17 +m18 +m19 +m20 +m21 +m22 +m23 +m24 +m25 +Low margin +0.59 0.63 0.66 +0.7 +0.74 0.78 0.81 0.85 0.89 0.93 0.96 +1 +Average margin 2.63 2.75 2.88 +3 +3.13 3.25 3.38 +3.5 +3.63 3.75 3.88 +4 +High margin +6.71 6.92 7.12 7.33 7.54 7.75 7.96 8.17 8.37 8.58 8.79 +9 +Figure 5 shows similar results for high margins. The EVAI for the robust policy remains +mostly below 10% for lower budget levels, but starts increasing rapidly when the budget +approaches Bopt (i.e., when approaching the unconstrained model). When the budget is less +restrictive, additional distributional information provides substantial value. In particular, +since the policy uses skewness information βi, it performs better (in expectation) for higher +budget levels than the robust ordering policy. We present some more performance plots +for the average margin setting and additional numerical experiments with mean-variance +information in EC.6. +We next quantify the value of MAD information by comparing the performance with +the situations when only the mean and range of demand is known. For the low margin +setting, Figure 6 shows the EVAI for the ordering policy with only mean-range informa- +tion. Like the mean-MAD policy, this policy follows from a discrete distribution, in this +case the extremal distribution on {a,b} with probabilities b−µ +b−a and µ−a +b−a that attains the +Edmundson-Madansky bound (see Ben-Tal and Hochman, 1972). That is, instead of the +worst-case three-point distribution, we take the expectation in (6) over this two-point dis- +tribution and find the robust mean-range ordering policy using the resulting LP. The plots +clearly demonstrate that knowledge on dispersion in terms of MAD improves performance +considerably. + +20 +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +0 +224 +449 +674 +0.00 +0.05 +0.09 +0.14 +Case 1: Uniform (10,50) +Mean-MAD +Mean-MAD- +0 +401 +802 +1204 +0.00 +0.05 +0.09 +0.14 +Case 2: Uniform (10,100) +0 +755 +1510 2265 +0.00 +0.05 +0.09 +0.14 +Case 3: Uniform (10,200) +0 +70 +140 +211 +0.00 +0.06 +0.11 +0.17 +Case 4: Beta (1,3) +0 +187 +374 +561 +0.00 +0.07 +0.13 +0.20 +Case 5: Beta (2,2) +0 +312 +624 +937 +0.00 +0.06 +0.12 +0.18 +Case 6: Beta (3,1) +0 +190 +380 +571 +0.00 +0.08 +0.15 +0.23 +Case 7: Triangular (10,50,18) +0 +236 +472 +709 +0.00 +0.07 +0.13 +0.20 +Case 8: Triangular (10,50,30) +0 +277 +554 +831 +0.00 +0.06 +0.11 +0.17 +Case 9: Triangular (10,50,42) +Figure 4 +The results for the low margin setting. The x-axis corresponds to B and the y-axis to the EVAI. +0 +370 +740 +1110 +0.00 +0.11 +0.23 +0.34 +Case 1: Uniform (10,50) +Mean-MAD +Mean-MAD- +0 +729 +1458 2187 +0.00 +0.11 +0.23 +0.34 +Case 2: Uniform (10,100) +0 +1446 2892 4339 +0.00 +0.12 +0.23 +0.35 +Case 3: Uniform (10,200) +0 +201 +403 +605 +0.00 +0.21 +0.42 +0.63 +Case 4: Beta (1,3) +0 +319 +638 +958 +0.00 +0.18 +0.37 +0.55 +Case 5: Beta (2,2) +0 +396 +792 +1188 +0.00 +0.11 +0.23 +0.34 +Case 6: Beta (3,1) +0 +306 +612 +918 +0.00 +0.18 +0.36 +0.54 +Case 7: Triangular (10,50,18) +0 +329 +658 +987 +0.00 +0.19 +0.38 +0.57 +Case 8: Triangular (10,50,30) +0 +361 +722 +1084 +0.00 +0.20 +0.41 +0.61 +Case 9: Triangular (10,50,42) +Figure 5 +The results for the high margin setting. The x-axis corresponds to B and the y-axis to the EVAI. + +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +21 +0 +224 +449 +674 +0.00 +0.26 +0.51 +0.77 +Case 1: Uniform (10,50) +Mean-MAD +Mean-MAD- +E-M +0 +401 +802 +1204 +0.00 +0.26 +0.51 +0.77 +Case 2: Uniform (10,100) +0 +755 +1510 2265 +0.00 +0.26 +0.51 +0.77 +Case 3: Uniform (10,200) +0 +70 +140 +211 +0.00 +0.16 +0.32 +0.48 +Case 4: Beta (1,3) +0 +187 +374 +561 +0.00 +0.45 +0.90 +1.35 +Case 5: Beta (2,2) +0 +312 +624 +937 +0.00 +1.03 +2.07 +3.10 +Case 6: Beta (3,1) +0 +190 +380 +571 +0.00 +0.34 +0.69 +1.03 +Case 7: Triangular (10,50,18) +0 +236 +472 +709 +0.00 +0.54 +1.09 +1.63 +Case 8: Triangular (10,50,30) +0 +277 +554 +831 +0.00 +0.63 +1.26 +1.89 +Case 9: Triangular (10,50,42) +Figure 6 +The results for the low margin setting. The x-axis corresponds to B and the y-axis to the EVAI. The +E-M performance plot refers to the model with only mean information. +5. +Conclusions +This paper establishes new ordering policies for the newsvendor with partial demand in- +formation (mean, MAD and range) with a budget constraint. The ordering policies follow +from a minimax approach, where we search for the order quantities with minimal costs +for the maximal (worst-case) cost function restricted to demand distributions that comply +with the partial information. +The minimax analysis for the multi-item setting gives rise to a knapsack problem, and +the solution of this knapsack problem in fact is the ordering policy. This policy prescribes +to sort items based on their marginal effect on the total costs, reminiscent of the greedy +algorithm that solves the continuous knapsack problem. The ordering policy only orders +the minimum, mean or maximum demand for each item. Hence, the decision maker can +rank the items based on their marginal effects, and then start ordering items according to +this list until the budget is spent. The fact that the ranking list is easy to generate, and +that the ‘order of ordering’ does not depend on the budget, makes the policy transparent +and easy to implement. Existing approaches for full and partial (such as mean-variance) +knowledge of the demand distribution lack this property of ‘budget-consistency’. + +22 +Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +The minimax approach provides robustness, with an ordering policy that protects against +all distributions that comply with the partial information. This approach avoids the need +to estimate the demand distribution, which can be a daunting process in practice and +is prone to errors. However, the minimax approach comes at the risk of being overly +conservative. Through extensive numerical experiments we compared the robust policies +for partial demand settings with the policies for full demand settings, and observed that +the proposed policies perform well. +At the heart of our analysis lies the idea to set up the robust minimax analysis with +MAD information. With MAD as dispersion measure we obtained a tractable optimization +model, with a solution in terms of a robust ordering policy that satisfies the budget- +consistency property. Using MAD to formulate solvable minimax problems can also be +applied to other inventory models. We demonstrate this idea in EC.7 for three extended +settings: the newsvendor with multiple contraints, the newsvendor with unreliable supply, +and the risk-averse newsvendor. In all three cases, the minimax analysis leads to a tractable +mathematical program, either a knapsack problem or a linear program. +References +Abdel-Malek, L., Montanari, R., and Morales, L. C. (2004). 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Mathematical Programming, 137(1):167–198. + +e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec1 +E-Companion to “Robust knapsack ordering for a +partially-informed newsvendor with budget constraint” +EC.1. +Proofs +Proof of Lemma 1. +In their original work, Ben-Tal and Hochman (1972) prove this +result for general convex functions by dividing the support into two intervals [a,µ] and [µ,b] +and then applying the Edmundson-Madansky bound to both subintervals. The following +proof uses semi-infinite programming duality and is taken from van Eekelen et al. (2022). +Consider a general convex function f(x) (this includes (x − q)+ as a special case). For +X ∼ P ∈ P(µ,δ), we solve +max +P(x)⩾0 +� b +a +f(x)dP(x) +s.t. +� b +a +dP(x) = 1, +� b +a +xdP(x) = µ, +� b +a +|x − µ|dP(x) = δ, +(EC.1) +Consider the dual of (EC.1), +min +λ0,λ1,λ2 +λ0 + λ1µ + λ2δ +s.t. +M(x) := λ0 + λ1x + λ2|x − µ| ⩾ f(x), ∀x ∈ [a,b]. +(EC.2) +The function M(x) has a ‘kink’ at x = µ. Since the dual problem (EC.2) has three variables, +the optimal M(x) touches f(x) at three points: x = a, µ and b. For this choice of M(x), +λ0 = f(a) − λ1a − λ2(µ − a), λ1 = 1 +2 +�f(b) − f(µ) +b − µ ++ f(µ) − f(a) +µ − a +� +, +λ2 = 1 +2 +�f(b) − f(µ) +b − µ +− f(µ) − f(a) +µ − a +� +. +Because the majorant is piecewise linear and convex, we can majorize every convex function +f(x) by letting M(x) touch at the boundary points a,b and at the kink point x = µ. +According to the complementary slackness property, these points constitute the support +of the extremal distribution, and the optimal probabilities follow from solving the linear +system resulting from the equations of (EC.1). This is a linear system of three unknown +probabilities and three equations, with the solution +pa = +δ +2(µ − a), +pµ = 1 − +δ +2(µ − a) − +δ +2(b − µ), +pb = +δ +2(b − µ). +Finally, for these primal and dual solutions, we verify that the objective values of problems +(EC.1) and (EC.2) agree, which confirms that strong duality holds. +□ + +ec2e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +Proof of Lemma 3. +We prove this result for general convex f(x). For a random variable +X with distribution P ∈ P(µ,d,β), the tight lower bound follows from +max +P(x)⩾0 +� b +a +f(x)dP(x) +s.t. +� b +a +dP(x) = 1, +� b +a +xdP(x) = µ, +� b +a +|x − µ|dP(x) = δ, +� b +a +1{x⩾µ}dP(x) = β. +(EC.3) +Consider the dual of (EC.3), +min +λ0,λ1,λ2 +λ0 + λ1µ + λ2δ + λ3β +s.t. +M(x) := λ0 + λ1x + λ2|x − µ| + λ31{x⩾µ} ⩽ f(x), ∀x ∈ [a,b]. +(EC.4) +Here M(x) has both a ‘kink’ and a jump discontinuity at x = µ. Let the function M(x) +touch the epigraph of f(x) in two points on opposite sides of µ. If we insert this knowledge, +the constraints in the dual problem reduce to two equality constraints. From the Karush- +Kuhn-Tucker conditions, we deduce the optimal tangent points: +x1 = µ + δ +2β , +x2 = µ − +δ +2(1 − β), +which correspond to υ1 and υ2. Substituting this solution and solving for λ0,λ1,λ2 and λ3 +gives +λ0 = f(υ2) + (λ1 − λ2)δ +2(1 − β) − λ1µ, +λ3 = f(υ1) − f(υ2) + +λ2δ +(1 − β) − (λ2 + λ1)δ +2β(1 − β) , +and hence the optimal value is given by βf(υ1) + (1 − β)f(υ2). To ensure the solution is +dual feasible, we assign suitable values to the two free decision variables. That is, we let +λ1 + λ2 and λ1 − λ2 equal the slope of f(x) at x = υ1 and υ2, respectively. The optimal +probabilities of (EC.3) are obtained by solving the linear system resulting from (EC.3). +□ +EC.2. +Known properties of MAD +We recall some well-known properties of the MAD; see e.g. Ben-Tal and Hochman (1985). +Denote by σ2 the variance of the random variable X, whose distribution is known to belong +to the set P(µ,δ). Then +δ2 +4β(1 − β) ⩽ σ2 ⩽ δ(b − a) +2 +. + +e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec3 +In particular, since +δ2 ⩽ 4β(1 − β)σ2 ⩽ σ2, +it holds that δ ⩽ σ. For a proof, we refer the reader to Ben-Tal and Hochman (1985). For +the distributions used in the paper, explicit formulas for δ are available: +• Uniform distribution on [a,b]: +δ = 1 +4(b − a) +• Beta distribution with parameters k,λ on support [a,b]: +δ = +2kkλλΓ(k + λ) +(k + λ)k+λ+1Γ(k)Γ(λ)(b − a) +• Triangular distribution on [a,b] with mode c: +δ = +� +� +� +� +� +2(b+c−2a)3 +81(a−b)(a−c), +for a + b < 2c, +2(a+c−2b)3 +81(a−b)(b−c), +for a + b > 2c +• Normal distribution N(µ,σ2): +δ = +� +2 +πσ +• Gamma distribution with parameters λ and k (for which µ = k/λ): +δ = +2kk +Γ(k)exp(k) +1 +λ. +The MAD is known to satisfy the bound +0 ⩽ δ ⩽ 2(b − µ)(µ − a) +b − a +. +(EC.5) +Let β = P(X ⩾ µ). For example, in the case of continuous symmetric distribution of X we +know that β = 0.5. This quantity is known to satisfy the bounds: +δ +2(b − µ) ⩽ β ⩽ 1 − +δ +2(µ − a). +(EC.6) +EC.3. +The knapsack problem +The knapsack problem (Kellerer et al., 2004) is an integer programming problem and can +be formulated as +max +x +� +i=1 +pixi +s.t. +n +� +i=1 +cixi ⩽ B, +xi ∈ {0,1}, +1 = 1,...,n. +(EC.7) + +ec4e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +for decision variable x, budget B, price p > 0 and costs c. Assume B < �n +i=1 ci. The contin- +uous version is obtained by considering the linear relaxation, i.e., we replace the integrality +constraints by 0 ⩽ xi ⩽ 1, i = 1,...,n. The so-called greedy choice algorithm produces an +optimal solution for the continuous knapsack problem. +We first renumber the items xi such that p1/c1 ⩾ ... ⩾ pn/cn. Hence, the first item causes +the largest increase in value relative to its costs. We now iterate over x1,...,xn and in each +iteration, set xi to its maximum capacity. When the budget constraint is violated, set +xi = B − +i−1 +� +i=1 +cixi. +This greedy choice algorithm produces the optimal solution to (EC.7). Below we will state +its proof, which is an adaptation from the proof in Kellerer et al. (2004). +Assume that without loss of generality that p1/c1 > ··· > pn/cn. If we would have pi/ci = +pi+1/ci+1 for some i, then we are indifferent between those items and the proof below can +be easily adapted to satisfy this. The greedy choice algorithm produces a solution such +that, for some index j, we have 1 = x1 = ··· = xj−1 > xj ⩾ xj+1 = ··· = xn = 0. Suppose +we would have a different feasible optimal solution y ̸= x. Since pi > 0 and �n +i=1 ci > B, it +must hold that �n +i=1 ciyi = B as otherwise we could spend additional capital to increase +the optimal value. Because p1/c1 ⩾ ... ⩾ pn/cn, there exists a smallest index k such that +yk < 1 and let l be the smallest index such that k < l and yl > 0. This solution must exists, +else we would have y = x. Now, we will increase the value of yk and decrease the value of +yl. By choosing ϵ = min{ck(1 − yk),clyl} > 0 and increasing yk by ϵ/ck and decreasing yl +by ϵ/cl, we maintain feasibility and preserve �n +i=1 ciyi = B. The solution value changes by +pkϵ/ck −plϵ/cl = ϵ(pk/ck − pl/cl) > 0. This contradicts the assumption that y is an optimal +solution. Therefore, x is optimal which concludes the proof. +EC.4. +DRO results +In Ben-Tal and Hochman (1972), the following result was proved (for a much larger class +of functions f(y,X) than in our case): +Proposition EC.1. If f(y,·) is convex, +sup +P∈P(µ,δ) +EP[f(y,X)] = gU(y) = +� +κ∈{1,2,3}n +n +� +i=1 +p(i) +κi f(y,ξ(1) +κ1 ,...,ξ(n) +κn ), +(EC.8) + +e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec5 +with p(i) +κi ,ξ(i) +κi defined as in Lemma 2. If f(y,·) is concave, +sup +P∈P(µ,δ,β) +EP[f(y,X)] = gL(y) = +� +κ∈{1,2}n +n +� +i=1 +ˆp(i) +κi f(y,υ(1) +κ1 ,...,υ(n) +κn ), +(EC.9) +with υ(i) +1 = µi + δi +2βi, υ(i) +2 = µi − +δi +2(1−βi) and ˆp(i) +1 = βi, ˆp(i) +2 = 1 − βi. +Hence, gU(·) in (EC.8) inherits the convexity in y from f(·,X) and its functional form +depends only on the form of f(·,X) (and similarly for gL(·)). The upper and lower bound +give a closed interval for +ValP(y) = EP[f(y,X)] +∀P ∈ P(µ,δ,β). +(EC.10) +Corollary EC.1. If f(y,·) is convex for all y then ValP(y) ∈ [gL(y),gU(y)] ∀P ∈ +P(µ,δ,β). If f(y,·) is concave for all y then ValP(y) ∈ [gU(y),gL(y)] ∀P ∈ P(µ,δ,β). +From Proposition EC.1 we see that the extremal distribution is independent of y. Hence, +we can substitute the 3n terms. This leads to a convex function in y, and hence the +minimization problem over y is tractable. +EC.5. +Robust analysis with mean-variance knowledge +EC.5.1. +Scarf’s result for single item +Scarf (1958) introduced a distribution-free analysis for the single-item newsvendor model +by assuming that the decision maker only knows the mean and variance of the demand. +Define the ambiguity set containing all distributions with the same mean and variance as +P(µ,σ) := {P|EP(D) = µ, EP(D2) = σ2 + µ2}. +Scarf (1958) determined an upper bound on the cost function C(q) by finding the worst- +case distribution in the ambiguity set. To find the order quantity that protects against the +ambiguity in P(µ,σ), the following minimax optimization problem is solved: +min +q +max +P∈P(µ,σ) dq + (m + d)EP(D − q)+. +Since +max +P∈P(µ,σ) EP(D − q)+ ⩽ +� +σ2 + (µ − q)2 + (µ − q) +2 +, + +ec6e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +this minimax optimization problem becomes minq maxP CS(q) with +CS(q) := d(q − µ) + (m + d) +� +σ2 + (µ − q)2 + (µ − q) +2 +. +(EC.11) +and solution +qS := argmin +q +CS(q) = µ + σ +2 +��m +d − +� +d +m +� +. +(EC.12) +The quantity qS is known as Scarf’s order quantity which prescribes to order more than +the expected demand when m > d, and less than the expected demand when d < m. +EC.5.2. +Gallego and Moon +When the model is based on mean-variance information, Gallego and Moon (1993) formu- +late the problem as +min +q CS(q) := +n +� +i=1 +ci +� +�di(qi − µi) + (mi + di) +� +σ2 +i + (qi − µi)2 − (qi − µi) +2 +� +� +s.t. +n +� +i=1 +ciqi ⩽ B, +(EC.13) +q ⩾ 0. +The optimal solution to problem (EC.13) is referred to as qS. Applying Scarf’s bound +for each item individually results in (EC.13). Similar to the full information setting with +a known distribution, this optimization problem can be solved with Lagrange multiplier +techniques. +EC.6. +Additional numerical experiments +This section presents additional numerical results. Section EC.6.1 presents the performance +plots for the average margin setting. We compare the mean-MAD and mean-variance +ordering policies in Section EC.6.2. +EC.6.1. +More mean-MAD results +Figure EC.2 depicts the results for the average profitability scenario. A quick glance re- +veals that these plots exhibit a different impression than the low profitability scenario. We +conclude that the mean-MAD EVAI remains below some bound for budget levels ranging +from zero to two-thirds of the maximum budget. For all cases, this bound on the EVAI +is around 10%.As the budget passes two-thirds of the maximum budget, the performance +starts to decrease. However, the mean-MAD-β EVAI decreases when approaching the max- +imal budget. + +e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec7 +10 +20 +30 +40 +50 +0.024 +0.025 +0.026 +Case 1: Uniform (10,50) +10 25 40 55 70 85 100 +0.0105 +0.0110 +0.0115 +Case 2: Uniform (10,100) +10 40 70 100130160190 +0.0050 +0.0052 +0.0054 +Case 3: Uniform (10,200) +10 +20 +30 +40 +50 +0.00 +0.03 +0.06 +0.09 +0.12 +Case 4: Beta (1,3) +10 +20 +30 +40 +50 +0.00 +0.03 +0.06 +0.09 +0.12 +Case 5: Beta (2,2) +10 +20 +30 +40 +50 +0.00 +0.03 +0.06 +0.09 +0.12 +Case 6: Beta (3,1) +10 +20 +30 +40 +50 +0.00 +0.02 +0.04 +Case 7: Triangular c = (18) +10 +20 +30 +40 +50 +0.00 +0.02 +0.04 +Case 8: Triangular c = (30) +10 +20 +30 +40 +50 +0.00 +0.02 +0.04 +Case 9: Triangular c = (42) +Figure EC.1 +Nine probability density functions used for multi-item performance analysis +0 +314 +628 +942 +0.00 +0.07 +0.13 +0.20 +Case 1: Uniform (10,50) +Mean-MAD +Mean-MAD- +0 +602 +1205 1808 +0.00 +0.07 +0.13 +0.20 +Case 2: Uniform (10,100) +0 +1180 2360 3540 +0.00 +0.07 +0.13 +0.20 +Case 3: Uniform (10,200) +0 +137 +275 +413 +0.00 +0.04 +0.08 +0.12 +Case 4: Beta (1,3) +0 +263 +527 +791 +0.00 +0.08 +0.15 +0.23 +Case 5: Beta (2,2) +0 +367 +735 +1103 +0.00 +0.07 +0.13 +0.20 +Case 6: Beta (3,1) +0 +252 +505 +758 +0.00 +0.05 +0.11 +0.16 +Case 7: Triangular (10,50,18) +0 +287 +574 +861 +0.00 +0.07 +0.14 +0.21 +Case 8: Triangular (10,50,30) +0 +330 +661 +992 +0.00 +0.11 +0.21 +0.32 +Case 9: Triangular (10,50,42) +Figure EC.2 +The results for the average margin setting. The x-axis corresponds to B and the y-axis to the EVAI. +EC.6.2. +Mean-variance comparison +We start the performance analysis for the low margin scenario. The x-axis refers to the +budget level B, and the y-axis refers to the EVAI. In each plot, the blue line corresponds + +ec8e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +to the EVAI for the mean-MAD model and the orange line to the mean-variance EVAI. +Figure EC.3 contains the performance plots for each of the nine cases we are considering. +0 +224 +449 +674 +0.00 +0.05 +0.09 +0.14 +Case 1: Uniform (10,50) +Mean-MAD +Mean-variance +0 +401 +802 +1204 +0.00 +0.05 +0.09 +0.14 +Case 2: Uniform (10,100) +0 +755 +1510 2265 +0.00 +0.05 +0.09 +0.14 +Case 3: Uniform (10,200) +0 +70 +140 +211 +0.00 +0.06 +0.11 +0.17 +Case 4: Beta (1,3) +0 +187 +374 +561 +0.00 +0.07 +0.13 +0.20 +Case 5: Beta (2,2) +0 +312 +624 +937 +0.00 +0.06 +0.12 +0.18 +Case 6: Beta (3,1) +0 +190 +380 +571 +0.00 +0.08 +0.15 +0.23 +Case 7: Triangular (10,50,18) +0 +236 +472 +709 +0.00 +0.07 +0.13 +0.20 +Case 8: Triangular (10,50,30) +0 +277 +554 +831 +0.00 +0.06 +0.11 +0.17 +Case 9: Triangular (10,50,42) +Figure EC.3 +The results for the low margin scenario. The x-axis corresponds to the budget level and the y-axis +to the EVAI. +In Figure EC.3 we compare the mean-MAD policy with the mean-variance ordering +policy in terms of EVAI for the scenario with low margins and a total of nine ground- +truth demand distributions. While both policies generally give low EVAIs, the EVAI of +the mean-variance policy is typically lower. We stress that this does not mean that the +mean-variance policy is better. Indeed, a fair numerical comparison is impossible, as the +respective ambiguity sets can contain vastly different distributions. While a finite variance +excludes distributions with infinite-second moment, MAD does not. In general, the worst- +case scenarios or extremal distributions are ‘more extreme’ for MAD than for variance. +This also offers a possible explanation for the slightly higher EVAI. +EC.7. +Extensions +We now present a distribution-free analysis for three extensions of the multi-item newsven- +dor model. Section EC.7.1 deals with multiple constraints, Section EC.7.2 considers uncer- +tain supply, and Section EC.7.3 discusses the risk-averse newsvendor where the conditional +value at risk (CVaR) is chosen as objective function. + +e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec9 +EC.7.1. +Multiple constraints +Lau and Lau (1996) consider the newsvendor problem with multiple constraints, and pro- +pose a numerical solution procedure that computes the Lagrange multipliers as roots of a +system of nonlinear equations. Perakis et al. (2020) also consider multiple capacity con- +straints in a retail environment, and distinguish between warehouse capacity and inventory +availability constraints. By exploiting Lagrangian duality the problem is decomposed into +two subproblems, which are solved iteratively by binary search. +We now argue that the distribution-free analysis developed in the present paper also +carries over to the setting with multiple constraints, and takes the form +min +q +n +� +i=1 +ci +� +di(qi − µi) + (mi + di) +� +p(i) +1 (ai − qi)+ + p(i) +2 (µi − qi)+ + p(i) +3 (bi − qi)+�� +s.t. +n +� +i=1 +ci,jqi ⩽ Bj +j = 1,...,m +qi ⩾ 0 +i = 1,...,n. +(EC.14) +By introducing dummy variables τ (i) +k , we reformulate problem (EC.14) as +min +q,τ +n +� +i=1 +ci +� +di(qi − µi) + (mi + di) +� +p(i) +1 τ (i) +1 + p(i) +2 τ (i) +2 + p(i) +3 τ (i) +3 +�� +s.t. +n +� +i=1 +ci,jqi ⩽ Bj, +j = 1,...,m, +τ (i) +k ⩾ ξ(i) +k − qi, +k = 1,2,3; i = 1,...,n, +τ (i) +k ⩾ 0, +k = 1,2,3; i = 1,...,n, +qi ⩾ 0, +i = 1,...,n, +(EC.15) +which remains a tractable LP, solvable for large-scale problems with interior-point meth- +ods. Moreover, by solving the dual problem of (EC.15), shadow prices of the m budget +constraints can be computed that quantify marginal expected net benefit of allocating an +additional unit of budget to Bj, j = 1,...m. +EC.7.2. +Supply and demand uncertainty +The newsvendor might take different decisions when the delivery of an order for q units is +not necessarily complete (uncertain supply). K¨aki et al. (2015) consider uncertain supply +and uncertain demand, when supply and demand are independent or follow a particular + +ec10e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +copula-based dependency structure. In the mean-variance setting and under the indepen- +dence assumption, Gallego and Moon (1993) solve the distribution-free newsvendor prob- +lem with random yield, but assume the yield is a binomial random variable that depends +on the order size q. That is, when an order for q units is made, each individual unit is +received with some fixed probability, or is not delivered at all. +As opposed to Gallego and Moon (1993), we do introduce an ambiguity set for the +random supply. Consider the setting with multiplicative yield Zi, where the random supply +is given by Zi·qi. Assume Zi has mean ˜µi, MAD ˜δi and support [˜ai,˜bi], where 0 ⩽ ˜ai ⩽ ˜bi ⩽ 1. +The distribution of Zi then resides in P(˜µi,˜δi). The extremal three-point distribution for Zi +has probabilities +˜p(i) +1 = +˜δi +2(˜µi − ˜ai), +˜p(i) +2 = 1 − +˜δi +2(˜µi − ˜ai) − +˜δi +2(˜bi − ˜µi) +, +˜p(i) +3 = +˜δi +2(˜bi − ˜µi) +, +and is supported on ζ(i) +1 = ˜ai, ζ(i) +2 = ˜µi, ζ(i) +3 = ˜bi, respectively. The multi-item newsvendor +with supply ambiguity is equivalent to +min +q +n +� +i=1 +max +P∈Pi EP +� +ci +� +di(Zi · qi − Di) + (mi + di)(Di − Zi · qi)+�� +s.t. +n +� +i=1 +ciqi ⩽ Bj, +j = 1,...,m, +qi ⩾ 0, +i = 1,...,n, +(EC.16) +with Pi := P(µi,δi) × P(˜µi,˜δi). Since the newsvendor problem is jointly convex in the pairwise +independent random variables Di and Zi, the distributions that maximize the objective +function of (EC.16) are the extremal three-point distributions. Applying these worst-case +distributions to (EC.16) results in +min +q +n +� +i=1 +ci +� +di( ˜µiqi − µi) + (mi + di) +� +κ∈{1,2,3}2 +p(i) +κ1 ˜p(i) +κ2τ (i) +κ +� +s.t. +n +� +i=1 +ciqi ⩽ B, +τ (i) +κ ⩾ ξ(i) +κ1 − ζ(i) +κ2 qi, +κ ∈ {1,2,3}2; i = 1,...,n, +τ (i) +κ ⩾ 0, +κ ∈ {1,2,3}2; i = 1,...,n, +qi ⩾ 0, +i = 1,...,n. +(EC.17) + +e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec11 +To demonstrate the distribution-free newsvendor with uncertain supply, consider the +one-dimensional case with random demand D with a uniform distribution on [20,80] and +multiplicative yield Z uniformly distributed on [0.65,0.95]. Figure EC.4 depicts the tight +lower and upper bounds that follow from optimizing over the ambiguity sets that contain +the distributions of D and Z. As the extremal distributions are discrete, the objective +function of (EC.17) admits a piecewise linear representation. +0 +20 +40 +60 +80 +100 +120 +Order quantity +15 +20 +25 +30 +35 +40 +45 +50 +Expected costs +Uniform distributions +Mean-MAD upper bound +Mean-MAD lower bound +Figure EC.4 +Tight bounds for the multi-item newsvendor with uncertain supply yield, where m = 1 and d = +0.8. The upper piecewise linear function is obtained by evaluating E[D − Z · q], with D following +the extremal distribution that lies in P(50,15,20,80) and Z the worst-case three-point distribution in +P(0.8,0.075,0.65,0.95). The lower bound follows from the best-case two-point distributions. The middle +curve depicts the ‘true’ costs, where D has a uniform distribution on [20,80], and Z is uniformly +distributed on [0.65,0.95]. +Because problem (EC.16) can be written in terms of a piecewise linear function, the +optimal solution follows from a knapsack algorithm similar to Theorem 2. Further, one can +gain additional insights by explicitly deriving the optimal order quantities for the robust +single-item model, as in Theorem 1. The problem is similar for additive yield, also resulting +in a three-point distribution for the worst case. Other directions for future research include +solving (EC.16) with multiple unreliable and non-identical suppliers (Dada et al., 2007) +and the newsvendor problem with fixed ordering costs and supplier capacity restrictions +(Merzifonluoglu and Feng, 2014). + +ec12e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +EC.7.3. +Risk aversion +We next consider a risk-averse decision maker, as in Chen et al. (2010), who makes decisions +based on CVaR. The decision maker no longer optimizes the expected costs, but instead +minimizes the average value of the costs exceeding the γth-quantile of the newsvendor’s +cost distribution. For the cost function G(q,D), CVaR can be calculated by solving a +convex minimization problem (Rockafellar and Uryasev, 2000): +min +θ∈R +� +θ + +1 +1 − γ E(G(q,D) − θ)+ +� +. +Calculating CVaR requires full knowledge of the demand distribution. However, in practice, +committing to a particular distribution might be problematic for the decision maker if +there is not enough data available. Hence, we consider the partial information setting as +in Zhu and Fukushima (2009); Delage and Ye (2010), and seek to solve +min +q:� +i ciqi⩽B,qi⩾0 max +P∈P(µ,δ) min +θ∈R +� +θ + +1 +1 − γ EP(G(q,D) − θ)+ +� +. +(EC.18) +Let us first consider the single-item model. Because the objective function of (EC.18) +is finite, P(µ,δ) is weakly compact as supp(D) is compact, and the objective function of +(EC.18) is linear in P and convex in θ, we are allowed to interchange the maximization and +minimization operators by virtue of the minimax theorem (Shapiro and Kleywegt, 2002). +Since (G(q,D) − θ)+ is a convex function of the uncertain demand, the three-point distri- +bution (10) also maximizes EP(G(q,D)−θ)+. When β = P(D ⩾ µ) is known, the two-point +distribution in Lemma 3 attains the matching lower bound. For the multivariate problem, +notice that (G(q,D) − θ)+ is again a convex function of the uncertain demand, where +D ∼ P ∈ P(µ,δ). By Proposition EC.1 and the reasoning above, the risk-averse newsvendor + +e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec13 +admits the following LP representation: +min +q,τ,η,θ θ + +1 +1 − γ +� +κ∈{1,2,3}n +n +� +i=1 +p(i) +κi ηκ +s.t. +n +� +i=1 +ciqi ⩽ B, +ηκ ⩾ +� +n +� +i=1 +ci +� +di(qi − ξ(i) +κi ) + (mi + di)τ (i) +κ +�� +− θ, +κ ∈ {1,2,3}n, +ηκ ⩾ 0, +κ ∈ {1,2,3}n, +τ (i) +κ ⩾ ξ(i) +κi − qi, +κ ∈ {1,2,3}n; i = 1,...,n, +τ (i) +κ ⩾ 0, +κ ∈ {1,2,3}n; i = 1,...,n, +qi ⩾ 0, +i = 1,...,n. +(EC.19) +We show in Figure EC.5 the bounds for the single-item model with demand having +support [10,50], µ = 30, δ = 20/3 and β = 1/2. Solving (EC.19) for γ = 0.75,0.95 and +different order sizes yields the upper bounds. We solve an analogous problem, but with the +expectation taken over the extremal two-point distribution, stated in Lemma 3, to obtain +the tight lower bounds. As a point of reference, we also plot the exact values of the CVaR +and expected costs when D follows a symmetric triangular distribution on [10,50]. +10 +15 +20 +25 +30 +35 +40 +CVaR99% +6 +8 +10 +12 +14 +16 +18 +20 +C(q) +q = 10 +q = 20 +q = 30 +q = 40 +q = 50 +Triangular +Mean-MAD upper bound +Mean-MAD lower bound +(a) Expected costs and CVaR +10 +15 +20 +25 +30 +35 +40 +45 +50 +Order quantity +5 +10 +15 +20 +25 +30 +CVaR75% +Triangular +Mean-MAD upper bound +Mean-MAD lower bound +(b) Mean-MAD bounds for CVaR +Figure EC.5 +An illustration of the tight mean-MAD bounds for the risk-averse newsvendor with CVAR as objec- +tive criterion, where m = 1, d = 0.8 and γ = 0.75,0.99. The middle curve corresponds to the CVaR +when D follows a symmetric triangular distribution on [10,50]. The upper and lower bounds follow +from optimizing over the ambiguity sets that contain this distribution. +Solving (EC.19) can be challenging since the objective function (G(q,D) − θ)+ is no +longer separable, thus resulting in an exponential number of variables and constraints. To + +ec14e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor +alleviate this computational difficulty, one might resort to sampling-based procedures such +as sample average approximation (Shapiro et al., 2009). +We also mention ambiguous chance constraints that can be conservatively approximated +by CVaR (Nemirovski and Shapiro, 2007). In the risk-averse newsvendor setting, the deci- +sion maker introduces an ambiguous chance constraint that restricts the probability of the +costs exceeding a certain threshold t to be less than 1 − γ, considering all distributions in +the ambiguity set. For the multi-item setting, this means ensuring +P(G(q,D) > t) ⩽ 1 − γ, +∀P ∈ P(µ,δ), +which is implied by +max +P∈P(µ,δ) CVaRγ[G(q,D)] ⩽ t. +In addition, the newsvendor might require a minimal probability that all customer orders +will be completely covered by the inventory on hand, i.e., the type-1 service level (Silver +et al., 1998). When several of these probabilistic constraints are interrelated, the decision +maker should conservatively approximate joint chance constraints. For this one can again +use CVaR; see Chen et al. (2010); Zymler et al. (2013); Roos and den Hertog (2020). Adding +ambiguous chance constraints to the models developed in this paper is a worthwhile topic +for further research. + diff --git a/D9E0T4oBgHgl3EQfywIT/content/tmp_files/load_file.txt b/D9E0T4oBgHgl3EQfywIT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0cf32001a3dce68d18420141cf9a88e93e01597b --- /dev/null +++ b/D9E0T4oBgHgl3EQfywIT/content/tmp_files/load_file.txt @@ -0,0 +1,1496 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf,len=1495 +page_content='Robust knapsack ordering for a partially-informed newsvendor with budget constraint Guus Boonstra Retail Consulting Department, IG&H Consultants, guus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='boonstra@igh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='com Wouter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' van Eekelen Department of Econometrics and Operations Research, Tilburg University, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='vaneekelen@tilburguniversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='edu Johan S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' van Leeuwaarden Department of Econometrics and Operations Research, Tilburg University, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='vanleeuwaarden@tilburguniversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='edu This paper studies the multi-item newsvendor problem with a constrained budget and information about demand limited to its range, mean and mean absolute deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We consider a minimax model that deter- mines order quantities by minimizing the expected overage and underage costs for the worst-case demand distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The resulting optimization problem turns out to be solvable by a method reminiscent of the greedy algorithm that solves the continuous knapsack problem, purchasing items in order of marginal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This method has lower computational complexity compared to directly solving the model and leads to a simple policy that (i) sorts items based on their marginal effect on the total cost and (ii) determines order quantities according to this ranking until the budget is spent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Key words : distributionally robust optimization, multi-item newsvendor model, knapsack problem, minimax analysis, inventory management History : This paper was first submitted on March 8, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Introduction The newsvendor model is one of the cornerstones of inventory management, introduced by Arrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (1951) for finding the order quantity that minimizes expected costs in view of unknown demand and the trade-off between leftovers and lost sales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The newsvendor model finds many applications in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' perishable food, fashion and high-tech industries, particularly when the total time span of production and lead times exceeds the market lifetime of a product;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' see Nahmias (1982) and Fisher and Raman (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Manufacturers and retailers need to decide how to employ the available budget or re- sources when determining the optimal order quantities of different products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' A budget constraint makes the problem multidimensional—as ordering more of one item leaves less budget for other items—and gives rise to a challenging optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Hadley and Whitin (1963) solve this problem with Lagrangian optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Abdel-Malek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (2004) and Lau and Lau (1996) provide alternative solution methods, Erlebacher (2000) estab- lishes closed-form solutions for special demand distributions and Nahmias and Schmidt 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='02662v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='OC] 5 Jan 2023 2 Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor (1984) develop heuristic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' All these works are for the full information setting, where the demand distributions for all items are fully specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In this paper we perform a distribution-free analysis of the multi-item newsvendor problem with budget constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This analysis does not rely on full specification of the demand distributions, but only re- quires for each item knowledge of the mean, mean absolute deviation (MAD) and range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Given this partial demand information, we obtain a robust ordering policy by employing distributionally robust optimization (DRO) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The newsvendor model in this paper seeks to minimize the expected costs as function of the order quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The cost function depends on the order quantity, but also on the demand, which is a random variable with some distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Given the demand distribu- tion, the single-item newsvendor model finds the optimal order quantity that minimizes the expected costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In traditional approaches, the demand distribution is fully specified, so that the expected costs can be calculated, and the optimal order quantity can be deter- mined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' A robust version of this problem assumes partial information, and only knows that the demand distribution belongs to some ambiguity set that contains all distributions that comply with this partial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We adopt a minimax strategy that can be viewed as a game between the newsvendor and nature: the newsvendor first picks the order quantity after which nature chooses a demand distribution that maximizes the expected costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The goal then becomes to solve this minimax problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The way we solve this minimax problem in this paper fits in a much richer class of DRO approaches that first calculate worst-case model performance, over the set of distributions satisfying some partial information, and then optimize against these worst-case circum- stances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Such DRO techniques found applications in many domains including scheduling (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Mak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', 2014), portfolio optimization (Popescu, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Delage and Ye, 2010), pricing (Elmachtoub et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Kleer and van Leeuwaarden, 2022), complex networks (van Leeuwaarden and Stegehuis, 2021), and inventory manage- ment (Scarf, 1958;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Gallego, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Perakis and Roels, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Ben-Tal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' A classic distributionally robust approach is due to Scarf (1958), who considered the single-item newsvendor problem with mean-variance demand information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Scarf was able to derive explicit expressions for the worst-case distribution, and solved the minimax problem to obtain the optimal order quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Whether a minimax problem is solvable depends on both the function to be optimized and the choice of ambiguity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' There are many ways Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor 3 to characterize a set of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In DRO, one can define ambiguity by using distance- based metrics, such as total variation or Kullback-Leibler distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Another popular class of ambiguity uses summary statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The ambiguity set studied in this paper contains all distributions with known mean and MAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The maximization part of the minimax problem can then be viewed as a semi-infinite linear optimization problem with three constraints, and an infinite number of variables (all distributions in the ambiguity set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In fact, such minimax problems are related to generalized moment bound problems, for which general theory says there exists an extremal distribution solving the maximization part with at most a number of support points equal to the number of moment constraints (Rogosinski, 1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' See Rahimian and Mehrotra (2019) for overviews of many more DRO applications and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For the multi-item newsvendor model in this paper, we solve the multi-dimensional mini- max problem with a random vector that describes the demand for all items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Compared with tractable one-dimensional problems such as the single-item newsvendor model, applying DRO techniques to such problems with multiple random variables might present consider- able challenges in terms of computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For example, given information on the mean and covariance of the demands, the distributionally robust multi-item newsvendor is significantly harder to solve than its single-item counterpart (Hanasusanto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' However, for the multi-item newsvendor model in conjunction with mean-MAD ambiguity, solving the minimax problem becomes tractable, and in fact has an elegant algorithmic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The key insight will prove to be that the worst-case demand distribution—the solution to the maximization part of the minimax problem—is identical for any order quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' As a result, the minimax problem reduces to a known-distribution optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This known distribution is in fact, for each item, a unique three-point distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In turn, the minimization problem with this known (discrete) distribution can be solved using a reduction to a knapsack problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The main contributions of this paper are as follows: (i) Solution of minimax problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We solve the minimax problem for mean-MAD ambi- guity and a budget constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We first show that the worst-case scenarios arise when item demands follow specific three-point distributions that comply with the partial demand information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We minimize the associated worst-case costs to obtain a robust 4 Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor ordering policy as the solution to a knapsack problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' As opposed to existing meth- ods for the newsvendor model under full demand information, the knapsack problem leads to an effective closed-form ordering policy, also for scenarios with many items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' As such, the present paper further develops DRO theory that uses MAD information to formulate tractable minimax problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (ii) Budget consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The robust ordering policy only depends on the minimal, mean and maximal demand for each item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Hence, the worst-case distributions are indepen- dent of all other model parameters, which makes the robust ordering policy ‘budget consistent’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' When the budget is increased, the orders for the original budget remain unaltered, while only the additional budget is further divided over the items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Such budget consistency is useful because the optimization model needs to be solved only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' That is, for the initial budget value the decision maker can generate an ordered list of items as the solution to the knapsack problem, using only standard spreadsheet software, and this solution is valid for all budget levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In contrast, most other exact and robust methods for the multi-item newsvendor model do not have this feature, which means that the decision maker has to recompute the optimal policy for each budget level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (iii) Performance of ordering policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Through a range of numerical examples we demon- strate the performance of the knapsack ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We draw comparisons with full infor- mation settings and other robust approaches that require partial demand information by assessing the so-called expected value of additional information (EVAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Overall, the performance of the robust policy only deviates a few percent from the optimal performance with full information availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We also quantify the value of MAD information by comparing the performance with the situations when only the mean and range of demand is known, and show that MAD indeed provides crucial infor- mation for providing good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In addition, we construct an ordering policy that attains the optimal value of a matching minimin problem which, in conjunction with the optimal value of the minimax problem, yields tight performance guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We next discuss some related literature on the newsvendor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Gallego and Moon (1993) consider the multi-item newsvendor model with budget constraint when the mean and variance of demand is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Gallego and Moon (1993) extend the ideas in Scarf (1958) to obtain an optimization problem that can be solved with Lagrange multiplier Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor 5 techniques, similar to the full information setting with a known distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In contrast, our minimax analysis with mean-MAD-range information yields a knapsack ordering pol- icy that generates a sorted list and prescribes to sort items successively according to that list, with order sizes equal to the minimal, mean or maximum demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Other related works that consider the multi-item newsvendor model under partial information include Vairaktarakis (2000), who assumes only the support of demand is known, and Ardestani- Jaafari and Delage (2016) who assume knowledge of partial moments and rephrase the robust optimization problem as a tractable linear program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Natarajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (2018) assume knowledge of mean, variance and semivariance, for which the newsvendor model is solvable in the single-item setting using a semi-infinite linear program, but largely intractable in the multi-item setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Natarajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (2018) therefore consider a relaxation that gives a semidefinite program (SDP) to find a lower bound (which is not tight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Hanasusanto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (2015) consider mean and covariance knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' They prove that the distributionally robust problem is NP-hard but admits a semidefinite programming formulation with an ex- ponential number of inequalities (that grows in the number of items).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (2018) and Natarajan and Teo (2017) present more tractable bounds for mean-covariance information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In the present paper we assume only marginal information is available, since covariance information and other dependency structures are difficult to estimate, and fixing covari- ance information often leads to difficult optimization problems with non-intuitive solutions (policies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The knapsack ordering policy that we obtain in this paper deals with the worst- case demand distributions among all demand distributions with a given mean, MAD and range, not conditioning on a specific dependency structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This approach makes the knap- sack ordering policy robust, but also suitable for scarce-data settings, as the mean, MAD and range are relatively easy to estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Section 2 introduces the single-item model and the multi-item model with budget, under the traditional assumption of full information about the demand distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In Section 3 we present our main results for the distributionally robust setting with partial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Section 4 presents a detailed numerical study that demonstrates the robust policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We present conclusions and several directions for future work in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Supplementary material appears in the Electronic Companion (EC), including several proofs, additional numerical experiments, and model extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 6 Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Classical newsvendor analysis We introduce the newsvendor model and several well-known results in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1 for the single-item setting, and in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2 for the multi-item setting with budget constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Classical single-item setting Consider an item with purchase price c and selling pricing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The decision maker places an order of size q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The demand for items is assumed to be the random variable D with distribution function FD(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Unsold items will be salvaged at the end of the period for salvage value s per item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The mark-up m > 0 represents the profit per sold item and satisfies p = c(1 + m) and the discount factor d > 0 captures the loss through s = (1 − d)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The expected costs consist of two terms: opportunity costs of lost sales and overage costs in case of overstocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This gives the cost function G(q,D) = � � � � � (p − c)(D − q) if q ⩽ D, (c − s)(q − D) if q > D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (1) The case q ⩽ D amounts to lost sales and q > D results in overstocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The objective is to order the quantity q of items that minimizes the expected costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Let E denote expectation, and define µ = E[D] and x+ = max(x,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Write the expected costs as C(q) := E[G(q,D)] = (c−s)q+(p−s)E(D−q)+−(c−s)µ = c � d(q − µ) + (m + d)E(D − q)+� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (2) To keep notation simple (and without loss of generality) set c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Then, the optimal order quantity q∗ = argmin q⩾0 C(q) ≡ argmin q⩾0 dq + (m + d)E(D − q)+, (3) is given by q∗ = inf � q : F(q) ⩾ m m + d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (4) A proof of (4) is provided in most standard textbooks on inventory management;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Hadley and Whitin (1963);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Nahmias (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Multi-item setting Consider n different items and order qi units for item i for a given period where i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For item i, the unit purchasing and selling price are ci and pi respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Possible leftovers will be salvaged at the end of the period for unit salvage value si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We define the model Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor 7 in terms of the mark-up mi > 0 and discount factor di > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The mark-up represents the profit per sold unit and the discount factor the loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' pi = ci(1 + mi) and si = (1 − di)ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The random demand for item i in one period is represented by the nonnegative random variable Di, distributed according to Fi(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' As in the single-item setting, we minimize the expected costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Define the multi-item cost function as G(q,D) := n � i=1 ci � di(qi − Di) + (mi + di)(Di − qi)+� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (5) We also introduce the budget constraint �n i=1 ciqi ⩽ B with B the available budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The multi-item newsvendor model, with decision vector q = (q1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',qn), is then given by min q C(q) := E[G(q,D)] = n � i=1 ci � di(qi − µi) + (mi + di)E(Di − qi)+� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' n � i=1 ciqi ⩽ B, qi ⩾ 0, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (6) Its solution, referred to as the optimal ordering policy, will be denoted by q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In the single-item setting the purchase costs had no influence on the objective function, but in the multi-item setting the optimal order quantity is affected by ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' It is well known that model (3) is a convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In (6) we take the summation over n convex functions, which preserves convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Moreover, the constraints form a convex set, so that (6) is a convex optimization problem (Boyd and Vandenberghe, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Proposed robust approach Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1 presents the robust ordering policy for the single-item setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This result serves as building block for the robust analysis of the multi-item setting in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2, which describes the optimal policy as the solution of a linear program (LP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='3 we show that this LP can be viewed as a knapsack problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' All these results are based on a tight upper bound for the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='4 we derive a matching tight lower bound for the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Distribution-free ordering policy for single item Let P denote a probability distribution, and write EP for E to emphasize that the expec- tation is taken with respect to the distribution P of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The MAD for random demand D 8 Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor is defined as δ := EP|D − µ|, where µ is the expected value of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Similar to the variance, the MAD is a measure of dispersion or variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We mention several properties of MAD in EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For the random variable D with mean µ, MAD δ, and (bounded) support [a,b], where 0 ⩽ a ⩽ b < ∞, the mean-MAD ambiguity set is defined as P(µ,δ) := {P|EP[D] = µ, EP|D − µ| = δ, supp(D) ⊆ [a,b]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We thus assume that the ‘true’ distribution ˜P of the random demand D is contained in this ambiguity set, that is, ˜P ∈ P(µ,δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' To obtain the robust order quantity, we solve min q max P∈P(µ,δ) dq + (m + d)EP(D − q)+, for which we first consider maxP∈P(µ,δ) EP(D − q)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' To characterize this tight bound, we apply a general upper bound for convex functions of a random variable by Ben-Tal and Hochman (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' To make this paper self-contained, we provide a proof of the following result in EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The extremal distribution that solves max P∈P(µ,δ) EP(D − q)+ is a three-point dis- tribution on the values a, µ and b that does not depend on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' From the proof of Lemma 1, it follows that the worst-case probability distribution of D, the extremal distribution that solves maxP∈P(µ,δ) EP(D − q)+, is a three-point distribution defined as P(D = x) = � � � � � � � � � � � � � δ 2(µ − a), for x = a, 1 − δ 2(µ − a) − δ 2(b − µ), for x = µ, δ 2(b − µ), for x = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (7) Applying this worst-case distribution, the robust order quantity follows from solving qU = argminq CU(q) with CU(q) := d(q − µ) + δ(m + d) 2(µ − a) (a − q)+ + (m + d) � 1 − δ 2(µ − a) − δ 2(b − µ) � (µ − q)+ + δ(m + d) 2(b − µ) (b − q)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (8) To illustrate the mean-MAD bound and robust order quantity qU, consider an example in which D is distributed according to a beta distribution with both shape parameters set Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor 9 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For a general beta distribution, a = 0 and b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In Figure 1a, we have m = 1 and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This leads to qU = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In Figure 1b, the mark-up increases to m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In this case the mean-MAD order quantity increases to qU = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' When computing this upper bound, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='0 Order quantity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='55 Expected costs Beta Mean-MAD bound Mean-variance bound (a) m = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='0 Order quantity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='6 Expected costs Beta Mean-MAD bound Mean-variance bound (b) m = 3 Figure 1 Mean-MAD and mean-variance bounds and corresponding ordering policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The upper curve corre- sponds to the mean-variance upper bound that follows from P(1/2,1/12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The middle curve depicts the mean-MAD upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The ‘true’ cost function assumes that D follows a beta distribution with both shape parameters equal to 1 (the lower curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' observe that the mean-MAD bound touches the ‘true’ cost function in the points a,µ and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This property actually holds in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Clearly, for q = a or b, it holds that CU(q) = C(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' When q = µ, the cost function equals C(µ) = d(µ − µ) + (m + d)E(D − µ)+ = δ(m + d) 2 = CU(µ), since E(D − µ)+ = E|D − µ|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' By analyzing (8) one can obtain an explicit ordering rule for qU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The objective func- tion of (8) is composed of piecewise linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' By exploiting this structure, we can construct an explicit ordering policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For scalars α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',αm,ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',νm ∈ R, f(x) = maxi=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',m{αix+νi} denotes a convex, piecewise linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The function CU(q) in (8) admits a representation of the form CU(q) = d(q − µ) + (m + d)E(D − q) = m(µ − q) =: f0(q), for q ∈ [0,a) and CU(q) = d(q − µ) + (m + d) � 1 − δ 2(µ − a) − δ 2(b − µ) � (µ − q) + δ(m + d) 2(b − µ) (b − q) = q(δ(m + d) 2(µ − a) − m) + ν1 =: f1(q), 10 Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor for q ∈ [a,µ), where ν1 is some constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For q ∈ [a,µ), the mean-MAD objective function is defined by the linear function f1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For the interval q ∈ [µ,b], we obtain CU(q) = d(q − µ) + δ(m + d) 2(b − µ) (b − q) = q � d − δ(m + d) 2(b − µ) � + ν2 =: f2(q) for some constant ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The cost function is thus the pointwise maximum of the three linear functions f0(q), f1(q) and f2(q): CU(q) = max{f0(q), f1(q), f2(q)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Since CU(q) = maxj=0,1,2{αjq + νj} is a convex function, it holds that α0 ⩽ α1 ⩽ α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Since we assume that m > 0, we know that α0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Therefore, from the derivatives α1, α2 of CU(q), we can derive an explicit order quantity by examining for which linear piece the slope turns positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This allows us to state Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Theorem 1 (Mean-MAD order quantity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The robust order quantity qU ∈ argminq CU(q) is given by (a) If m < δd 2(µ − a) − δ, then qU = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (b) If δd 2(µ − a) − δ < m < d(2(b − µ) − δ) δ , then qU = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (c) If d(2(b − µ) − δ) δ < m, then qU = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (d) If m = δd 2(µ − a) − δ and m = d(2(b − µ) − δ) δ , then qU ∈ [a,µ] and qU ∈ [µ,b], respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' According to Theorem 1, the robust order quantity qU for mean-MAD-range information consists of three predictable values (minimal, mean, maximum demand) that do not depend on the mark-up m and discount factor d, whereas the conditions that dictate how much to order do depend on them (in addition to the demand mean, MAD and range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Multiple items and budget constraint A distribution-free analysis of the multi-item model requires a multivariate ambiguity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' As in the single-item case, the partial information is the mean µi, MAD δi and support supp(Di) = [ai,bi] for each random variable Di, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The mean-MAD ambiguity set is defined as P(µ,δ) := {P|EP (Di) = µi, EP |Di − µi| = δi, supp(Di) ⊆ [ai,bi], ∀i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (9) Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor 11 We henceforth assume that the distribution of the vector of random variables D = (D1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',Dn) belongs to this ambiguity set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', P ∈ P(µ,δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Since the objective function in (6) is separable, one can apply the single-item bound to each term E(Di − qi)+ in the summation individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The following result, for the multi-item problem, is then a direct consequence of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The extremal distribution that solves max P∈P(µ,δ) EP[G(q,D)] consists for each Di of a three-point distribution with values ξ(i) 1 = ai, ξ(i) 2 = µi, ξ(i) 3 = bi and probabilities p(i) 1 = δi 2(µi − ai), p(i) 2 = 1 − δi 2(µi − ai) − δi 2(bi − µi), p(i) 3 = δi 2(bi − µi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (10) For the multi-item newsvendor model based on mean-MAD ambiguity, we use Lemma 2 to solve the maximization part of min q:� i ciqi⩽B,qi⩾0 max P∈P(µ,δ) EP � n � i=1 cidi(qi − µi) + ci(mi + di)(Di − qi)+ � , (11) and obtain min q n � i=1 ci � di(qi − µi) + (mi + di) � p(i) 1 (ai − qi)+ + p(i) 2 (µi − qi)+ + p(i) 3 (bi − qi)+�� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' n � i=1 ciqi ⩽ B, qi ⩾ 0, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (12) The objective function of (12) has a piecewise linear structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Moreover, because of this result and since the constraints are linear, (12) can be cast as a linear program (LP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In particular, as explained below, the robust ordering policy qU can be found by solving min q n � i=1 max j=0,1,2{αi,jqi + νi,j} s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' n � i=1 ciqi ⩽ B, qi ⩾ 0, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n, (13) where αi,0 = −cimi, νi,0 = cimiµi, αi,1 = ci �δi(mi + di) 2(µi − ai) − mi � , νi,1 = ci(mi + di) � µi − δiai 2(µi − ai) � − cidiµi, αi,2 = ci � di − δi(mi + di) 2(bi − µi) � , νi,2 = ciδi(mi + di)bi 2(bi − µi) − cidiµi, for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 12 Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor Let fi,j(x) = αi,jx + νi,j for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n and j = 0,1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' From the single-item case, we know that the objective, for each item i, can be written as maxj=0,1,2{fi,j(qi)} with αi,0 ⩽ αi,1 ⩽ αi,2, and thus the objective functions of (12) and (13) are equal, which makes the two models equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Since we know from linear programming theory that convex, piecewise linear objective functions can be written as linear constraints, problem (13) admits an LP representation (Boyd and Vandenberghe, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Knapsack algorithm It turns out that problem (13) is intimately related to the continuous knapsack problem, thus making available efficient sorting-based algorithms to solve (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We next describe an efficient algorithm that determines the robust ordering policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Define the linear funtion fi,j for each item i, and let αi,j represent its derivative with respect to qi, for items i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n and linear pieces j = 0,1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' That is, dfi,j(qi) dqi = αi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For each item i, fi,0, fi,1 and fi,2 represent the marginal effect on the value of (13) when we increase qi to ai,µi and bi respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The parameter αi,j represents the slope of these linear functions and an order quantity is increased only when αi,j < 0, because otherwise it will not reduce the expected costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We consecutively allocate budget to the item that causes the largest relative decrease in expected costs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' that is, item k with the smallest negative derivative αk,i relative to its cost ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Define the set of all items as N = {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Since only order quantities that decrease the expected costs are considered, define the ordered set: G := {(i,j) | αi,j < 0,i ∈ N,j ∈ {0,1,2}}, (14) where the ordering is determined according to the value of αi,j/ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For m = |G|, this ordering is represented by the sequence (i1,j1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',(im,jm) for which it holds that αi1,j1/ci1 ⩽ ··· ⩽ αim,jm/cim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Here G contains tuples (i,j) for which i represents an item in the newsvendor model and j a linear piece of the piecewise function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' As these functions are convex, the linear pieces appear for each item i in increasing order in the set G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We can now state the knapsack algorithm for the distribution-free multi-item newsvendor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor 13 Algorithm 1 (Knapsack algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For a budget level B ⩾ 0, the ordering policy qU is found by the following procedure: (i) Initialize by setting q = (0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',0), and construct G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Continue to (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (ii) Select the first element (i,j) ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' If the set G is empty, the optimal solution is qU = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Otherwise, continue to (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (iii) If j = 0, set qi = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' If j = 1, set qi = µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' If j = 2, set qi = bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Continue to (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (iv) Determine whether the budget constraint �n i=1 ciqi ⩽ B is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' If so, set qi such that ciqi = B − � k∈N|k̸=i ckqk, and the optimal solution is qU = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Otherwise, remove element (i,j) from G and return to step (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This algorithm yields an optimal solution to (13), as asserted in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Theorem 2 (Knapsack ordering policy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The robust ordering policy qU that solves the multi-item newsvendor model (13) is determined by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' To prove that this algorithm produces an optimal solution, we construct a con- tinuous knapsack problem that solves (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In the following, (ik,jk) corresponds to the kth entry of the ordered sequence of items in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Define the following auxiliary model: min x m � k=1 pkxk s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' m � k=1 ckxk ⩽ B, 0 ⩽ xk ⩽ uk ∀k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',m, (15) where uk = � � � � � � � aik, for jk = 0 µik − aik, for jk = 1 bik − µik, for jk = 2 and pk = αik,jk and ck = cik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' From the order of the sequence, it follows that p1/c1 ⩽ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' ⩽ pm/cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Assume that (x∗ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',x∗ m) is an optimal solution to optimization problem (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For i ∈ N, let qU i = � k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',m|i=ik x∗ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Since αi,0 ⩽ αi,1 ⩽ αi,2, the pieces jk appear in G in increasing order for each item i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Thus, in an optimal solution, uik,jk will only be attained if its predecessor uik,jl is also attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' By construction, qU is feasible for (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Moreover, the objective values of problems (13) and (15) only differ by a constant term, so both problems have the same optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For the continuous knapsack problem, a greedy allocation produces an optimal solution (see EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Hence, qU = (qU 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',qU n ) is optimal for (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' □ 14 Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor Theorem 2 shows that there exists a ranking for the selection of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Take an initial budget B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' If we increase the budget B by some small value, we first increase item i to ai for the item that has the highest mark-up mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This makes sense intuitively because the product with the highest mark-up is most profitable and, since qi < ai, we have no risk of overstocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We successively select the items with the greatest marginal benefit αi,j/ci, and increase the order quantity consecutively to either ai, µi or bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This procedure continues until we have spent the entire budget, or reached the uncapacitated optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Items that are ordered in the beginning of this procedure have the largest impact on the decrease in costs for the multi-item newsvendor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' As the main complexity of the knapsack algorithm in Theorem 2 stems from sorting the set G, the greedy approach is of computational complexity O(nlog n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Moreover, the solution can be found in O(n) time by first identifying the critical element (is,js) that will violate the budget constraint, as proposed by Balas and Zemel (1980) for the continuous knapsack problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' One then compares each αi,j/ci with the ratio of the critical element to determine the optimal allocation of budget to the items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The optimal solution can also be found through the LP (13), which we solve with the simplex method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We remark that a single iteration of the simplex method takes O(n2) arithmetic operations (Ill´es and Terlaky, 2002), which exceeds the time requirement of the knapsack algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' A matching lower bound The robust analysis so far was based on finding a tight upper bound on the cost function when we know the mean, MAD and range of the demand distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' When additional information is available, we can also construct a matching lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We include the skewness information βi = P(Di ⩾ µi) in the mean-MAD ambiguity set to obtain the tight lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For the random variables D = (D1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',Dn), define the ambiguity set as P(µ,δ,β) := {P|P ∈ P(µ,δ), P(Di ⩾ µi) = βi, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n} with P(µ,δ,β) ⊆ P(µ,δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The proof of the following result is identical to that of Lemma 2, but now uses the tight lower bound for a convex function of random variables discussed in Ben-Tal and Hochman (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' To make this paper self-contained, a proof for the univariate case is provided in EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This is sufficient since the univariate result can be applied to each term of the summation in G(q,D) separately, as with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor 15 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The extremal distribution that solves min P∈P(µ,δ,β) EP[G(q,D)] consists for each Di of a two-point distribution with values µi + δi 2βi, µi − δi 2(1−βi) and probabilities βi, 1 − βi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Using this result, we obtain min q CL(q) := n � i=1 ci � di(qi − µi) + (mi + di) � βi(µi + δi 2βi − qi)+ + (1 − βi)(µi − δi 2(1 − βi) − qi)+ �� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t n � i=1 ciqi ⩽ B, qi ⩾ 0, for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n, (16) as a model to provide a lower bound for the multi-item newsvendor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' As the objective function in problem (16) also consists of piecewise linear functions, there exists an LP representation and knapsack algorithm for (16) similar to the results for problem (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We can now solve (13) and (16) to obtain tight performance intervals for the multi-item newsvendor model, using recent DRO results (see EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='4 and Postek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For all feasible ordering policies q and P ∈ P(µ,δ,β), it holds that C(q) ∈ � CL(q),CU(q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In addition, for the optimal solutions to the newsvendor problem and its distributionally robust counterparts, C(q∗) ∈ � CL(qL),CU(qU) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' One can find the tightest upper and lower bounds, based on mean-MAD ambiguity, for the multi-item newsvendor model by calculating the optimal solutions to models (12) and (16), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Numerical examples of robust ordering We will now illustrate and visualize the robust ordering policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' To demonstrate the ‘budget-consistency’ property, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1 applies the knapsack algorithm for a setting where the budget is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2 we contrast the performance of the knapsack policy for partial demand information against that of the optimal solution for the full information setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Our code is made available in the form of an online supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 16 Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Numerical illustration of the ‘budget-consistency’ property We illustrate the knapsack algorithm and the process of allocating budget to different order quantities for items in the newsvendor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Consider n = 5 identically distributed items with support a = 10, b = 50 and mean µ = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' From Figure 2, we can infer that item 1 is the most profitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Low budget levels are allocated to this item such that we obtain q1 = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Item number 3 is the last item to which the budget is allocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Hence, it is the least profitable item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Table 1 displays the ordered set G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' From this table, we can indeed infer that item 1 has the smallest value for αi,0/ci and therefore is increased first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 0 50 100 150 200 250 300 Budget 0 10 20 30 40 50 Order quantity Item 1 Item 2 Item 3 Item 4 Item 5 Figure 2 Development of the order quantities when the budget increases according to the knapsack algorithm Table 1 Table containing αi,j/ci and corresponding information of the ordered set G G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 αi,j/ci 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='92 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='75 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='72 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='49 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='15 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='08 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='7 Function piece 0 1 0 1 0 0 1 2 1 0 1 2 2 2 2 Item 1 1 2 2 4 5 4 1 5 3 3 5 2 4 3 Figure 2 nicely illustrates that when the budget is increased, the orders for the original budget remain unaltered, while only the additional budget is further divided over the items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' To further illustrate the ‘budget-consistency’ property, consider the multi-item newsvendor model for which n = 2, m2 = 2, the remaining cost parameters equal 1, and demand is identically distributed according to a symmetric triangle distribution supported on [10,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In Figure 3 we plot the expected costs and order quantities for various budget levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor 17 Figure 3a contains the allocation between both order quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For low budget values, one first increases the order quantity of item one, the most profitable item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Figure 3b shows the upper bound (12) and lower bound (16) that together lead to a tight performance interval for the expected costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For the sake of comparison, we also show results for the partial demand information setting considered in Gallego and Moon (1993), assuming that the mean and variance of demands are known;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' see EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='5 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The results of Gallego and Moon (1993) de- pend (non-trivially) on all model parameters, including the budget B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This lack of budget- consistency forces the decision maker to solve an optimization problem, see (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13), for each budget level separately, and explains the smooth curve in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In contrast, our knapsack algorithm generates a sorted ordering list that does not depend on B, and prescribes to sort items successively according to that list, with order sizes equal to the minimal, mean or maximum demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 0 5 10 15 20 25 30 Order quantity of item 1 0 5 10 15 20 25 30 35 Order quantity of item 2 Optimal order quantity Mean-MAD policy Mean-variance policy (a) Ordering policy 0 10 20 30 40 50 60 Budget 10 20 30 40 50 60 70 80 90 Expected costs Triangular Mean-MAD lower bound Mean-MAD upper bound Mean-variance bound (b) Newsvendor costs Figure 3 Mean-variance and mean-MAD bounds and ordering policies for the newsvendor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The mean- variance curves are obtained through solving (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The mean-MAD policy corresponds to the optimal solution of (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The mean-MAD upper and lower bounds correspond to the extremal three- and two- point distributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The ‘true’ cost function assumes that D follows a symmetric triangular distribution on [10,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We emphasize that these results are not meant to numerically compare the mean-MAD and mean-variance policies, because the displayed differences merely express different ways of dealing with ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Indeed, it is hard to compare both policies as the respective ambiguity sets can contain vastly different distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For instance, a finite variance excludes distributions with an infinite second moment, while finite MAD does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For 18 Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor our purposes, MAD and variance are equally adequate descriptors of dispersion, and both are easily calibrated on data using basic statistical estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The crucial difference in the DRO context of this paper is that MAD leads to a simple, budget-consistent ordering policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Expected value of additional information We introduce as performance measure the expected value of additional information (EVAI), defined as EVAI(qU B) = C(qU B) − C(q∗ B) C(q∗ B) , where qU B is the robust ordering policy and q∗ B is the optimal ordering policy when the joint demand distribution is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We let B run from 0 to �n i=1 q∗ i =: Bopt, and consider nine different demand distributions, listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Table 2 Nine distributions used for multi-item performance analysis Case Case Case 1 Uniform[10,50] 4 Beta(1,3) on [0,50] 7 Triangular(10,50,18) 2 Uniform[10,100] 5 Beta(2,2) on [0,50] 8 Triangular(10,50,30) 3 Uniform[10,200] 6 Beta(3,1) on [0,50] 9 Triangular(10,50,42) We consider n = 25 items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For each item i, let ci = di = 1 and assume identically dis- tributed demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For example, in Case 2 the demand Di for each item i follows the uniform distribution with parameters ai = 10 and bi = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Table 3 provides an overview for the mark-up, representing low, average and high margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For the low margin regime, Figure 4 shows results for each of the nine cases, for both the robust ordering policy with mean-MAD-range information, and for the policy that uses the additional information βi = P(Di ⩾ µi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For the former, the worst performance over all nine cases has a maximum deviation of approximately 23% compared to the optimal order quantity q∗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Overall, the performance of the robust policy only deviates a few percent from the optimal performance with full information availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For the uniformly distributed cases (Cases 1-3), the performance decreases when the range increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For beta distributed demand (Cases 4-6), right-tailed distributions perform worse than left-tailed distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This effect is also observed for the triangular distributions (Cases 7-9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The policy with additional information βi = P(Di ⩾ µi) performs somewhat better in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor 19 Table 3 Mark-up values for all 25 items in the newsvendor model Mark-up m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11 m12 m13 Low margin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='55 Average margin 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='88 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='5 High margin 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='42 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='63 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='83 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='46 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='67 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='88 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='5 Mark-up m14 m15 m16 m17 m18 m19 m20 m21 m22 m23 m24 m25 Low margin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='96 1 Average margin 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='88 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='88 4 High margin 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='71 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='92 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='12 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='33 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='54 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='96 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='17 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='37 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='58 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='79 9 Figure 5 shows similar results for high margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The EVAI for the robust policy remains mostly below 10% for lower budget levels, but starts increasing rapidly when the budget approaches Bopt (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', when approaching the unconstrained model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' When the budget is less restrictive, additional distributional information provides substantial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In particular, since the policy uses skewness information βi, it performs better (in expectation) for higher budget levels than the robust ordering policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We present some more performance plots for the average margin setting and additional numerical experiments with mean-variance information in EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We next quantify the value of MAD information by comparing the performance with the situations when only the mean and range of demand is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For the low margin setting, Figure 6 shows the EVAI for the ordering policy with only mean-range informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Like the mean-MAD policy, this policy follows from a discrete distribution, in this case the extremal distribution on {a,b} with probabilities b−µ b−a and µ−a b−a that attains the Edmundson-Madansky bound (see Ben-Tal and Hochman, 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' That is, instead of the worst-case three-point distribution, we take the expectation in (6) over this two-point dis- tribution and find the robust mean-range ordering policy using the resulting LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The plots clearly demonstrate that knowledge on dispersion in terms of MAD improves performance considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 20 Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor 0 224 449 674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='14 Case 1: Uniform (10,50) Mean-MAD Mean-MAD- 0 401 802 1204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='14 Case 2: Uniform (10,100) 0 755 1510 2265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='14 Case 3: Uniform (10,200) 0 70 140 211 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='17 Case 4: Beta (1,3) 0 187 374 561 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='20 Case 5: Beta (2,2) 0 312 624 937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='18 Case 6: Beta (3,1) 0 190 380 571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='23 Case 7: Triangular (10,50,18) 0 236 472 709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='20 Case 8: Triangular (10,50,30) 0 277 554 831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='17 Case 9: Triangular (10,50,42) Figure 4 The results for the low margin setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The x-axis corresponds to B and the y-axis to the EVAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 0 370 740 1110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='34 Case 1: Uniform (10,50) Mean-MAD Mean-MAD- 0 729 1458 2187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='34 Case 2: Uniform (10,100) 0 1446 2892 4339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='35 Case 3: Uniform (10,200) 0 201 403 605 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='63 Case 4: Beta (1,3) 0 319 638 958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='55 Case 5: Beta (2,2) 0 396 792 1188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='34 Case 6: Beta (3,1) 0 306 612 918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='54 Case 7: Triangular (10,50,18) 0 329 658 987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='57 Case 8: Triangular (10,50,30) 0 361 722 1084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='61 Case 9: Triangular (10,50,42) Figure 5 The results for the high margin setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The x-axis corresponds to B and the y-axis to the EVAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor 21 0 224 449 674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='77 Case 1: Uniform (10,50) Mean-MAD Mean-MAD- E-M 0 401 802 1204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='77 Case 2: Uniform (10,100) 0 755 1510 2265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='77 Case 3: Uniform (10,200) 0 70 140 211 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='48 Case 4: Beta (1,3) 0 187 374 561 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='35 Case 5: Beta (2,2) 0 312 624 937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='10 Case 6: Beta (3,1) 0 190 380 571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='69 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='03 Case 7: Triangular (10,50,18) 0 236 472 709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='63 Case 8: Triangular (10,50,30) 0 277 554 831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='89 Case 9: Triangular (10,50,42) Figure 6 The results for the low margin setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The x-axis corresponds to B and the y-axis to the EVAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The E-M performance plot refers to the model with only mean information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Conclusions This paper establishes new ordering policies for the newsvendor with partial demand in- formation (mean, MAD and range) with a budget constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The ordering policies follow from a minimax approach, where we search for the order quantities with minimal costs for the maximal (worst-case) cost function restricted to demand distributions that comply with the partial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The minimax analysis for the multi-item setting gives rise to a knapsack problem, and the solution of this knapsack problem in fact is the ordering policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This policy prescribes to sort items based on their marginal effect on the total costs, reminiscent of the greedy algorithm that solves the continuous knapsack problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The ordering policy only orders the minimum, mean or maximum demand for each item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Hence, the decision maker can rank the items based on their marginal effects, and then start ordering items according to this list until the budget is spent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The fact that the ranking list is easy to generate, and that the ‘order of ordering’ does not depend on the budget, makes the policy transparent and easy to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Existing approaches for full and partial (such as mean-variance) knowledge of the demand distribution lack this property of ‘budget-consistency’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 22 Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor The minimax approach provides robustness, with an ordering policy that protects against all distributions that comply with the partial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This approach avoids the need to estimate the demand distribution, which can be a daunting process in practice and is prone to errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' However, the minimax approach comes at the risk of being overly conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Through extensive numerical experiments we compared the robust policies for partial demand settings with the policies for full demand settings, and observed that the proposed policies perform well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' At the heart of our analysis lies the idea to set up the robust minimax analysis with MAD information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' With MAD as dispersion measure we obtained a tractable optimization model, with a solution in terms of a robust ordering policy that satisfies the budget- consistency property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Using MAD to formulate solvable minimax problems can also be applied to other inventory models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We demonstrate this idea in EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='7 for three extended settings: the newsvendor with multiple contraints, the newsvendor with unreliable supply, and the risk-averse newsvendor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In all three cases, the minimax analysis leads to a tractable 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', and Rustem, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Distributionally robust joint chance constraints with second- order moment information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Mathematical Programming, 137(1):167–198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec1 E-Companion to “Robust knapsack ordering for a partially-informed newsvendor with budget constraint” EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Proofs Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In their original work, Ben-Tal and Hochman (1972) prove this result for general convex functions by dividing the support into two intervals [a,µ] and [µ,b] and then applying the Edmundson-Madansky bound to both subintervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The following proof uses semi-infinite programming duality and is taken from van Eekelen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Consider a general convex function f(x) (this includes (x − q)+ as a special case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For X ∼ P ∈ P(µ,δ), we solve max P(x)⩾0 � b a f(x)dP(x) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' � b a dP(x) = 1, � b a xdP(x) = µ, � b a |x − µ|dP(x) = δ, (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1) Consider the dual of (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1), min λ0,λ1,λ2 λ0 + λ1µ + λ2δ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' M(x) := λ0 + λ1x + λ2|x − µ| ⩾ f(x), ∀x ∈ [a,b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2) The function M(x) has a ‘kink’ at x = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Since the dual problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2) has three variables, the optimal M(x) touches f(x) at three points: x = a, µ and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For this choice of M(x), λ0 = f(a) − λ1a − λ2(µ − a), λ1 = 1 2 �f(b) − f(µ) b − µ + f(µ) − f(a) µ − a � , λ2 = 1 2 �f(b) − f(µ) b − µ − f(µ) − f(a) µ − a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Because the majorant is piecewise linear and convex, we can majorize every convex function f(x) by letting M(x) touch at the boundary points a,b and at the kink point x = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' According to the complementary slackness property, these points constitute the support of the extremal distribution, and the optimal probabilities follow from solving the linear system resulting from the equations of (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This is a linear system of three unknown probabilities and three equations, with the solution pa = δ 2(µ − a), pµ = 1 − δ 2(µ − a) − δ 2(b − µ), pb = δ 2(b − µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Finally, for these primal and dual solutions, we verify that the objective values of problems (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1) and (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2) agree, which confirms that strong duality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' □ ec2e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We prove this result for general convex f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For a random variable X with distribution P ∈ P(µ,d,β), the tight lower bound follows from max P(x)⩾0 � b a f(x)dP(x) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' � b a dP(x) = 1, � b a xdP(x) = µ, � b a |x − µ|dP(x) = δ, � b a 1{x⩾µ}dP(x) = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='3) Consider the dual of (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='3), min λ0,λ1,λ2 λ0 + λ1µ + λ2δ + λ3β s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' M(x) := λ0 + λ1x + λ2|x − µ| + λ31{x⩾µ} ⩽ f(x), ∀x ∈ [a,b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='4) Here M(x) has both a ‘kink’ and a jump discontinuity at x = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Let the function M(x) touch the epigraph of f(x) in two points on opposite sides of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' If we insert this knowledge, the constraints in the dual problem reduce to two equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' From the Karush- Kuhn-Tucker conditions, we deduce the optimal tangent points: x1 = µ + δ 2β , x2 = µ − δ 2(1 − β), which correspond to υ1 and υ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Substituting this solution and solving for λ0,λ1,λ2 and λ3 gives λ0 = f(υ2) + (λ1 − λ2)δ 2(1 − β) − λ1µ, λ3 = f(υ1) − f(υ2) + λ2δ (1 − β) − (λ2 + λ1)δ 2β(1 − β) , and hence the optimal value is given by βf(υ1) + (1 − β)f(υ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' To ensure the solution is dual feasible, we assign suitable values to the two free decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' That is, we let λ1 + λ2 and λ1 − λ2 equal the slope of f(x) at x = υ1 and υ2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The optimal probabilities of (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='3) are obtained by solving the linear system resulting from (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' □ EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Known properties of MAD We recall some well-known properties of the MAD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Ben-Tal and Hochman (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Denote by σ2 the variance of the random variable X, whose distribution is known to belong to the set P(µ,δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Then δ2 4β(1 − β) ⩽ σ2 ⩽ δ(b − a) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec3 In particular, since δ2 ⩽ 4β(1 − β)σ2 ⩽ σ2, it holds that δ ⩽ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For a proof, we refer the reader to Ben-Tal and Hochman (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For the distributions used in the paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' explicit formulas for δ are available: Uniform distribution on [a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='b]: δ = 1 4(b − a) Beta distribution with parameters k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='λ on support [a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='b]: δ = 2kkλλΓ(k + λ) (k + λ)k+λ+1Γ(k)Γ(λ)(b − a) Triangular distribution on [a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='b] with mode c: δ = � � � � � 2(b+c−2a)3 81(a−b)(a−c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' for a + b < 2c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 2(a+c−2b)3 81(a−b)(b−c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' for a + b > 2c Normal distribution N(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='σ2): δ = � 2 πσ Gamma distribution with parameters λ and k (for which µ = k/λ): δ = 2kk Γ(k)exp(k) 1 λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The MAD is known to satisfy the bound 0 ⩽ δ ⩽ 2(b − µ)(µ − a) b − a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='5) Let β = P(X ⩾ µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For example, in the case of continuous symmetric distribution of X we know that β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This quantity is known to satisfy the bounds: δ 2(b − µ) ⩽ β ⩽ 1 − δ 2(µ − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='6) EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The knapsack problem The knapsack problem (Kellerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', 2004) is an integer programming problem and can be formulated as max x � i=1 pixi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' n � i=1 cixi ⩽ B, xi ∈ {0,1}, 1 = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='7) ec4e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor for decision variable x, budget B, price p > 0 and costs c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Assume B < �n i=1 ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The contin- uous version is obtained by considering the linear relaxation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', we replace the integrality constraints by 0 ⩽ xi ⩽ 1, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The so-called greedy choice algorithm produces an optimal solution for the continuous knapsack problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We first renumber the items xi such that p1/c1 ⩾ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' ⩾ pn/cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Hence, the first item causes the largest increase in value relative to its costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We now iterate over x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',xn and in each iteration, set xi to its maximum capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' When the budget constraint is violated, set xi = B − i−1 � i=1 cixi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This greedy choice algorithm produces the optimal solution to (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Below we will state its proof, which is an adaptation from the proof in Kellerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Assume that without loss of generality that p1/c1 > ··· > pn/cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' If we would have pi/ci = pi+1/ci+1 for some i, then we are indifferent between those items and the proof below can be easily adapted to satisfy this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The greedy choice algorithm produces a solution such that, for some index j, we have 1 = x1 = ··· = xj−1 > xj ⩾ xj+1 = ··· = xn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Suppose we would have a different feasible optimal solution y ̸= x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Since pi > 0 and �n i=1 ci > B, it must hold that �n i=1 ciyi = B as otherwise we could spend additional capital to increase the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Because p1/c1 ⩾ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' ⩾ pn/cn, there exists a smallest index k such that yk < 1 and let l be the smallest index such that k < l and yl > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This solution must exists, else we would have y = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Now, we will increase the value of yk and decrease the value of yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' By choosing ϵ = min{ck(1 − yk),clyl} > 0 and increasing yk by ϵ/ck and decreasing yl by ϵ/cl, we maintain feasibility and preserve �n i=1 ciyi = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The solution value changes by pkϵ/ck −plϵ/cl = ϵ(pk/ck − pl/cl) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This contradicts the assumption that y is an optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Therefore, x is optimal which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' DRO results In Ben-Tal and Hochman (1972), the following result was proved (for a much larger class of functions f(y,X) than in our case): Proposition EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' If f(y,·) is convex, sup P∈P(µ,δ) EP[f(y,X)] = gU(y) = � κ∈{1,2,3}n n � i=1 p(i) κi f(y,ξ(1) κ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',ξ(n) κn ), (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='8) e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec5 with p(i) κi ,ξ(i) κi defined as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' If f(y,·) is concave, sup P∈P(µ,δ,β) EP[f(y,X)] = gL(y) = � κ∈{1,2}n n � i=1 ˆp(i) κi f(y,υ(1) κ1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',υ(n) κn ), (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='9) with υ(i) 1 = µi + δi 2βi, υ(i) 2 = µi − δi 2(1−βi) and ˆp(i) 1 = βi, ˆp(i) 2 = 1 − βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Hence, gU(·) in (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='8) inherits the convexity in y from f(·,X) and its functional form depends only on the form of f(·,X) (and similarly for gL(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The upper and lower bound give a closed interval for ValP(y) = EP[f(y,X)] ∀P ∈ P(µ,δ,β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='10) Corollary EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' If f(y,·) is convex for all y then ValP(y) ∈ [gL(y),gU(y)] ∀P ∈ P(µ,δ,β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' If f(y,·) is concave for all y then ValP(y) ∈ [gU(y),gL(y)] ∀P ∈ P(µ,δ,β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' From Proposition EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1 we see that the extremal distribution is independent of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Hence, we can substitute the 3n terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This leads to a convex function in y, and hence the minimization problem over y is tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Robust analysis with mean-variance knowledge EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Scarf’s result for single item Scarf (1958) introduced a distribution-free analysis for the single-item newsvendor model by assuming that the decision maker only knows the mean and variance of the demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Define the ambiguity set containing all distributions with the same mean and variance as P(µ,σ) := {P|EP(D) = µ, EP(D2) = σ2 + µ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Scarf (1958) determined an upper bound on the cost function C(q) by finding the worst- case distribution in the ambiguity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' To find the order quantity that protects against the ambiguity in P(µ,σ), the following minimax optimization problem is solved: min q max P∈P(µ,σ) dq + (m + d)EP(D − q)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Since max P∈P(µ,σ) EP(D − q)+ ⩽ � σ2 + (µ − q)2 + (µ − q) 2 , ec6e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor this minimax optimization problem becomes minq maxP CS(q) with CS(q) := d(q − µ) + (m + d) � σ2 + (µ − q)2 + (µ − q) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='11) and solution qS := argmin q CS(q) = µ + σ 2 ��m d − � d m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='12) The quantity qS is known as Scarf’s order quantity which prescribes to order more than the expected demand when m > d, and less than the expected demand when d < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Gallego and Moon When the model is based on mean-variance information, Gallego and Moon (1993) formu- late the problem as min q CS(q) := n � i=1 ci � �di(qi − µi) + (mi + di) � σ2 i + (qi − µi)2 − (qi − µi) 2 � � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' n � i=1 ciqi ⩽ B, (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13) q ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The optimal solution to problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13) is referred to as qS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Applying Scarf’s bound for each item individually results in (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Similar to the full information setting with a known distribution, this optimization problem can be solved with Lagrange multiplier techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Additional numerical experiments This section presents additional numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Section EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1 presents the performance plots for the average margin setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We compare the mean-MAD and mean-variance ordering policies in Section EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' More mean-MAD results Figure EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2 depicts the results for the average profitability scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' A quick glance re- veals that these plots exhibit a different impression than the low profitability scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We conclude that the mean-MAD EVAI remains below some bound for budget levels ranging from zero to two-thirds of the maximum budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For all cases, this bound on the EVAI is around 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='As the budget passes two-thirds of the maximum budget, the performance starts to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' However, the mean-MAD-β EVAI decreases when approaching the max- imal budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec7 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='026 Case 1: Uniform (10,50) 10 25 40 55 70 85 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='0105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='0110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='0115 Case 2: Uniform (10,100) 10 40 70 100130160190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='0052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='0054 Case 3: Uniform (10,200) 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='12 Case 4: Beta (1,3) 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='12 Case 5: Beta (2,2) 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='12 Case 6: Beta (3,1) 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='04 Case 7: Triangular c = (18) 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='04 Case 8: Triangular c = (30) 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='04 Case 9: Triangular c = (42) Figure EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1 Nine probability density functions used for multi-item performance analysis 0 314 628 942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='20 Case 1: Uniform (10,50) Mean-MAD Mean-MAD- 0 602 1205 1808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='20 Case 2: Uniform (10,100) 0 1180 2360 3540 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='20 Case 3: Uniform (10,200) 0 137 275 413 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='12 Case 4: Beta (1,3) 0 263 527 791 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='23 Case 5: Beta (2,2) 0 367 735 1103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='20 Case 6: Beta (3,1) 0 252 505 758 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='16 Case 7: Triangular (10,50,18) 0 287 574 861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='21 Case 8: Triangular (10,50,30) 0 330 661 992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='32 Case 9: Triangular (10,50,42) Figure EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2 The results for the average margin setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The x-axis corresponds to B and the y-axis to the EVAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Mean-variance comparison We start the performance analysis for the low margin scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The x-axis refers to the budget level B, and the y-axis refers to the EVAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In each plot, the blue line corresponds ec8e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor to the EVAI for the mean-MAD model and the orange line to the mean-variance EVAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Figure EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='3 contains the performance plots for each of the nine cases we are considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 0 224 449 674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='14 Case 1: Uniform (10,50) Mean-MAD Mean-variance 0 401 802 1204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='14 Case 2: Uniform (10,100) 0 755 1510 2265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='14 Case 3: Uniform (10,200) 0 70 140 211 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='17 Case 4: Beta (1,3) 0 187 374 561 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='20 Case 5: Beta (2,2) 0 312 624 937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='18 Case 6: Beta (3,1) 0 190 380 571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='23 Case 7: Triangular (10,50,18) 0 236 472 709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='20 Case 8: Triangular (10,50,30) 0 277 554 831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='17 Case 9: Triangular (10,50,42) Figure EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='3 The results for the low margin scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The x-axis corresponds to the budget level and the y-axis to the EVAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In Figure EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='3 we compare the mean-MAD policy with the mean-variance ordering policy in terms of EVAI for the scenario with low margins and a total of nine ground- truth demand distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' While both policies generally give low EVAIs, the EVAI of the mean-variance policy is typically lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We stress that this does not mean that the mean-variance policy is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Indeed, a fair numerical comparison is impossible, as the respective ambiguity sets can contain vastly different distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' While a finite variance excludes distributions with infinite-second moment, MAD does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In general, the worst- case scenarios or extremal distributions are ‘more extreme’ for MAD than for variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' This also offers a possible explanation for the slightly higher EVAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Extensions We now present a distribution-free analysis for three extensions of the multi-item newsven- dor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Section EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1 deals with multiple constraints, Section EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2 considers uncer- tain supply, and Section EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='3 discusses the risk-averse newsvendor where the conditional value at risk (CVaR) is chosen as objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec9 EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Multiple constraints Lau and Lau (1996) consider the newsvendor problem with multiple constraints, and pro- pose a numerical solution procedure that computes the Lagrange multipliers as roots of a system of nonlinear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Perakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (2020) also consider multiple capacity con- straints in a retail environment, and distinguish between warehouse capacity and inventory availability constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' By exploiting Lagrangian duality the problem is decomposed into two subproblems, which are solved iteratively by binary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We now argue that the distribution-free analysis developed in the present paper also carries over to the setting with multiple constraints, and takes the form min q n � i=1 ci � di(qi − µi) + (mi + di) � p(i) 1 (ai − qi)+ + p(i) 2 (µi − qi)+ + p(i) 3 (bi − qi)+�� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' n � i=1 ci,jqi ⩽ Bj j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',m qi ⩾ 0 i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='14) By introducing dummy variables τ (i) k , we reformulate problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='14) as min q,τ n � i=1 ci � di(qi − µi) + (mi + di) � p(i) 1 τ (i) 1 + p(i) 2 τ (i) 2 + p(i) 3 τ (i) 3 �� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' n � i=1 ci,jqi ⩽ Bj, j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',m, τ (i) k ⩾ ξ(i) k − qi, k = 1,2,3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n, τ (i) k ⩾ 0, k = 1,2,3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n, qi ⩾ 0, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n, (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='15) which remains a tractable LP, solvable for large-scale problems with interior-point meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Moreover, by solving the dual problem of (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='15), shadow prices of the m budget constraints can be computed that quantify marginal expected net benefit of allocating an additional unit of budget to Bj, j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Supply and demand uncertainty The newsvendor might take different decisions when the delivery of an order for q units is not necessarily complete (uncertain supply).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' K¨aki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (2015) consider uncertain supply and uncertain demand, when supply and demand are independent or follow a particular ec10e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor copula-based dependency structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In the mean-variance setting and under the indepen- dence assumption, Gallego and Moon (1993) solve the distribution-free newsvendor prob- lem with random yield, but assume the yield is a binomial random variable that depends on the order size q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' That is, when an order for q units is made, each individual unit is received with some fixed probability, or is not delivered at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' As opposed to Gallego and Moon (1993), we do introduce an ambiguity set for the random supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Consider the setting with multiplicative yield Zi, where the random supply is given by Zi·qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Assume Zi has mean ˜µi, MAD ˜δi and support [˜ai,˜bi], where 0 ⩽ ˜ai ⩽ ˜bi ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The distribution of Zi then resides in P(˜µi,˜δi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The extremal three-point distribution for Zi has probabilities ˜p(i) 1 = ˜δi 2(˜µi − ˜ai), ˜p(i) 2 = 1 − ˜δi 2(˜µi − ˜ai) − ˜δi 2(˜bi − ˜µi) , ˜p(i) 3 = ˜δi 2(˜bi − ˜µi) , and is supported on ζ(i) 1 = ˜ai, ζ(i) 2 = ˜µi, ζ(i) 3 = ˜bi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The multi-item newsvendor with supply ambiguity is equivalent to min q n � i=1 max P∈Pi EP � ci � di(Zi · qi − Di) + (mi + di)(Di − Zi · qi)+�� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' n � i=1 ciqi ⩽ Bj, j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',m, qi ⩾ 0, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n, (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='16) with Pi := P(µi,δi) × P(˜µi,˜δi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Since the newsvendor problem is jointly convex in the pairwise independent random variables Di and Zi, the distributions that maximize the objective function of (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='16) are the extremal three-point distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Applying these worst-case distributions to (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='16) results in min q n � i=1 ci � di( ˜µiqi − µi) + (mi + di) � κ∈{1,2,3}2 p(i) κ1 ˜p(i) κ2τ (i) κ � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' n � i=1 ciqi ⩽ B, τ (i) κ ⩾ ξ(i) κ1 − ζ(i) κ2 qi, κ ∈ {1,2,3}2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n, τ (i) κ ⩾ 0, κ ∈ {1,2,3}2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n, qi ⩾ 0, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='17) e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec11 To demonstrate the distribution-free newsvendor with uncertain supply, consider the one-dimensional case with random demand D with a uniform distribution on [20,80] and multiplicative yield Z uniformly distributed on [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='65,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Figure EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='4 depicts the tight lower and upper bounds that follow from optimizing over the ambiguity sets that contain the distributions of D and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' As the extremal distributions are discrete, the objective function of (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='17) admits a piecewise linear representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 0 20 40 60 80 100 120 Order quantity 15 20 25 30 35 40 45 50 Expected costs Uniform distributions Mean-MAD upper bound Mean-MAD lower bound Figure EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='4 Tight bounds for the multi-item newsvendor with uncertain supply yield, where m = 1 and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The upper piecewise linear function is obtained by evaluating E[D − Z · q], with D following the extremal distribution that lies in P(50,15,20,80) and Z the worst-case three-point distribution in P(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='8,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='075,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='65,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The lower bound follows from the best-case two-point distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The middle curve depicts the ‘true’ costs, where D has a uniform distribution on [20,80], and Z is uniformly distributed on [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='65,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Because problem (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='16) can be written in terms of a piecewise linear function, the optimal solution follows from a knapsack algorithm similar to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Further, one can gain additional insights by explicitly deriving the optimal order quantities for the robust single-item model, as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The problem is similar for additive yield, also resulting in a three-point distribution for the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Other directions for future research include solving (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='16) with multiple unreliable and non-identical suppliers (Dada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', 2007) and the newsvendor problem with fixed ordering costs and supplier capacity restrictions (Merzifonluoglu and Feng, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' ec12e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Risk aversion We next consider a risk-averse decision maker, as in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (2010), who makes decisions based on CVaR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The decision maker no longer optimizes the expected costs, but instead minimizes the average value of the costs exceeding the γth-quantile of the newsvendor’s cost distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For the cost function G(q,D), CVaR can be calculated by solving a convex minimization problem (Rockafellar and Uryasev, 2000): min θ∈R � θ + 1 1 − γ E(G(q,D) − θ)+ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Calculating CVaR requires full knowledge of the demand distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' However, in practice, committing to a particular distribution might be problematic for the decision maker if there is not enough data available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Hence, we consider the partial information setting as in Zhu and Fukushima (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Delage and Ye (2010), and seek to solve min q:� i ciqi⩽B,qi⩾0 max P∈P(µ,δ) min θ∈R � θ + 1 1 − γ EP(G(q,D) − θ)+ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='18) Let us first consider the single-item model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Because the objective function of (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='18) is finite, P(µ,δ) is weakly compact as supp(D) is compact, and the objective function of (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='18) is linear in P and convex in θ, we are allowed to interchange the maximization and minimization operators by virtue of the minimax theorem (Shapiro and Kleywegt, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Since (G(q,D) − θ)+ is a convex function of the uncertain demand, the three-point distri- bution (10) also maximizes EP(G(q,D)−θ)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' When β = P(D ⩾ µ) is known, the two-point distribution in Lemma 3 attains the matching lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For the multivariate problem, notice that (G(q,D) − θ)+ is again a convex function of the uncertain demand, where D ∼ P ∈ P(µ,δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' By Proposition EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='1 and the reasoning above, the risk-averse newsvendor e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec13 admits the following LP representation: min q,τ,η,θ θ + 1 1 − γ � κ∈{1,2,3}n n � i=1 p(i) κi ηκ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' n � i=1 ciqi ⩽ B, ηκ ⩾ � n � i=1 ci � di(qi − ξ(i) κi ) + (mi + di)τ (i) κ �� − θ, κ ∈ {1,2,3}n, ηκ ⩾ 0, κ ∈ {1,2,3}n, τ (i) κ ⩾ ξ(i) κi − qi, κ ∈ {1,2,3}n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n, τ (i) κ ⩾ 0, κ ∈ {1,2,3}n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n, qi ⩾ 0, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='19) We show in Figure EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='5 the bounds for the single-item model with demand having support [10,50], µ = 30, δ = 20/3 and β = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Solving (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='19) for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='75,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='95 and different order sizes yields the upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We solve an analogous problem, but with the expectation taken over the extremal two-point distribution, stated in Lemma 3, to obtain the tight lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' As a point of reference, we also plot the exact values of the CVaR and expected costs when D follows a symmetric triangular distribution on [10,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' 10 15 20 25 30 35 40 CVaR99% 6 8 10 12 14 16 18 20 C(q) q = 10 q = 20 q = 30 q = 40 q = 50 Triangular Mean-MAD upper bound Mean-MAD lower bound (a) Expected costs and CVaR 10 15 20 25 30 35 40 45 50 Order quantity 5 10 15 20 25 30 CVaR75% Triangular Mean-MAD upper bound Mean-MAD lower bound (b) Mean-MAD bounds for CVaR Figure EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='5 An illustration of the tight mean-MAD bounds for the risk-averse newsvendor with CVAR as objec- tive criterion, where m = 1, d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='8 and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='75,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The middle curve corresponds to the CVaR when D follows a symmetric triangular distribution on [10,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' The upper and lower bounds follow from optimizing over the ambiguity sets that contain this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Solving (EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='19) can be challenging since the objective function (G(q,D) − θ)+ is no longer separable, thus resulting in an exponential number of variables and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' To ec14e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor alleviate this computational difficulty, one might resort to sampling-based procedures such as sample average approximation (Shapiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' We also mention ambiguous chance constraints that can be conservatively approximated by CVaR (Nemirovski and Shapiro, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In the risk-averse newsvendor setting, the deci- sion maker introduces an ambiguous chance constraint that restricts the probability of the costs exceeding a certain threshold t to be less than 1 − γ, considering all distributions in the ambiguity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For the multi-item setting, this means ensuring P(G(q,D) > t) ⩽ 1 − γ, ∀P ∈ P(µ,δ), which is implied by max P∈P(µ,δ) CVaRγ[G(q,D)] ⩽ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' In addition, the newsvendor might require a minimal probability that all customer orders will be completely covered by the inventory on hand, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', the type-1 service level (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' When several of these probabilistic constraints are interrelated, the decision maker should conservatively approximate joint chance constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' For this one can again use CVaR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' see Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Zymler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Roos and den Hertog (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} +page_content=' Adding ambiguous chance constraints to the models developed in this paper is a worthwhile topic for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E0T4oBgHgl3EQfywIT/content/2301.02662v1.pdf'} diff --git a/DNFQT4oBgHgl3EQf_zdP/content/tmp_files/2301.13459v1.pdf.txt b/DNFQT4oBgHgl3EQf_zdP/content/tmp_files/2301.13459v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b21481ec9d7d53abeda89adc2f3330e4331392b1 --- /dev/null +++ b/DNFQT4oBgHgl3EQf_zdP/content/tmp_files/2301.13459v1.pdf.txt @@ -0,0 +1,1163 @@ +Learning Generalized Hybrid Proximity Representation for +Image Recognition +1st Zhiyuan Li +Department of Computer Science +University of Cincinnati +Cincinnati, OH, United States +li3z3@mail.uc.edu, +2nd Anca Ralescu +Department of Computer Science +University of Cincinnati +Cincinnati, OH, United States +ralescal@ucmail.uc.edu +Abstract—Recently, deep metric learning techniques received +attentions, as the learned distance representations are useful to +capture the similarity relationship among samples and further +improve the performance of various of supervised or unsuper- +vised learning tasks. We propose a novel supervised metric +learning method that can learn the distance metrics in both +geometric and probabilistic space for image recognition. In +contrast to the previous metric learning methods which usually +focus on learning the distance metrics in Euclidean space, our +proposed method is able to learn better distance representation +in a hybrid approach. To achieve this, we proposed a Generalized +Hybrid Metric Loss (GHM-Loss) to learn the general hybrid +proximity features from the image data by controlling the trade- +off between geometric proximity and probabilistic proximity. +To evaluate the effectiveness of our method, we first provide +theoretical derivations and proofs of the proposed loss function, +then we perform extensive experiments on two public datasets +to show the advantage of our method compared to other state- +of-the-art metric learning methods. +Index Terms—Deep metric learning, proximity, probability +distribution, representation learning, image classification +I. INTRODUCTION +Metric learning takes input data to learn the similar and +dissimilar features between samples. The learned distance +metric provides a meaningful and robust representation to +discriminate the proximity or distance between samples and +can be further utilized for both supervised and unsupervised +learning tasks [1]. Recently, deep learning-based metric learn- +ing algorithms, i.e., deep metric learning, were widely applied +in the computer vision area by developing either a novel +network architecture or an intuitive and efficient loss function +[2]–[4]. Some typical works, such as the Siamese network [5], +Triplet network [2], SupCon [6], aim to formulate an instance +discrimination task to learn a useful feature representation +by optimizing the proximity function in the Euclidean space, +i.e., geometric distance or Cosine proximity between the +feature embeddings. In this paper, we seek to address the +inadequacies of geometric proximity of recent state-of-the- +art metric learning methods by reconsidering an alternative +Copyright (c) 2022 IEEE. Personal use of this material is permitted. +Permission from IEEE must be obtained for all other uses, in any current or +future media, including reprinting/republishing this material for advertising or +promotional purposes, creating new collective works, for resale or redistribu- +tion to servers or lists, or reuse of any copyrighted component of this work +in other works. +approach in which learned distance metrics are not biased to +only geometric proximity. +Metric learning methods have shown an excellent classifi- +cation performance in image recognition applications, due to +their extraordinary ability to discriminate similar information +between samples [1], [4], [7], [8]. Such metric learning tasks +can be either supervised or self-supervised. In supervised +metric learning, the model learns to pull together the samples +from the same classes and push away the samples from dif- +ferent classes [9]. Self-supervised metric learning, also named +contrastive learning, requires a data augmentation step to cre- +ate some pseudo-ground-truth from the data itself, where the +augmentation from the same sample is included in “positive” +pairs and the augmentation from different samples is included +in “negative” pairs [10]. Similar to supervised metric learning, +the self-supervised model learns similar representations from +the positive pairs and should be different than the representa- +tions of the negative pairs. Various types of metric/contrastive +learning works have been developed for image pattern recogni- +tion applications, including image classification [6], [11]–[13], +image clustering [14], [15], image segmentation [16], [17], +image reconstruction [18], [19], and object detection [20]. All +these works used geometric proximity (e.g., Cosine similarity +or Euclidean distance) as the proximity function in the training +an objective loss to learn the geometric representation of the +samples. However, the probability distribution of the samples +should not be ignored. +To overcome these limitations and boost the prediction per- +formance of metric learning, we proposed a novel supervised +metric learning method to learn the hybrid proximity that +combines the proximity in both geometric and probabilistic +space. To achieve it, we defined a supervised Generalized +Hybrid Metric Loss (GHM-Loss) to better learn the distance +representations in both geometric and probabilistic space. We +noticed that even if the geometric distance is small, the +probabilistic distance can be large when the sample variance +is large (Figure 1). This observation reminds us to reconsider +that the model may not sufficiently learn the distance features +based only on the geometric distance between data points. +Thus, enabling the model to partially learn the probabilistic +distance controls the trade-off between two types of distance +representation (geometric and probabilistic). +arXiv:2301.13459v1 [cs.CV] 31 Jan 2023 + +𝑋2 +𝑋3 +𝐷 𝜇𝑋1, 𝜇𝑋2 > 𝐷 𝜇𝑋2, 𝜇𝑋3 +𝐾𝐿 𝑃𝑋1||𝑃𝑋2 < 𝐾𝐿 𝑃𝑋2||𝑃𝑋3 +𝑋1 +Fig. 1. +Geometric distance vs. probabilistic distance between probability +distributions X1, X2 and X3. Without considering the variances, the geomet- +ric distance between mean values of µX1, µX2, and µX3 cannot represent +their probabilistic distance. D: Geometric distance; KL: Kullback–Leibler +divergence. +Our proposed GHM-Loss is formulated by a hybrid di- +vergence underlying the geometric and probabilistic space, +which is a generalized distance loss form for many defined +metric learning methods, including Triplet [2], N-pairs [21], +Max Margin [22], NTXent [13], SupCon [6], etc. We first +theoretically showing the advantage of using the probabilistic +distance in metric learning compared to the geometric-based +distance, and we employed two public datasets to show the +effectiveness of our method compared to other state-of-the- +art metric/contrastive learning methods. We also investigated +the superiority of the proposed GHM-Loss with other met- +ric/contrastive learning loss functions. To sum up, our main +findings and contributions to this work are as follows: +1) We propose a novel supervised metric learning method +for enhancing the performance of image recognition +by defining a Generalized Hybrid Metric Loss (GHM- +Loss). The proposed GHM-Loss is able to learn better +distance representation that controls the trade-off be- +tween the geometric-based and the probabilistic distance +from feature embeddings. +2) We define two proximity functions with certain proper- +ties in geometric and probabilistic space, respectively, +and provide proof for each property. Meanwhile, we +theoretically show the advantage of the GHM-Loss by +including the probabilistic proximity for learning the +distance between distributions. +3) Our approach is supported both by a theoretical discus- +sion and by extensive experiments performed on two +common image classification tasks to demonstrate the +effectiveness of our method compared to other state-of- +the-art metric learning methods. +II. RELATED WORK +In this section, we first discuss some state-of-the-art meth- +ods of deep metric learning and some of its applications in +the computer vision domain. We further review related works +on contrastive learning. +A. Metric Learning +1) Traditional Metric Learning: The early stage of the ma- +chine learning techniques requires a hand-crafted processing +step, i.e., feature engineering, such as feature selection and +feature extraction before training a machine learning model +for supervised (e.g., classification) or unsupervised learning +(e.g., clustering) tasks [7], [23], [24]. These methods, including +linear projections, i.e., principal component analysis (PCA) +[25], decomposition, i.e., non-negative matrix factorization +(NMF) [26] to extract useful feature information and are not +directly within the classification structure, resulting in a limited +performance on the certain complex structure, such as high- +dimensional data and non-linearity. Unlike traditional machine +learning approaches, metric learning performs the learning +process on the data to learn a distance feature representation +by decreasing the distance between similar samples and in- +creasing the distance between dissimilar ones in a embedding +space. The learned distance features will have a high ability +to discriminate the classes of the sample data. Usually, metric +learning approaches apply linear transformation techniques +to the input data and map it to a new feature space with +a higher-class separation [27]. However, these methods lack +the generalization capability and nonlinear knowledge of the +attributes [28]. +2) Deep Metric Learning: Unlike traditional metric learn- +ing methods, deep metric learning relies on training deep +neural networks with activation functions that capture nonlin- +ear properties [4], and it has dominated metric representation +learning in the image recognition community [2], [3], [5], [6], +[29]–[31]. For example, the Siamese network [5] used two +identical convolutional neural networks (CNNs) to encode a +pair of input samples and minimize the contrastive loss to learn +the representative distance features. Similar to the Siamese +network, Hoffer et al [2] proposed a Triplet network, including +the anchor, positive (similar), and negative (dissimilar) sample, +which learns the inequality that the positive sample stays closer +to the anchor compared to the negative sample. Afterward, +Wang et al [3] defined a new Angular loss to constrain +the angle at the negative sample from the Triplet network. +Later, Sohn [21] proposed an N-pair loss to address the +slow convergence problem of the Triplet loss. More recently, +Khosla et al [6] developed a supervised contrastive learning +framework with the more general form of metric learning loss, +i.e., SupCon, and showed the effectiveness of classification +performance compared to the Triplet loss and the N-pair loss. +These existing works have shown great promise for metric +and feature representation learning in a variety of image +classification tasks. Nevertheless, to the best of our knowledge, +most previous studies are focused on learning the geometric- +based metrics of the embedding space, while the probabilistic- +based metrics are usually ignored. In this work, our method +is able to learn the meaningful metrics of both geometric and +probabilistic space. + +B. Contrastive Representation Learning +The main purpose of deep metric learning and contrastive +learning is to train a deep learning model to learn the distance +feature representations in an embedding space. The main +difference between these two methods is that contrastive learn- +ing is closely related to the self-supervised learning domain, +which contains a data augmentation step to generalize an +arbitrary number of positive and negative sample pairs from +each sample [32]. Given the stunning achievement of self- +supervised representation learning, many contrastive learning +methods have been developed for various computer vision +tasks [33]–[36]. For example, Ye et al [37] proposed an +embedding contrastive learning method with the Siamese +network to learn the invariant features of embedding space. +Chen et al [13] developed a famous contrastive learning +framework, SimCLR, in which the model is pretrained to +discriminate the positive pairs of data augmentation from the +same source image, demonstrating superior performance in +ImageNet classification. Similarly, He et al [12] proposed the +Moco v1, to maximize the proximity between the positive +pairs based on the monument network encoder. Additional +studies that are similar to SimCLR and Moco, including +BYOL [38], SimSam [39] Barlow Twins [40], etc., show the +exceptional performance of learned feature representation for +further supervised or unsupervised tasks. +… +𝑥1 +𝑥2 +𝑥𝑁−1 +𝑥𝑁 +𝐟1 +𝐟2 +𝐟𝑁−1 +𝐟𝑁 +… +𝐟1 +𝐟2 +𝐟𝑁−1 +𝐟𝑁 +Pull +Push +Learning Hybrid Metrics +Softmax +Encoder F(∙; 𝜽) +Classification +Normal +Normal +… +Fibrosis +Glaucoma +MLP +Feature Extraction +MLP +Input +𝐿2 +Fig. 2. The overview of our proposed framework (example of fundus disease +diagnosis). We use a pretrained convolutional neural network (CNN) and a +multi-layer perceptron (MLP) to encode each image to the embedded feature. +Afterward, we propose a metric learning branch that is supervised with the +proposed GHM-Loss which is trained together with the cross-entropy loss of +a classification branch in a multi-task scheme. +III. METHODOLOGY +A. Overview +Our proposed supervised metric learning framework is il- +lustrated in Figure 2. We first denote a training image dataset +D = {xi, yi}N +i=1, where yi is the label of image xi, a set +of indices of all the positive samples for a randomly selected +image xi in a batch, U(i) = {j ∈ Θ|yj = yi, j ̸= i}, a set of +indices of all negative samples for a randomly selected image +xi in a batch, V (i) := {j ∈ Θ|yj ̸= yi, i ̸= j}. The problem +formulation is to learn a network F(·; θ) that maps each input +xi to a L2 normalized d-dimensional feature embedding fi, +such as fi = F(xi; θ) ∈ Rd. To achieve this, we use a pre- +trained CNN, i.e., ResNet18, followed by a MLP to produce N +high-level feature vectors and perform two supervised learning +branches. The first branch is a metric learning task, which aims +to learn the robust metrics by pulling all the samples with +indices U(i) and pushing away all the samples with indices +V (i). Meanwhile, the embedded feature fi is connected to +another MLP layer with a Softmax to generate the predicted +probability for the class label yi and is supervised with cross- +entropy loss to perform a classification task. Below, we will +elaborate on the procedure of the each branch, including the +definition of GHM-Loss and its advantage, and other network +details. +B. Generalized Hybird Metric Loss +1) General Loss Form: To perform the metric learning +branch, we propose a general form metric loss function, +in which the network can learn the proximity information +between the embedded features {f1, f2, . . . , fN} by optimizing +the loss. Let S(·) denote the proximity function for two input +vectors fi and fj. That is, for fi, fj ∈ Rd, S(fi, fj) : Rd → R1. +Thus, the probability of xi, xu, u ∈ U(i) is being recognized +as yi is defined by +p(yi|xi, xu) = +exp [S(fi, fu)] +� +j∈Θ,j̸=i exp [S(fi, fj)] +(1) +On the other hand, the probability of xi, xv, v ∈ V (i) is not +being recognized as yi is defined by +p(yi|xi, xv) = +exp [S(fi, fv)] +� +j∈Θ,j̸=i exp [S(fi, fj)] +(2) +Next, assume that all the probabilities of different images be- +ing recognized as image xi are independent, let q(yi|xi, xv) = +1−p(yi|xi, xv) thus, the objective likelihood function that we +are interested is defined by +ℓi = +� +u∈U(i) +� +v∈V (i) +p(yi|xi, xu)q(yi|xi, xv) +(3) +Correspondingly, the negative log likelihood over all the data +points indexed by Θ yields: +L∗ = − +� +i∈Θ +∥V (i)∥ +� +u∈U(i) +log p(yi|xi, xu) +− +� +i∈Θ +∥U(i)∥ +� +v∈V (i) +log q(yi|xi, xv) +(4) +where ∥U(i)∥ and ∥V (i)∥ denotes the size of the set U(i) +and V (i), respectively. +2) Geometric Proximity: We first consider the proximity +in the geometric space. Given a pair vectors fi and fj, the +proximity function Sg(fi, fj) satisfies the following properties: +1 Sg(fi, fj) ∈ [0, 1]; +2 Sg(fi, fj) = Sg(fj, fi); +3 ∀c ∈ [fi, fj], Sg(fi, fj) ≤ min{Sg(fi, c), Sg(c, fj)}. +The proximity measures that satisfy the above properties +include Cosine similarity. + +Proof. Using Cosine similarity as the proximity metrics, such +that Sg(fi, fj) = fi · fj/∥fi∥∥fj∥ satisfies each property above. +1. Obviously, Sg(fi, fj) ∈ [0, 1]. +2. Obviously, this property is true. +3. Let c ∈ [fi, fj], we have |fi−c| ≤ |fi−fj|, |c−fj| ≤ |fi− +fj|, which means that Sg(fi, c) ≥ Sg(a, b), Sg(c, fj) ≥ +S(a, b). Thus, Sg(fi, fj) ≤ min{Sg(fi, c), Sg(c, fj)}∀c ∈ +[fi, fj]. +3) Probabilistic Proximity: Instead of only using geometric +proximity, which ignores the sampling probability distribution, +we consider the probabilistic proximity to summarize the +distribution of the embedded features {fi, f2, . . . , fN}. Given +a pair vectors fi and fj, with size of |fi|, |fj|, the probabilistic +proximity function satisfies the following properties: +1. Sp(fi, fj) ∈ [0, 1]; +2. Sp(fi, fj) = Sp(fj, fi); +3. Sp(fi, fj) = 0 if and only if fi = fj; +4. Sp(fi, fj) ≤ Sp(fi, fc) + Sp(fc, fj) under the certain +condition, in which |fc| = |fi| = |fj|. +We use a Gaussian mixture model (GMM) to represent the +empirical distribution fi, which is defined by +p(fi) = +� +k∈K +wkN(fi; µk, σ2 +k) +(5) +where wk is a latent variable followed by a categorical distri- +bution, denoting the k-th component, and N is the Gaussian +probability density function with parameters µk and σk, which +is defined as +N(fi; µk, σ2 +k) = +1 +� +2πσ2 +k +exp +� +− 1 +2σ2 +k +(fi − µk)2 +� +(6) +Using this model, the probabilistic distance between p(fi) +and p(fj) is chosen with the symmetric divergence, i.e., +Jensen–Shannon (JS)-divergence. For simplicity, we use pi +and pj to denotes the probability distributions of fi and fj, +respectively. Therefore, the Sp(fi, fj) is denoted by +Sp(fi, fj) = 1 +2 [dKL(pi∥¯pij) + dKL(pj∥¯pij)] +(7) +where ¯pij = (pi + pj) /2, dKL(·) presents a function of the +Kullback–Leibler (KL)-divergence. +Proof. We prove that Sp(fi, fj) satisfies the properties of +defined probabilistic proximity function. +1. The range of the JS-divergence is within 0 and 1, thus +Sp(fi, fj) ∈ [0, 1] is true. +2. Obviously, based on Eq (7), it is easy to have Sp(fi, fj) = +1 +2 [dKL(pi∥¯pij) + dKL(pj∥¯pij)] = Sp(fi, fj). +3. Sp(fi, fj) ≥ 0, as a sum of nonnegative terms. To have +Sp(fi, fj) = 0, each term of Sp(fi, fj) must be 0. Thus, +Sp(fi, fj) = 0 if and only if dKL(pi∥¯pij) = dKL(pj∥¯pij). +Since dKL(pi∥pj) = 0 if and only if pi = pj, thus, +Sp(fi, fj) = 0 if and only if fi = fj. +Now we prove property 4. Using the Shannon entropy, +H(pi) = − � +pi∈pi pi log pi, the explicit form of Sp(fi, fj) +can be written as +Sp(fi, fj) = H(¯pij) − 1 +2 [H(pi) + H(pj)] +Assume that H(¯pic) + H(¯pcj) ≥ H(¯pij) + H(¯pc), thus, +Sp(fi, fc) + Sp(fc, fj) − Sp(fi, fj) can be rewritten as +H(¯pic) − H(¯pc) + H(¯pcj) − H(¯pij) ≥ 0 +Thus, Sp(fi, fc) + Sp(fc, fj) ≥ Sp(fi, fj) is true if and only if +H(¯pic) + H(¯pcj) ≥ H(¯pij) + H(¯pc). +C. Learning Hybrid Proximity +1) Generalized Hybrid Metric Loss: The learning objective +loss function of the metric learning branch is the convex +combination of geometric proximity loss and probabilistic +proximity loss. As such, the objective is denoted by +L∗ +GHM = λL∗ +g − (1 − λ)L∗ +p +(8) +where λ ∈ [0, 1] indicates the weighting factor to control the +geometric proximity loss, L∗ +g, and the probabilistic proximity +loss, L∗ +p. In this way, the network is able to capture both +geometric and probabilistic information during the training +process. +2) Comparing With Geometric Proximity: To show the +advantage of including the probabilistic proximity loss in the +metric learning branch using the probabilistic view, we com- +pare the geometric proximity and the probabilistic proximity +between two probability distribution. +Consider a KL-divergence between fi and fj. For simplicity, +we use p(x) and q(x) to represent p(fi) and p(fj), respectively, +and assume x ∼ N(µ, σ2). Thus, the expanded form of +dKL(p∥q) for two Gaussians is denoted as +dKL(p∥q) = +� +x +p(x) log p(x) dx − +� +x +p(x) log q(x) dx +(9) +In here, we derive the result using the fact of [41]. For the +first term, +� +x p(x) log p(x) dx can be expanded as +− +� +x +p(x) log +� +2πσ2p dx − +� +x +p(x)(x − µp)2 +2σ2p +dx += − log +� +2πσ2p − +1 +2σ2p +� +x +p(x)(x − µp)2 dx +(10) +Next, we expand the quadratic form: +− log +� +2πσ2p − +1 +2σ2p +� +Ep(x2) − Ep(x)2� += − log +� +2πσ2p − 1 +2 +(11) +Following the same derivation, +� +x p(x) log q(x) dx can be +expand by +� +x +p(x) log q(x) dx = − log +� +2πσ2q − +� +σ2 +q + (µp − µq)2� +2σ2q +(12) + +Assume σ2 +p = σ2 +q = c, c is a constant, based on Eq (10)-(12), +the KL-divergence between p(x) and q(x) is given by +dKL(p∥q) = −1 +2 − 1 +2c + 1 +2c (µp − µq)2 +(13) +that is a linear function consisting of L2 distance between +two mean values, showing that the probabilistic proximity also +considers the variation of the sampling distribution, while the +geometric proximity does not. This derivation also supports +the phenomenon in Figure 1. +D. Network Implementation Details +As illustrated in Figure 2, the proposed framework consists +of a feature extraction backbone and two supervised learning +branches: one for metric learning and one for classification. We +used a pretrained ResNet18 [42], following the same setting as +the previous work [36]. We used max pooling on the attention +map after the last layer of the residual block in ResNet18. +Then, we flatted the output to a vector and sequentially connect +it with a MLP layer, batch normalization, and ReLU to reduce +the feature dimension to 128. Next, each fi 1) was connected +with a L2 normalization layer, i.e., ∥fi∥ = 1 to calculate the +hybrid proximity of the metric learning branch, and 2) connect +to another MLP layer and Softmax for classification. +The classification branch is to take the input batch {x}b +i=1 to +generate a prediction output. We optimized the cross-entropy +loss, LCE, together with the metric learning loss, L∗ +GHM, in +a multi-task learning scheme. Thus, we defined our total +objective loss as the weighted combination of a metric learning +branch and a classification branch. The learning objective loss +is denoted by +Ltotal = βL∗ +GHM + LCE +(14) +where β indicates the weighting factor to control the impor- +tance of the GHM-Loss In our experiments, we set β = 1 and +λ = 0.5, we also analyze the effects of both β and λ using +a grid search. Each input image of a batch was randomly +scaled within a factor range of [0.3, 1.0], and cropped into +patches of size 224 x 224. We set the batch size b = 8 in +the experiment and trained our framework using an Adam +optimization, the learning rate and weight decay are set to +0.0001. We train our network for 2000 epochs. The whole +framework was implemented using python 3.8, Scikit-Learn +0.24.1, Pytorch 1.9.1, and Cuda 11.1 with a NVIDIA GeForce +GTX 1660 SUPER GPU. +IV. DATA AND EXPERIMENTS +A. Datasets +To show the effectiveness of our method, same as Li +et al [36], we perform two binary (normal and abnormal) +classification tasks by diagnosing pathological myopia (PM) +and age-related macular degeneration (AMD) on two public +ophthalmic disease datasets of iChallenge-PM and iChallenge- +AMD. +1) iChallenge-PM: iChallenge-PM [43] contains 1200 an- +notated retinal fundus images in which 50% are PM subjects. +More details of the iChallenge-PM dataset can be found on +the [43]. We perform a 10-fold cross-validation to evaluate our +method. +2) iChallenge-AMD: There is a total of 1200 color fundus +images of the iChallenge-AMD dataset [44], in which 77% are +non-AMD subjects and 23% are AMD subjects. It provides the +disc boundaries and fovea locations, as well as the boundaries +of kinds of lesions. More details of the iChallenge-AMD +dataset can be found on [44]. Note, that we only used the +training dataset (400 fundus images) since only the training +dataset is released with annotations. We perform a 10-fold +cross-validation to evaluate our method. +B. Model Comparison Setting +1) Evaluation Metrics: We used AUC, accuracy, precision, +recall, and F1-score to assess the classification performance. +AUC stands for Area Under the Receiver Operating Character- +istic (ROC) curve. The definition of accuracy, precision, recall, +and F1-score are denoted by: +Accuracy = (TP + TN)/(TP + TN + FP + FN) +Precision = TP/(TP + FP) +Recall = TP/(TP + FN) +F1 = 2 ∗ (Precision ∗ Recall)/(Precision + Recall) +where TP, TN, FP, and FN indicate the true positive, true +negative, false positive, and false negative, respectively. +To provide the statistical analysis of our method, we con- +ducted a non-parametric Wilcoxon test [45] with a α level of +0.05. A p-value less than 0.05 is considered as statistical sig- +nificant for all inference. All statistical tests in the experiments +were performed using R-4.0.3 (RStudio, Boston, MA, USA). +2) Competing State-of-the-Art Methods: To have a fair +comparison, we trained all peer methods with the pretrained +ResNet18 with the same hyperparameters, network architec- +tures, and optimizer under the 10-fold cross-validation. Since +our framework consists of metric learning and classification +branches, we fix the classification branch and only modify +the metric learning part when compared with other metric +learning methods in the experiment. Our proposed method was +compared with other deep metric learning methods, Siamese +[5], Triplet [2], SupCon [6], N-pair [21], and InfoNCE [46]. +We run these metric learning methods with the code released +on iChallenge-PM and iChallenge-AMD datasets. We also +provided a supervised ‘Baseline’ method by modifying the +output layer of the last fully connected layer of the ResNet18 +to 2 and trained with cross-entropy loss. +C. Comparison on the iChallenge-PM Dataset +We compared with other state-of-the-art methods on the +iChallenge-PM Dataset. The results are shown in Table I. +We found that each method can achieve over 95% prediction +performance on all evaluation metrics, which indicates that the +patterns of pathological myopia in color fundus images are + +obvious. We can see that N-pair [21] achieved a limited result +and is due to this method requires large, annotated training +data that may not be suitable for the color fundus images. +Notably, our method significantly outperformed other peer +metric learning methods with 99.08% (p<0.0001) on AUC +and 99.01% (p<0.0001) on accuracy for PM diagnosis. These +results further demonstrate the effectiveness of our method +compared to other state-of-the-art metric learning methods. +TABLE I +MODEL COMPARISONS WITH OTHER DEEP METRIC LEARNING METHODS +ON THE ICHALLENGE-PM DATASET (UNIT: %). +AUC +Accuracy +Precision +Recall +F1 +Baseline +96.01 +95.45 +94.51 +97.25 +95.34 +Siamese [5] +97.45 +97.30 +96.15 +96.60 +96.58 +Triplet [2] +97.95 +98.64 +97.49 +96.14 +97.21 +SupCon [6] +98.06 +98.22 +98.36 +97.29 +97.64 +N-pair [21] +95.36 +95.83 +96.41 +97.25 +96.12 +InfoNCE [46] +98.11 +97.91 +96.83 +97.59 +97.36 +Ours +99.08 +99.01 +98.08 +99.12 +98.40 +D. Comparison on the iChallenge-AMD Dataset +We compared with other state-of-the-art methods on the +iChallenge-AMD Dataset. As shown in Table II, we can +see that our method achieved the best prediction performance +among other competing metric learning methods. Compared +to the second-best method, InfoNCE [46], our method signif- +icantly improved the performance, i.e., 78.69% vs. 76.75% +(p<0.0001) on AUC and 88.04 % vs. 86.51% (p<0.0001) +on accuracy. Notably, our method also outperformed the +supervised ‘Baseline’ method on all evaluation metrics. These +results demonstrated the effectiveness of the proposed method. +TABLE II +MODEL COMPARISONS WITH OTHER DEEP METRIC LEARNING METHODS +ON THE ICHALLENGE-AMD DATASET (UNIT: %). +AUC +Accuracy +Precision +Recall +F1 +Baseline +76.51 +84.16 +82.54 +76.18 +78.86 +Siamese [5] +67.58 +82.45 +72.54 +68.26 +70.14 +Triplet [2] +69.52 +84.29 +76.87 +72.48 +73.21 +SupCon [6] +73.24 +85.64 +78.42 +74.15 +76.05 +N-pair [21] +69.58 +83.41 +75.14 +70.54 +71.86 +InfoNCE [46] +76.75 +86.51 +85.36 +72.35 +77.95 +Ours +78.69 +88.04 +82.95 +75.28 +78.24 +E. Comparison with Transfer Learning Models +To show the robustness of learned features of our method, +we compared our method with the ImageNet pretrained mod- +els, including VGG-19 [47], InceptionNet v1 [48], and Effi- +cientNet B0 [49] on the iChallenge-AMD dataset. We modified +the output channel of the last fully connected layer in each +pretrained model to 2 and trained them with cross-entropy +loss. To have a fair comparison, all the models were trained +with the same number of epochs, learning rate, and weight +decay term on a 10-fold cross validation. The results are shown +in Table III. We can see that Efficient Net achieves the best +prediction performance among the transfer learning models. +Compared to Efficient Net, it is observed that our method can +achieve a higher prediction performance with around 1.5% +(p<0.0001) on AUC and 7% (p<0.0001) on accuracy. Note, +we trained our method with only 400 color fundus images +and performed better than ImageNet models, which were +pretrained with more than 1 million natural images. With this +observation, the results further show the practical value of our +method. +TABLE III +MODEL COMPARISONS WITH IMAGENET TRANSFER LEARNING MODELS +ON THE ICHALLENGE-AMD DATASET (UNIT: %). +AUC +Accuracy +Precision +Recall +F1 +VGG-19 [47] +74.14 +81.52 +76.54 +72.36 +73.89 +Inception v1 [48] +76.32 +77.35 +78.39 +75.54 +76.28 +Efficient B0 [49] +77.25 +81.52 +80.32 +79.25 +79.52 +Ours +78.69 +88.04 +82.95 +75.28 +78.24 +F. Analytical Study +TABLE IV +THE IMPORTANCE OF THE GHM-LOSS IN THE METRIC LEARNING +BRANCH ON THE ICHALLENGE-AMD DATASET (UNIT: %). +AUC +Accuracy +Precision +Recall +F1 +β = 0.0 +75.41 +83.21 +80.54 +72.88 +76.15 +β = 0.5 +76.85 +85.42 +80.95 +74.54 +77.28 +β = 1.0 +78.69 +88.04 +82.95 +75.28 +78.24 +β = 2.0 +72.45 +79.41 +77.66 +70.23 +73.59 +1) Importance of the GHM-Loss: The proposed method +consists of metric learning branch and classification branch in +a multi-task scheme, in which we trained GHM-Loss together +with the cross-entropy loss. In this section, we analyzed +the importance of the GHM-Loss of our method on the +iChallenge-AMD dataset. We first fix the λ = 0.5 in GHM- +Loss and trained our framework with different β in Eq (14), +where β is the importance of the metric learning branch. +β = 0.0 denotes that the framework is only trained with +the cross-entropy loss. As β increases, the more weight or +importance of the GHM-Loss in the network training. +The results are shown in Table IV. As we can see, when +β = 0.0, the network only learns the classification branch +and the result is 75.41% on AUC and 83.21% on accuracy. +As β increases, we found that the prediction performance +improves to the best when β reached 1 (e.g., 78.69% on AUC, +88.04% on accuracy). However, when β continues increasing, +the prediction performance starts to drop apparently from +78.69% to 72.45% on AUC. The comparison shows that both +the metric learning branch and classification branch equally +contributed to our framework for PM diagnosis. +2) Effects of Weighting Factors in the GHM-Loss: We +analyzed the effects of the weighting factors, i.e., β, λ in +the GHM-Loss on the iChallenge-AMD Dataset, in which +β indicates the importance of the metric learning branch of +our method and λ controls the weight size between geomet- +ric proximity and probabilistic proximity of the GHM-Loss. +Note, λ = 0.0 denotes that only probabilistic proximity was + +considered between fi and fj. As we can see in Figure 3, +for each fix β, the classification performance increases to the +best performance when λ is reached 0.5 and drops apparently +as it continues to increase. These results demonstrate that 1) +both metric learning and classification branches are useful of +our method and 2) both geometric and probabilistic proximity +should be captured between fi and fj in the training. +Fig. 3. +Classification performance comparison on the iChallenge-AMD +Dataset with different weighting factors β and λ of the GHM-Loss. We use +AUC to choose the optimal β and λ using a grid search. +3) Visualization of the Feature Distribution: We visualized +the feature embedding distribution, i.e., f1 (red line) and f2, +after ResNet18 for a positive pair color fundus image on +the iChallenge AMD dataset. The feature distributions are +shown in Figure 4. Before optimization, we can see that the +distribution of feature embeddings from a positive pair sample +are independent without overlaps. However, the probabilistic +distance of f1 (red line) and f2 is reduced and stays close +to each other after optimizing the network. Since we use +GMM to approximate the empirical distribution of each feature +embedding, the probability parameters of µ and σ of all +the images with the same label should be closed to each +other, thus, resulting in the similar probability densities. This +visualization also demonstrate that the proposed GHM-Loss +can efficiently capture the probabilistic patterns during the +training process. +V. DISCUSSION +Metric learning is an important technique in visual repre- +sentation area by learning the distance metric, which can be +further used to perform supervised and unsupervised learning +tasks, such as image classification [6], [12], [13], image +clustering [14], [50], and object detection [20], [51], etc. With +the advances of deep learning techniques, deep metric learn- +ing has been widely studied in the metric learning research +community. Although promising results were obtained on +previous works [2], [5], [6], [21], [46], these methods usually +ignore the probability distribution of the feature embeddings +during the training process, which may lead an inaccurate +Before Optimization +After Optimization +Fig. 4. The feature distribution between a positive pair of f1 (red line) and +f2 (blue line) of color fundus images during the training process on the +iChallenge AMD dataset. We applied Gaussian mixture model (GMM) to +approximate the empirically distribution of these features. The probabilistic +proximity between f1 (red line) and f2 are reduced after optimization. +prediction. In this work, we present a novel supervised metric +learning method that consists of learning both geometric and +probabilistic proximity for image recognition. We formulate a +Generalized Hybrid Metric Loss (GHM-Loss) to better learn +the distance representation, where geometric-based distance +and probabilistic-based distance are learned. Our method is +validated on two public ophthalmic disease datasets (e.g., +iChallenge-PM and iChallenge-AMD), in which our method +can significantly outperform other state-of-the-art metric learn- +ing methods. With a convex combination of the geometric +proximity and probabilistic proximity, our method consistently +achieves the best prediction performance than the individual +proximity. +Although our method outperforms other state-of-the-art +metric learning methods, it comes with limitations. Our +method is a supervised learning approach, which relies on +a large number of annotated training data, and it is costly +to obtain. In future, we will investigate the unsupervised +metric learning approach or self-supervised learning approach +to address the human effort issue on image recognition com- +munities. The exploration of probabilistic unsupervised/self- +supervised metric learning would be our future work. +VI. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='edu, 2nd Anca Ralescu Department of Computer Science University of Cincinnati Cincinnati, OH, United States ralescal@ucmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='edu Abstract—Recently, deep metric learning techniques received attentions, as the learned distance representations are useful to capture the similarity relationship among samples and further improve the performance of various of supervised or unsuper- vised learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We propose a novel supervised metric learning method that can learn the distance metrics in both geometric and probabilistic space for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' In contrast to the previous metric learning methods which usually focus on learning the distance metrics in Euclidean space, our proposed method is able to learn better distance representation in a hybrid approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' To achieve this, we proposed a Generalized Hybrid Metric Loss (GHM-Loss) to learn the general hybrid proximity features from the image data by controlling the trade- off between geometric proximity and probabilistic proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' To evaluate the effectiveness of our method, we first provide theoretical derivations and proofs of the proposed loss function, then we perform extensive experiments on two public datasets to show the advantage of our method compared to other state- of-the-art metric learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Index Terms—Deep metric learning, proximity, probability distribution, representation learning, image classification I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' INTRODUCTION Metric learning takes input data to learn the similar and dissimilar features between samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The learned distance metric provides a meaningful and robust representation to discriminate the proximity or distance between samples and can be further utilized for both supervised and unsupervised learning tasks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Recently, deep learning-based metric learn- ing algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', deep metric learning, were widely applied in the computer vision area by developing either a novel network architecture or an intuitive and efficient loss function [2]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Some typical works, such as the Siamese network [5], Triplet network [2], SupCon [6], aim to formulate an instance discrimination task to learn a useful feature representation by optimizing the proximity function in the Euclidean space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', geometric distance or Cosine proximity between the feature embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' In this paper, we seek to address the inadequacies of geometric proximity of recent state-of-the- art metric learning methods by reconsidering an alternative Copyright (c) 2022 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribu- tion to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' approach in which learned distance metrics are not biased to only geometric proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Metric learning methods have shown an excellent classifi- cation performance in image recognition applications, due to their extraordinary ability to discriminate similar information between samples [1], [4], [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Such metric learning tasks can be either supervised or self-supervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' In supervised metric learning, the model learns to pull together the samples from the same classes and push away the samples from dif- ferent classes [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Self-supervised metric learning, also named contrastive learning, requires a data augmentation step to cre- ate some pseudo-ground-truth from the data itself, where the augmentation from the same sample is included in “positive” pairs and the augmentation from different samples is included in “negative” pairs [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Similar to supervised metric learning, the self-supervised model learns similar representations from the positive pairs and should be different than the representa- tions of the negative pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Various types of metric/contrastive learning works have been developed for image pattern recogni- tion applications, including image classification [6], [11]–[13], image clustering [14], [15], image segmentation [16], [17], image reconstruction [18], [19], and object detection [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' All these works used geometric proximity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', Cosine similarity or Euclidean distance) as the proximity function in the training an objective loss to learn the geometric representation of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' However, the probability distribution of the samples should not be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' To overcome these limitations and boost the prediction per- formance of metric learning, we proposed a novel supervised metric learning method to learn the hybrid proximity that combines the proximity in both geometric and probabilistic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' To achieve it, we defined a supervised Generalized Hybrid Metric Loss (GHM-Loss) to better learn the distance representations in both geometric and probabilistic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We noticed that even if the geometric distance is small, the probabilistic distance can be large when the sample variance is large (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' This observation reminds us to reconsider that the model may not sufficiently learn the distance features based only on the geometric distance between data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Thus, enabling the model to partially learn the probabilistic distance controls the trade-off between two types of distance representation (geometric and probabilistic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='13459v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='CV] 31 Jan 2023 𝑋2 𝑋3 𝐷 𝜇𝑋1, 𝜇𝑋2 > 𝐷 𝜇𝑋2, 𝜇𝑋3 𝐾𝐿 𝑃𝑋1||𝑃𝑋2 < 𝐾𝐿 𝑃𝑋2||𝑃𝑋3 𝑋1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Geometric distance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' probabilistic distance between probability distributions X1, X2 and X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Without considering the variances, the geomet- ric distance between mean values of µX1, µX2, and µX3 cannot represent their probabilistic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' D: Geometric distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' KL: Kullback–Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Our proposed GHM-Loss is formulated by a hybrid di- vergence underlying the geometric and probabilistic space, which is a generalized distance loss form for many defined metric learning methods, including Triplet [2], N-pairs [21], Max Margin [22], NTXent [13], SupCon [6], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We first theoretically showing the advantage of using the probabilistic distance in metric learning compared to the geometric-based distance, and we employed two public datasets to show the effectiveness of our method compared to other state-of-the- art metric/contrastive learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We also investigated the superiority of the proposed GHM-Loss with other met- ric/contrastive learning loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' To sum up, our main findings and contributions to this work are as follows: 1) We propose a novel supervised metric learning method for enhancing the performance of image recognition by defining a Generalized Hybrid Metric Loss (GHM- Loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The proposed GHM-Loss is able to learn better distance representation that controls the trade-off be- tween the geometric-based and the probabilistic distance from feature embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 2) We define two proximity functions with certain proper- ties in geometric and probabilistic space, respectively, and provide proof for each property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Meanwhile, we theoretically show the advantage of the GHM-Loss by including the probabilistic proximity for learning the distance between distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 3) Our approach is supported both by a theoretical discus- sion and by extensive experiments performed on two common image classification tasks to demonstrate the effectiveness of our method compared to other state-of- the-art metric learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' RELATED WORK In this section, we first discuss some state-of-the-art meth- ods of deep metric learning and some of its applications in the computer vision domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We further review related works on contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Metric Learning 1) Traditional Metric Learning: The early stage of the ma- chine learning techniques requires a hand-crafted processing step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', feature engineering, such as feature selection and feature extraction before training a machine learning model for supervised (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', classification) or unsupervised learning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', clustering) tasks [7], [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' These methods, including linear projections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', principal component analysis (PCA) [25], decomposition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', non-negative matrix factorization (NMF) [26] to extract useful feature information and are not directly within the classification structure, resulting in a limited performance on the certain complex structure, such as high- dimensional data and non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Unlike traditional machine learning approaches, metric learning performs the learning process on the data to learn a distance feature representation by decreasing the distance between similar samples and in- creasing the distance between dissimilar ones in a embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The learned distance features will have a high ability to discriminate the classes of the sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Usually, metric learning approaches apply linear transformation techniques to the input data and map it to a new feature space with a higher-class separation [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' However, these methods lack the generalization capability and nonlinear knowledge of the attributes [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 2) Deep Metric Learning: Unlike traditional metric learn- ing methods, deep metric learning relies on training deep neural networks with activation functions that capture nonlin- ear properties [4], and it has dominated metric representation learning in the image recognition community [2], [3], [5], [6], [29]–[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' For example, the Siamese network [5] used two identical convolutional neural networks (CNNs) to encode a pair of input samples and minimize the contrastive loss to learn the representative distance features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Similar to the Siamese network, Hoffer et al [2] proposed a Triplet network, including the anchor, positive (similar), and negative (dissimilar) sample, which learns the inequality that the positive sample stays closer to the anchor compared to the negative sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Afterward, Wang et al [3] defined a new Angular loss to constrain the angle at the negative sample from the Triplet network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Later, Sohn [21] proposed an N-pair loss to address the slow convergence problem of the Triplet loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' More recently, Khosla et al [6] developed a supervised contrastive learning framework with the more general form of metric learning loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', SupCon, and showed the effectiveness of classification performance compared to the Triplet loss and the N-pair loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' These existing works have shown great promise for metric and feature representation learning in a variety of image classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Nevertheless, to the best of our knowledge, most previous studies are focused on learning the geometric- based metrics of the embedding space, while the probabilistic- based metrics are usually ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' In this work, our method is able to learn the meaningful metrics of both geometric and probabilistic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Contrastive Representation Learning The main purpose of deep metric learning and contrastive learning is to train a deep learning model to learn the distance feature representations in an embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The main difference between these two methods is that contrastive learn- ing is closely related to the self-supervised learning domain, which contains a data augmentation step to generalize an arbitrary number of positive and negative sample pairs from each sample [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Given the stunning achievement of self- supervised representation learning, many contrastive learning methods have been developed for various computer vision tasks [33]–[36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' For example, Ye et al [37] proposed an embedding contrastive learning method with the Siamese network to learn the invariant features of embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Chen et al [13] developed a famous contrastive learning framework, SimCLR, in which the model is pretrained to discriminate the positive pairs of data augmentation from the same source image, demonstrating superior performance in ImageNet classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Similarly, He et al [12] proposed the Moco v1, to maximize the proximity between the positive pairs based on the monument network encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Additional studies that are similar to SimCLR and Moco, including BYOL [38], SimSam [39] Barlow Twins [40], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', show the exceptional performance of learned feature representation for further supervised or unsupervised tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' … 𝑥1 𝑥2 𝑥𝑁−1 𝑥𝑁 𝐟1 𝐟2 𝐟𝑁−1 𝐟𝑁 … 𝐟1 𝐟2 𝐟𝑁−1 𝐟𝑁 Pull Push Learning Hybrid Metrics Softmax Encoder F(∙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 𝜽) Classification Normal Normal … Fibrosis Glaucoma MLP Feature Extraction MLP Input 𝐿2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The overview of our proposed framework (example of fundus disease diagnosis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We use a pretrained convolutional neural network (CNN) and a multi-layer perceptron (MLP) to encode each image to the embedded feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Afterward, we propose a metric learning branch that is supervised with the proposed GHM-Loss which is trained together with the cross-entropy loss of a classification branch in a multi-task scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Overview Our proposed supervised metric learning framework is il- lustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We first denote a training image dataset D = {xi, yi}N i=1, where yi is the label of image xi, a set of indices of all the positive samples for a randomly selected image xi in a batch, U(i) = {j ∈ Θ|yj = yi, j ̸= i}, a set of indices of all negative samples for a randomly selected image xi in a batch, V (i) := {j ∈ Θ|yj ̸= yi, i ̸= j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The problem formulation is to learn a network F(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' θ) that maps each input xi to a L2 normalized d-dimensional feature embedding fi, such as fi = F(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' θ) ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' To achieve this, we use a pre- trained CNN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', ResNet18, followed by a MLP to produce N high-level feature vectors and perform two supervised learning branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The first branch is a metric learning task, which aims to learn the robust metrics by pulling all the samples with indices U(i) and pushing away all the samples with indices V (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Meanwhile, the embedded feature fi is connected to another MLP layer with a Softmax to generate the predicted probability for the class label yi and is supervised with cross- entropy loss to perform a classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Below, we will elaborate on the procedure of the each branch, including the definition of GHM-Loss and its advantage, and other network details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Generalized Hybird Metric Loss 1) General Loss Form: To perform the metric learning branch, we propose a general form metric loss function, in which the network can learn the proximity information between the embedded features {f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' , fN} by optimizing the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Let S(·) denote the proximity function for two input vectors fi and fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' That is, for fi, fj ∈ Rd, S(fi, fj) : Rd → R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' the probability of xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' u ∈ U(i) is being recognized as yi is defined by p(yi|xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' xu) = exp [S(fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' fu)] � j∈Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='j̸=i exp [S(fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' fj)] (1) On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' the probability of xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' xv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' v ∈ V (i) is not being recognized as yi is defined by p(yi|xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' xv) = exp [S(fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' fv)] � j∈Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='j̸=i exp [S(fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' fj)] (2) Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' assume that all the probabilities of different images be- ing recognized as image xi are independent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' let q(yi|xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' xv) = 1−p(yi|xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' xv) thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' the objective likelihood function that we are interested is defined by ℓi = � u∈U(i) � v∈V (i) p(yi|xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' xu)q(yi|xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' xv) (3) Correspondingly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' the negative log likelihood over all the data points indexed by Θ yields: L∗ = − � i∈Θ ∥V (i)∥ � u∈U(i) log p(yi|xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' xu) − � i∈Θ ∥U(i)∥ � v∈V (i) log q(yi|xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' xv) (4) where ∥U(i)∥ and ∥V (i)∥ denotes the size of the set U(i) and V (i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 2) Geometric Proximity: We first consider the proximity in the geometric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Given a pair vectors fi and fj, the proximity function Sg(fi, fj) satisfies the following properties: 1 Sg(fi, fj) ∈ [0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 2 Sg(fi, fj) = Sg(fj, fi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 3 ∀c ∈ [fi, fj], Sg(fi, fj) ≤ min{Sg(fi, c), Sg(c, fj)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The proximity measures that satisfy the above properties include Cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Using Cosine similarity as the proximity metrics, such that Sg(fi, fj) = fi · fj/∥fi∥∥fj∥ satisfies each property above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Obviously, Sg(fi, fj) ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Obviously, this property is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Let c ∈ [fi, fj], we have |fi−c| ≤ |fi−fj|, |c−fj| ≤ |fi− fj|, which means that Sg(fi, c) ≥ Sg(a, b), Sg(c, fj) ≥ S(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Thus, Sg(fi, fj) ≤ min{Sg(fi, c), Sg(c, fj)}∀c ∈ [fi, fj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 3) Probabilistic Proximity: Instead of only using geometric proximity, which ignores the sampling probability distribution, we consider the probabilistic proximity to summarize the distribution of the embedded features {fi, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' , fN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Given a pair vectors fi and fj, with size of |fi|, |fj|, the probabilistic proximity function satisfies the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Sp(fi, fj) ∈ [0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Sp(fi, fj) = Sp(fj, fi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Sp(fi, fj) = 0 if and only if fi = fj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Sp(fi, fj) ≤ Sp(fi, fc) + Sp(fc, fj) under the certain condition, in which |fc| = |fi| = |fj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We use a Gaussian mixture model (GMM) to represent the empirical distribution fi, which is defined by p(fi) = � k∈K wkN(fi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' µk, σ2 k) (5) where wk is a latent variable followed by a categorical distri- bution, denoting the k-th component, and N is the Gaussian probability density function with parameters µk and σk, which is defined as N(fi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' µk, σ2 k) = 1 � 2πσ2 k exp � − 1 2σ2 k (fi − µk)2 � (6) Using this model, the probabilistic distance between p(fi) and p(fj) is chosen with the symmetric divergence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', Jensen–Shannon (JS)-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' For simplicity, we use pi and pj to denotes the probability distributions of fi and fj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Therefore, the Sp(fi, fj) is denoted by Sp(fi, fj) = 1 2 [dKL(pi∥¯pij) + dKL(pj∥¯pij)] (7) where ¯pij = (pi + pj) /2, dKL(·) presents a function of the Kullback–Leibler (KL)-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We prove that Sp(fi, fj) satisfies the properties of defined probabilistic proximity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The range of the JS-divergence is within 0 and 1, thus Sp(fi, fj) ∈ [0, 1] is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Obviously, based on Eq (7), it is easy to have Sp(fi, fj) = 1 2 [dKL(pi∥¯pij) + dKL(pj∥¯pij)] = Sp(fi, fj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Sp(fi, fj) ≥ 0, as a sum of nonnegative terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' To have Sp(fi, fj) = 0, each term of Sp(fi, fj) must be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Thus, Sp(fi, fj) = 0 if and only if dKL(pi∥¯pij) = dKL(pj∥¯pij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Since dKL(pi∥pj) = 0 if and only if pi = pj, thus, Sp(fi, fj) = 0 if and only if fi = fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Now we prove property 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Using the Shannon entropy, H(pi) = − � pi∈pi pi log pi, the explicit form of Sp(fi, fj) can be written as Sp(fi, fj) = H(¯pij) − 1 2 [H(pi) + H(pj)] Assume that H(¯pic) + H(¯pcj) ≥ H(¯pij) + H(¯pc), thus, Sp(fi, fc) + Sp(fc, fj) − Sp(fi, fj) can be rewritten as H(¯pic) − H(¯pc) + H(¯pcj) − H(¯pij) ≥ 0 Thus, Sp(fi, fc) + Sp(fc, fj) ≥ Sp(fi, fj) is true if and only if H(¯pic) + H(¯pcj) ≥ H(¯pij) + H(¯pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Learning Hybrid Proximity 1) Generalized Hybrid Metric Loss: The learning objective loss function of the metric learning branch is the convex combination of geometric proximity loss and probabilistic proximity loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' As such, the objective is denoted by L∗ GHM = λL∗ g − (1 − λ)L∗ p (8) where λ ∈ [0, 1] indicates the weighting factor to control the geometric proximity loss, L∗ g, and the probabilistic proximity loss, L∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' In this way, the network is able to capture both geometric and probabilistic information during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 2) Comparing With Geometric Proximity: To show the advantage of including the probabilistic proximity loss in the metric learning branch using the probabilistic view, we com- pare the geometric proximity and the probabilistic proximity between two probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Consider a KL-divergence between fi and fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' For simplicity, we use p(x) and q(x) to represent p(fi) and p(fj), respectively, and assume x ∼ N(µ, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Thus, the expanded form of dKL(p∥q) for two Gaussians is denoted as dKL(p∥q) = � x p(x) log p(x) dx − � x p(x) log q(x) dx (9) In here, we derive the result using the fact of [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' For the first term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' � x p(x) log p(x) dx can be expanded as − � x p(x) log � 2πσ2p dx − � x p(x)(x − µp)2 2σ2p dx = − log � 2πσ2p − 1 2σ2p � x p(x)(x − µp)2 dx (10) Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' we expand the quadratic form: − log � 2πσ2p − 1 2σ2p � Ep(x2) − Ep(x)2� = − log � 2πσ2p − 1 2 (11) Following the same derivation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' � x p(x) log q(x) dx can be expand by � x p(x) log q(x) dx = − log � 2πσ2q − � σ2 q + (µp − µq)2� 2σ2q (12) Assume σ2 p = σ2 q = c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' c is a constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' based on Eq (10)-(12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' the KL-divergence between p(x) and q(x) is given by dKL(p∥q) = −1 2 − 1 2c + 1 2c (µp − µq)2 (13) that is a linear function consisting of L2 distance between two mean values,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' showing that the probabilistic proximity also considers the variation of the sampling distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' while the geometric proximity does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' This derivation also supports the phenomenon in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Network Implementation Details As illustrated in Figure 2, the proposed framework consists of a feature extraction backbone and two supervised learning branches: one for metric learning and one for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We used a pretrained ResNet18 [42], following the same setting as the previous work [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We used max pooling on the attention map after the last layer of the residual block in ResNet18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Then, we flatted the output to a vector and sequentially connect it with a MLP layer, batch normalization, and ReLU to reduce the feature dimension to 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Next, each fi 1) was connected with a L2 normalization layer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', ∥fi∥ = 1 to calculate the hybrid proximity of the metric learning branch, and 2) connect to another MLP layer and Softmax for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The classification branch is to take the input batch {x}b i=1 to generate a prediction output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We optimized the cross-entropy loss, LCE, together with the metric learning loss, L∗ GHM, in a multi-task learning scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Thus, we defined our total objective loss as the weighted combination of a metric learning branch and a classification branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The learning objective loss is denoted by Ltotal = βL∗ GHM + LCE (14) where β indicates the weighting factor to control the impor- tance of the GHM-Loss In our experiments, we set β = 1 and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='5, we also analyze the effects of both β and λ using a grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Each input image of a batch was randomly scaled within a factor range of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0], and cropped into patches of size 224 x 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We set the batch size b = 8 in the experiment and trained our framework using an Adam optimization, the learning rate and weight decay are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We train our network for 2000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The whole framework was implemented using python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='8, Scikit-Learn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='1, Pytorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='1, and Cuda 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='1 with a NVIDIA GeForce GTX 1660 SUPER GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' DATA AND EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Datasets To show the effectiveness of our method, same as Li et al [36], we perform two binary (normal and abnormal) classification tasks by diagnosing pathological myopia (PM) and age-related macular degeneration (AMD) on two public ophthalmic disease datasets of iChallenge-PM and iChallenge- AMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 1) iChallenge-PM: iChallenge-PM [43] contains 1200 an- notated retinal fundus images in which 50% are PM subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' More details of the iChallenge-PM dataset can be found on the [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We perform a 10-fold cross-validation to evaluate our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 2) iChallenge-AMD: There is a total of 1200 color fundus images of the iChallenge-AMD dataset [44], in which 77% are non-AMD subjects and 23% are AMD subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' It provides the disc boundaries and fovea locations, as well as the boundaries of kinds of lesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' More details of the iChallenge-AMD dataset can be found on [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Note, that we only used the training dataset (400 fundus images) since only the training dataset is released with annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We perform a 10-fold cross-validation to evaluate our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Model Comparison Setting 1) Evaluation Metrics: We used AUC, accuracy, precision, recall, and F1-score to assess the classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' AUC stands for Area Under the Receiver Operating Character- istic (ROC) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The definition of accuracy, precision, recall, and F1-score are denoted by: Accuracy = (TP + TN)/(TP + TN + FP + FN) Precision = TP/(TP + FP) Recall = TP/(TP + FN) F1 = 2 ∗ (Precision ∗ Recall)/(Precision + Recall) where TP, TN, FP, and FN indicate the true positive, true negative, false positive, and false negative, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' To provide the statistical analysis of our method, we con- ducted a non-parametric Wilcoxon test [45] with a α level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' A p-value less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='05 is considered as statistical sig- nificant for all inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' All statistical tests in the experiments were performed using R-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='3 (RStudio, Boston, MA, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 2) Competing State-of-the-Art Methods: To have a fair comparison, we trained all peer methods with the pretrained ResNet18 with the same hyperparameters, network architec- tures, and optimizer under the 10-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Since our framework consists of metric learning and classification branches, we fix the classification branch and only modify the metric learning part when compared with other metric learning methods in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Our proposed method was compared with other deep metric learning methods, Siamese [5], Triplet [2], SupCon [6], N-pair [21], and InfoNCE [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We run these metric learning methods with the code released on iChallenge-PM and iChallenge-AMD datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We also provided a supervised ‘Baseline’ method by modifying the output layer of the last fully connected layer of the ResNet18 to 2 and trained with cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Comparison on the iChallenge-PM Dataset We compared with other state-of-the-art methods on the iChallenge-PM Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The results are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We found that each method can achieve over 95% prediction performance on all evaluation metrics, which indicates that the patterns of pathological myopia in color fundus images are obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We can see that N-pair [21] achieved a limited result and is due to this method requires large, annotated training data that may not be suitable for the color fundus images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Notably, our method significantly outperformed other peer metric learning methods with 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='08% (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0001) on AUC and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='01% (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0001) on accuracy for PM diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' These results further demonstrate the effectiveness of our method compared to other state-of-the-art metric learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' TABLE I MODEL COMPARISONS WITH OTHER DEEP METRIC LEARNING METHODS ON THE ICHALLENGE-PM DATASET (UNIT: %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' AUC Accuracy Precision Recall F1 Baseline 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='01 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='45 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='51 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='25 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='34 Siamese [5] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='45 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='30 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='15 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='60 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='58 Triplet [2] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='95 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='64 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='49 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='14 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='21 SupCon [6] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='06 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='22 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='36 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='29 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='64 N-pair [21] 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='36 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='83 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='41 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='25 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='12 InfoNCE [46] 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='11 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='91 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='83 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='59 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='36 Ours 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='08 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='01 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='08 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='12 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='40 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Comparison on the iChallenge-AMD Dataset We compared with other state-of-the-art methods on the iChallenge-AMD Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' As shown in Table II, we can see that our method achieved the best prediction performance among other competing metric learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Compared to the second-best method, InfoNCE [46], our method signif- icantly improved the performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='69% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='75% (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0001) on AUC and 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='04 % vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='51% (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0001) on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Notably, our method also outperformed the supervised ‘Baseline’ method on all evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' These results demonstrated the effectiveness of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' TABLE II MODEL COMPARISONS WITH OTHER DEEP METRIC LEARNING METHODS ON THE ICHALLENGE-AMD DATASET (UNIT: %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' AUC Accuracy Precision Recall F1 Baseline 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='51 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='16 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='54 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='18 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='86 Siamese [5] 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='58 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='45 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='54 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='26 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='14 Triplet [2] 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='52 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='29 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='87 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='48 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='21 SupCon [6] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='24 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='64 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='42 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='15 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='05 N-pair [21] 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='58 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='41 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='14 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='54 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='86 InfoNCE [46] 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='75 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='51 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='36 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='35 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='95 Ours 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='69 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='04 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='95 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='28 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='24 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Comparison with Transfer Learning Models To show the robustness of learned features of our method, we compared our method with the ImageNet pretrained mod- els, including VGG-19 [47], InceptionNet v1 [48], and Effi- cientNet B0 [49] on the iChallenge-AMD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We modified the output channel of the last fully connected layer in each pretrained model to 2 and trained them with cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' To have a fair comparison, all the models were trained with the same number of epochs, learning rate, and weight decay term on a 10-fold cross validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The results are shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We can see that Efficient Net achieves the best prediction performance among the transfer learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Compared to Efficient Net, it is observed that our method can achieve a higher prediction performance with around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='5% (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0001) on AUC and 7% (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0001) on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Note, we trained our method with only 400 color fundus images and performed better than ImageNet models, which were pretrained with more than 1 million natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' With this observation, the results further show the practical value of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' TABLE III MODEL COMPARISONS WITH IMAGENET TRANSFER LEARNING MODELS ON THE ICHALLENGE-AMD DATASET (UNIT: %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' AUC Accuracy Precision Recall F1 VGG-19 [47] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='14 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='52 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='54 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='36 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='89 Inception v1 [48] 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='32 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='35 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='39 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='54 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='28 Efficient B0 [49] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='25 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='52 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='32 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='25 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='52 Ours 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='69 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='04 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='95 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='28 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='24 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Analytical Study TABLE IV THE IMPORTANCE OF THE GHM-LOSS IN THE METRIC LEARNING BRANCH ON THE ICHALLENGE-AMD DATASET (UNIT: %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' AUC Accuracy Precision Recall F1 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='41 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='21 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='54 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='88 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='15 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='85 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='42 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='95 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='54 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='28 β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='69 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='04 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='95 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='28 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='24 β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='45 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='41 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='66 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='23 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='59 1) Importance of the GHM-Loss: The proposed method consists of metric learning branch and classification branch in a multi-task scheme, in which we trained GHM-Loss together with the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' In this section, we analyzed the importance of the GHM-Loss of our method on the iChallenge-AMD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We first fix the λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='5 in GHM- Loss and trained our framework with different β in Eq (14), where β is the importance of the metric learning branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0 denotes that the framework is only trained with the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' As β increases, the more weight or importance of the GHM-Loss in the network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The results are shown in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' As we can see, when β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0, the network only learns the classification branch and the result is 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='41% on AUC and 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='21% on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' As β increases, we found that the prediction performance improves to the best when β reached 1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='69% on AUC, 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='04% on accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' However, when β continues increasing, the prediction performance starts to drop apparently from 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='69% to 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='45% on AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The comparison shows that both the metric learning branch and classification branch equally contributed to our framework for PM diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 2) Effects of Weighting Factors in the GHM-Loss: We analyzed the effects of the weighting factors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', β, λ in the GHM-Loss on the iChallenge-AMD Dataset, in which β indicates the importance of the metric learning branch of our method and λ controls the weight size between geomet- ric proximity and probabilistic proximity of the GHM-Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Note, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='0 denotes that only probabilistic proximity was considered between fi and fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' As we can see in Figure 3, for each fix β, the classification performance increases to the best performance when λ is reached 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='5 and drops apparently as it continues to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' These results demonstrate that 1) both metric learning and classification branches are useful of our method and 2) both geometric and probabilistic proximity should be captured between fi and fj in the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Classification performance comparison on the iChallenge-AMD Dataset with different weighting factors β and λ of the GHM-Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We use AUC to choose the optimal β and λ using a grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 3) Visualization of the Feature Distribution: We visualized the feature embedding distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', f1 (red line) and f2, after ResNet18 for a positive pair color fundus image on the iChallenge AMD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The feature distributions are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Before optimization, we can see that the distribution of feature embeddings from a positive pair sample are independent without overlaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' However, the probabilistic distance of f1 (red line) and f2 is reduced and stays close to each other after optimizing the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Since we use GMM to approximate the empirical distribution of each feature embedding, the probability parameters of µ and σ of all the images with the same label should be closed to each other, thus, resulting in the similar probability densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' This visualization also demonstrate that the proposed GHM-Loss can efficiently capture the probabilistic patterns during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' DISCUSSION Metric learning is an important technique in visual repre- sentation area by learning the distance metric, which can be further used to perform supervised and unsupervised learning tasks, such as image classification [6], [12], [13], image clustering [14], [50], and object detection [20], [51], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' With the advances of deep learning techniques, deep metric learn- ing has been widely studied in the metric learning research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Although promising results were obtained on previous works [2], [5], [6], [21], [46], these methods usually ignore the probability distribution of the feature embeddings during the training process, which may lead an inaccurate Before Optimization After Optimization Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The feature distribution between a positive pair of f1 (red line) and f2 (blue line) of color fundus images during the training process on the iChallenge AMD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We applied Gaussian mixture model (GMM) to approximate the empirically distribution of these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The probabilistic proximity between f1 (red line) and f2 are reduced after optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' In this work, we present a novel supervised metric learning method that consists of learning both geometric and probabilistic proximity for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' We formulate a Generalized Hybrid Metric Loss (GHM-Loss) to better learn the distance representation, where geometric-based distance and probabilistic-based distance are learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Our method is validated on two public ophthalmic disease datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', iChallenge-PM and iChallenge-AMD), in which our method can significantly outperform other state-of-the-art metric learn- ing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' With a convex combination of the geometric proximity and probabilistic proximity, our method consistently achieves the best prediction performance than the individual proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Although our method outperforms other state-of-the-art metric learning methods, it comes with limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Our method is a supervised learning approach, which relies on a large number of annotated training data, and it is costly to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' In future, we will investigate the unsupervised metric learning approach or self-supervised learning approach to address the human effort issue on image recognition com- munities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The exploration of probabilistic unsupervised/self- supervised metric learning would be our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' CONCLUSION In this paper, we present a novel supervised metric learning method for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Our main idea is to learn a hybrid proximity that consists of both geometric-based metric and probabilistic-based metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' The geometric proximity of proposed GHM-Loss helps the model learn the similarity information under the Euclidean space and the probabilistic proximity proposed GHM-Loss learns the similarity under the empirical probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' With extensive experiments, our method consistently achieves the excellent prediction performance compared with the other state-of-the-art metric learning methods, showing the effectiveness of learned dis- tance features of our method in image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' REFERENCES [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=' Kulis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNFQT4oBgHgl3EQf_zdP/content/2301.13459v1.pdf'} +page_content=', “Metric learning: A survey,” Foundations and Trends® in Machine Learning, vol.' metadata={'source': 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a/E9E1T4oBgHgl3EQfEgPe/content/tmp_files/2301.02892v1.pdf.txt b/E9E1T4oBgHgl3EQfEgPe/content/tmp_files/2301.02892v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..345461980d1511db4181535371251c7cf45023b9 --- /dev/null +++ b/E9E1T4oBgHgl3EQfEgPe/content/tmp_files/2301.02892v1.pdf.txt @@ -0,0 +1,1366 @@ +Disorder-induced finite center-of-mass momentum Cooper pairing and its consequences to the +critical temperature and superconducting gap of overdoped cuprates +Victor Velasco1 and Marcello B. Silva Neto1 +1Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, Brazil +One of the most studied classes of unconventional high-temperature superconductors is the hole-doped +cuprates, where special attention is given to those doped with extra interstitial oxygens. In this context, the +formation of spatially inhomogeneous agglomerates of dopant oxygen atoms in the form of nanosized puddles +is not only relevant, but also subject of intense recent experimental and theoretical surveys. Following these +efforts, in this work we show the consequences of the presence of networks of oxygen puddles in the supercon- +ducting state of overdoped cuprates. Starting from the inhomogeneous disordered background brought by the +network of puddles, we show that an effective interaction between electrons can be mediated by the local vibra- +tional degrees of freedom of each puddle, but the pairs arising from this interaction have a finite center-of-mass +momentum p, thus breaking up the Cooper channel. Furthermore, we derive an analytical expression for the +amplitude of the superconducting gap ∆k in terms of disorder and finite center-of-mass momentum and show +that amplitude fluctuations are induced in the superconducting state by the presence of the puddles, where both +the gap and the critical temperature are affect and reduced by disorder and finite momentum pairs. Finally, we +discuss our findings in the context of networks of superconducting oxygen nano-puddles in cuprates. +I. +INTRODUCTION +It is a well known fact within the Bardeen-Cooper- +Schrieffer (BCS) theory of superconductivity that the two +quasi-particles forming the bound states that constitute the +superconductor, named Cooper pairs, have momentum k and +−k, near the Fermi surface, with oposite spins ↑ and ↓, form- +ing a singlet with zero center-of-mass momentum [1], in +what is usually called the Cooper channel. However, the ex- +istence of a finite-momentum superconducting ground state +has recently been raised theoretically [2–7] and supported +by several experiments in correlated quantum materials [8– +11]. Moreover, the possibility of emergent finite-momentum +pair states, in the form of pair density waves, in a variety +of well-established superconducting compounds, for example +transition-metal dichalcogenides and in cuprates [12], points +to the importance of understanding the intrinsic characteris- +tics of these states and its interplay with other common fea- +tures present in these systems, such as disorder [13] and in the +presence of magnetic fields [14]. +Although condensed matter models usually start from the +notion of a perfect crystal, a plethora of notable effects are +only accessible when this notion is no longer true. One fa- +mous example is the problem of the high-Tc superconductiv- +ity on cuprates, in which a region of d-wave pairing occurs in +the form of a dome-shaped area and as a function of doping in +its phase diagram. Here, doping, either intentional or acciden- +tal, usually takes place, for example, via chemical substitution +in La2−xSrxCuO4 [15], or via inclusion of interstitial dopant +oxygen atoms (Oi) in Bi2Sr2CaCu2O8+δ [16], La2CuO4+y +[17] or YBa2Cu3O6.5+y [18], which can be treated as point- +like scattering centers as well as extended defects that intro- +duce disorder and deviate the neighboring atoms from their +crystallographic positions. This brings to light a fundamen- +tal question regarding the context of the dome-shaped area +of high temperature superconductivity in cuprates, on what +mechanism is responsible for the reduction in Tc upon over- +doping as well as to the subsequent disappearance of super- +conductivity at a critical doping. Usually, this is ascribed to +intrinsic effects, in which pairing correlations diminish with +doping, due to screening of local Coulomb interactions [19], +but some authours have also addressed the role of disorder in +surpressing superconductivity [20–22]. Disorder, however, is +usually incorporated as random on-site energies in Hubbard- +like models that can lead to Anderson localization phenomena +[23–25], thus it is important to extend this effects to include +also the possiblity of severe structural disorder within finite +regions of the crystal. +One of the most significant results from the study of disor- +der effects in superconductivity is the well known Anderson’s +theorem, which states that both the transition temperature, Tc, +and the isotropic gap, ∆0, of s−wave superconductors are in- +sensitive to the presence of weak disorder at the mean-field +level of BCS-like models [26–28]. One of the requirements +of the theorem is that the density of states remains unchanged +when compared to the pure metal case. As such, if the in- +fluence of disorder is strong enough to deplete the density +of states, the theorem no longer holds, and disorder dramat- +ically affects superconductivity [29]. In this case of strong +disorder and high concentration of impurity centers, the su- +perconducting correlation length is comparable to the disor- +der correlation length, and the mean-field equations can lead +to self-organized granularity where fluctuations of the local +order parameter are present [30]. This is likely to be the case +for overdoped cuprate superconductors with high concentra- +tion of interstitial oxygens that can lead to the formation of +nanosized oxygen puddles, regions with agglomeration of Oi, +that support superconductivity [17, 18, 31–33]. +The case of unconventional high temperature d−wave su- +perconductivity in hole-doped cuprates is of experimental and +theoretical relevance since its discovery [34]. Apart from sev- +eral different physical characteristics, one of the main differ- +ences between these materials and the conventional BCS su- +perconductors is that the superconducting gap amplitude is +not homogenous when the system undergoes the supercon- +ducting transition. This is evidenced by scanning tunneling +microscopy (STM) spectra in Bi2Sr2CaCu2O8+δ at different +arXiv:2301.02892v1 [cond-mat.supr-con] 7 Jan 2023 + +2 +doping levels, where the inhomogenous gap in the supercon- +ducting regime is revealed to be represented by a variety of +gap sizes and amplitues occuring in all samples as the con- +centration of dopants is varied [16]. Most remarkably, there +is a clear correlation between the position of Oi agglomer- +ates and the amplitudes of the gaps, since regions with larger +groups of dopants are observed to correspond to regions of +larger gap amplitudes [16]. Paralelly, Oi dopants have been +observed to self-organize into nanosize regions, or puddles, +as mentioned above, via µXRS in HgBa2CuO4+δ [35], as +well as in other cuprate compounds [36]. Remarkably, it has +been observed that spatial variations in the self-organization +of the nanosized Oi-rich puddles have a direct effect on su- +perconductivity, through variations in the critical temperature +[37]. Therefore, it is of paramount importance a deeper un- +derstanding, from a theoretical perspective, of the role of the +oxygen puddles in the physics of hole-doped cuprates. +In this work, we aim to investigate the effects of how struc- +tural disorder caused by the agglomeration of Oi in puddles +is responsible for the appearence of finite (nonzero) center-of- +mass (CM) momentum Cooper pairs in overdoped cuprates. +This will be done by making use of a previously reported +model proposed to describe how superconductivity rises in +cuprates, on the underdoped side of the phase diagram, in +terms of the phase synchronization of networks of nanosized +superconducting puddles, rich in interstitial dopant oxygens +[38]. Following, we extend the puddle model to derive analyt- +ical expressions showing how the superconducting gap, and +thus the critical temperature, are affected by the presence of +Cooper pairs with finite CM momentum and structural dis- +order. +Finally, we show numerically that both Tc and ∆0 +decrease with increasing disorder, thus pointing to a simple +physical mechanism to explain the closing of the supercon- +ducting dome-shaped area of the phase diagram, as being due +to the reduction of the available phase space for Cooper pair- +ing due to the development of a nonzero, finite CM momen- +tum Cooper pairs. +This paper is divided as following: in Sec. II we explain +the puddle model, which is the base for the calculations pre- +sented in this work, and derive the effective interaction be- +tween electrons and the network of puddles, giving rise to a +finite CM momentum pair state. Following, in Sec. III we de- +scribe the effects of structural disorder that the agglomeration +of Oi within each puddle causes to the system. In Sec. IV +we derive the the self-consistent equation for the amplitude of +the superconducting gap in terms of disorder and finite CM +momentum Cooper pairs. Then Sec. V is devoted to the nu- +merical calculations. Finally, we discuss the implications of +our results within the framework of networks of nano-sized +puddles and summarize our findings in Sec. VI. +II. +INHOMOGENEOUS OXYGEN PUDDLES +The oxygen rich nanopuddles have different elastic proper- +ties than their surroundings, and can therefore be considered +as elastic insertions in an otherwise homogeneous medium, +with its own vibrational mode, forming a network of super- +Figure 1. Pictorical view of the disordered background introduced by +the network of puddles (blue) in the system. The network consists of +puddles of different sizes, defined by the radius of each insertion. +Electrons (black and red) scatter in each puddle and, in the super- +conducting state, percolate within the network. +conducting nanoscale puddles, as shown in Fig. 1, which is +the starting point for the model that captured how supercon- +ductivity may arise in cuprates due to the phase synchroniza- +tion of each nanopuddle [38]. In terms of the Kuramoto model +for sychronization of phase oscillators [39, 40], each nano- +sized puddle is assigned to a phase, that in the underdoped +regime evolves independently of the others, giving rise to lo- +calized patches of superconductivity, as revaled by STM and +other techniques. With increased concentration of Oi through +doping, the superfluid density is responsible for the enhance- +ment of the interactions between the puddles and, in terms +of the Kuramoto model, to lock their phases in a synchronous +way. Following a BCS-like procedure, the order parameter for +synchronization is connected to the amplitude of the bulk su- +perconductor gap, that is non zero only after the locking of the +global phase in the synchronized phase. The synchronization +and the large frequency of the global network of puddles is +also responsible for large values of Tc in the optimally doped +cuprates within the model. +Inspired by these experimental and theoretical findings, +we introduce a model Hamiltonian that captures the inter- +action between electrons and localized vibrations that arise +from the agglomeration of interstitial oxygens in one pud- +dle. This interaction must be local, since each electron will +only interact with the quantized vibration whenever it is in +the region defined by the puddle (see Fig. 1). The minimal +model that captures this physical situation can be divided in +H = Hel + Hp + Hel−p, with +Hel = +� +k,σ +ξkc† +k,σck,σ + +� +k,k′ +Tk,k′c† +k′,σck,σ, +where the first term represents a band of electrons with dis- +persion ξk measured relative to the chemical potential, with +creation c† +k,σ and annihilation ck,σ fermionic operators. The +second term represents the scattering of electrons in each in- + +3 +homogeneity described by the puddles, with strenght con- +troled by the spin-preserving momentum transfer disorder ma- +trix Tk,k′. The oxygen puddles are described by local phonon +modes +Hp = +� +q +ℏωqa† +qaq, +with frequencies ωq and the creation (a† +q) and annihilation +(aq) bosonic operators, responsible for the description of the +localized vibration of each puddle. Finally, the interaction +term can be described as +Hel−p = +� +r,R,σ +g(r − R)c† +r,σcr,σ +� +a† +R + aR +� +, +where r and R are the electron and puddle locations, respec- +tively. The puddle is a finite size region in space, thus R de- +fines the center of this region that can be modeled as a sphere. +The interaction strenght g(r − R) is only relevant whenever +the electron is in the region around the puddle, which can +be modeled using a Gogny-type short range interaction that +is dependent on the radius of the oxygen agglomeration re- +gion [41]. After performing the transformation to momentum +space, the interaction term is written as +Hel−p = +� +k,k′,σ,q +M(q, k − k′)c† +k,σck′,σ +� +a† +−q + aq +� +, (1) +where +M(q, k − k′) = +� +R +g(k − k′) exp [i(q − [k − k′]) · R] +is associated with the fact that the puddles are not present in all +sites, rather they are inhomogeneously distributed around the +system, thus the summation has to be retained only to these +regions, which is relevant for the case of Bi2Sr2CaCu2O8+δ, +since locations of dopant oxygens are observed to be consis- +tent with the position inferred from local strain analysis of the +incommensurate structure, as imaged by scanning transmis- +sion electron microscopy (STEM) [42], which means that the +crucial oxygen dopants are periodically distributed in corre- +lation with local strain. However, not all strained regions are +occupied with dopant oxygen atoms, that is the distribution of +Oi is inhomogeneous, which justifies our approximation and +is consistent with STM measurements [43]. In the limits of a +clean or a totally doped system, this term can be treated ex- +actly. The factor g(k − k′) is the Fourier transform of the in- +teracting potential between the electrons and the puddles and +controls the momentum transfer between the incoming and +scattered electron. +One can see from Eq. (1) that the presence of a finite den- +sity of puddles spread around the systems give rise to a off- +diagonal term associated with the momentum transfer k − k′ +that comes from the interacting potential. In the limit that the +summation over M(q, k − k′) can be made exactly, one re- +covers the usual definition of an electron-phonon interaction, +where the momentum transfer is the momentum of the local +phononic mode q, as in the Frohlich [44] and Holstein [45] +models, for example. In order to explore the effects of this +kind of interaction in the form of pairing, we introduce an +unitary transformation H′ = e−SHeS, with an ansatz for the +transformation matrix +S = +� +k,k′,σ,q,Q +M(q, k − k′)c† +k,σck′,σ +� +xa† +−q + yaq +� +, +where x and y are factors determined a posteriori. After the +transformation (see Appendix A for details), we end with an +effective interaction written as +Heff = +� +k,k′ +� +p,p′ +V (k, k′)f(p, p′)c† +k,↑c† +p−k,↓cp′−k′,↓ck′,↑ +(2) +with V (k, k′) = D(k, k′)|g(k − k′)|2 being the potential +arising from the interaction between electrons and puddles, +D(k, k′) the phononic propagator associated with the lo- +cal phonon modes produced by the vibrating puddles and +f(p, p′) = � +R e−i(p−p′)·R the phase factor controlling mo- +mentum transfer between the interacting electrons. +In the +regime where the phononic propagator is negative, given that +ξk ≈ ξk′, we have an effective attractive interaction between +the electrons mediated by the nanopuddles. Remarkably, this +interaction leads to the formation of finite center-of-mass mo- +mentum Cooper pairs represented by p and p′. Therefore, +from the perspective of inhomogeneously distributed puddles +bringing disorder to an otherwise clean medium, a bound state +between two electrons can be formed with a finite center-of- +mass momentum that is associated with the strenght of the +interaction between the electrons forming the pair and the ag- +glomeration of interstitial dopant oxygens in one nanopuddle. +It is important to notice that the states arising from the ef- +fective Hamiltonian in Eq. (2) are different from other pro- +posed pair states with finite CM momentum, as for example +the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, where fi- +nite center-of-mass momentum Cooper pairs can be stabilized +under a finite magnetic field via the Zeeman coupling [46, 47], +and the recently proposed current driven FFLO state [48]. +Moreover, it has been shown that, even without the presence +of a magnetic field or other external potentials, a finite CM +momentum Cooper pair can be stable in a superconducting +ground state as pointed in Ref. [7], but the authors do not ex- +plore the effects that can give rise to this kind of state. Here +we start from the fact that nanosized puddles are formed via +doping and the responsible for the CM momentum of the pairs +is disorder induced by the puddles in the system. +Eventhough we are not considering any specific form for +the interaction potential g(k − k′), it is important to com- +ment that the only requirement is that it must be a finite size +potential in real space, which means that is not a point-like +disorder center that is scattering the electrons in the interac- +tion term of Eq. (1), rather is a region in space defined by +the agglomeration of oxygen interstitials. In this case, we can +point to potentials like the Woods-Saxon potential [49] that is + +4 +Figure 2. Top: Structure factor for disordered media from Hose- +mann’s paracrystalline theory [58], given by eq. (6) in the text. The +pristine case corresponds to the ℓ → ∞ limit, where the structure +factor is given by delta-peaks at reciprocal lattice vectors and mo- +mentum is conserved (here ℓ is a measure of disorder and for this +reason should be inversely related to the residual resistivity shift due +to structural disorder, ℓ ∝ 1/δρ0). Bottom left:: − Bragg diffraction +pattern for a structually disordered medium, showing Bragg peaks at +the central region and Bragg rings at the outter region; Bottom right +− plot of the structure factor as a function of momentum transfer, +∆Q, showing well defined Bragg peaks, for small momentum trans- +fer, at the reciprocal lattice vectors, G, while the Bragg peaks be- +come ever broader, at larger momentum transfer, eventually merging +into rings. +used to describe the forces applied on protons and neutrons in +the atomic nucleous or the Gogny-type interactions [50–52], +which is another kind of nucleon-nucleon potential that has +also found applications in astrophysics [53], as possible can- +didates to describe the electron-puddle interaction. However, +a precise and detailed description of such potential would re- +quired more knowledege about the formation of the nanosized +puddles and its effects on the crystal structure of the host ma- +terial, which would affect the electronic degrees of freedom +[54], but this is outside the scope of the present study. +III. +STRUCTURAL DISORDER +Before we proceed to the characterization of the supercon- +ducting state that arises from the effective Hamiltonian de- +rived in the last section, it is important to briefly discuss which +kind of disorder is giving the Cooper pairs a finite CM mo- +mentum. In order to do that, we introduce concepts arising +from the study of structural disorder, which is the kind of per- +turbation that the agglomeration of Oi causes in the crystalline +structure of different cuprate systems, as for example by tilt- +ing the CuO6 octahedra in La2CuO4+δ [55] and by altering +the distance between the apical oxygen and the planar copper +atom in Bi2Sr2CaCu2O8+δ [56]. +Translational invariance is one of the most fundamental +properties of pristine crystals. +The concept of a Brillouin +zone, that repeats itself by translations of reciprocal lattice +vectors, allows us to organize electrons in energy bands, +ϵn(k), labeled by a band index, n, and function of a quasi- +momentum (wave-vector) quantum number, k, in terms of +which periodic Bloch wave-functions, un,k(r), are defined. +A perfect crystal is characterized by very intense and sharp +peaks in the Fraunhofer diffraction pattern of Bragg scatter- +ing experiments. The existence of such sharp peaks follows +directly from Heisenberg’s uncertainty principle and their lo- +cation is determined by the crystalline-lattice structure factor. +Simply put, an extended Bloch wave with well defined mo- +mentum state, k, that interacts with ions located at arbitrary +positions, ri, of the crystal (infinite uncertainty ∆r → ∞), +scatters into another extended Bloch wave with momentum +state, k′, with zero uncertainty, ∆k → 0. The entire process +carries a phase +φ(k′ − k) = +1 +√ +N +� +ri +friei(k′−k)·ri, +(3) +where N is the number of lattice sites in the crystal and fri is +an atomic form factor that gives the probability that an atom +is located at a certain crystallographic position. The scattered +intensity is proportional to |φ(k′−k)|2 and is thus determined +by the lattice structure factor, +S(k′ − k) = 1 +N +� +ri,rj +frifrjei(k′−k)·(ri−rj). +(4) +For a pristine crystal all atoms are at their ideal locations, +fri = frj = 1 and thus S(k′ − k) = � +g δk′−k,g, where g is +a reciprocal lattice vector. The Fraunhoffer diffraction pattern +in this case thus corresponds to δ−like peaks as shown in Fig. +2 and the kinematic constraint of quasi-momentum conserva- +tion, +k′ = k + g, +(5) +forms the basis for Bloch’s theorem. In the opposite limit of a +random atom gas, however, an extended Bloch wave with well +defined momentum state, k, that interacts with ions located at +a particular, well defined position, ri, of the crystal (zero un- +certainty ∆r → 0), scatters into another extended Bloch wave +with momentum state, k′, with infinite uncertainty, ∆k → ∞. +In this case, frifrj = δri,rj, and S(q) = 1. There are no +kinematic constraints whatsoever relating k and k′ to g and +the Fraunhoffer diffraction pattern in this case corresponds to +an isotropic disc of even intensity, as shown in Fig. 2. +Interpolating between the pristine and random limits de- +scribed above by increasing disorder is pivotal to the descrip- +tion of inherently inhomogeneous systems, such as the one +of ramdom oxygen puddles described in the present work. If +disorder is of the first type, namely weak disorder, all atoms +deviate only slightly from their ideal positions in the crys- +tal, independently of the deviations of their neighbors [57]. +This is the case of pointlike defects, thermal vibrations or +micro-mechanical strains, and this kind of disorder preserves +long range crystalline order. In this case the widths of the +peaks in the Fraunhoffer diffraction pattern are not affected, + +Bragg diffraction patterns +lα1/po +measures the amount of distortions +Smax(g) +S℃(k'- k) = k'-k-q,0 +1＋l2(k'kqg) +g0 +20 +Structure factor Sq(k'-k) +15 +10 +5 +0 +0 +2 +4 +Reciprocal vector AQ=k-k-q (units of G) +uncertainty in reciprocal lattice +breakdown momentum conservation5 +and only their intensity is slightly reduced since for uncor- +related Gaussian disorder, frifrj = D2 < 1, where D2 +is the Debye-Waller factor. The structure factor is given by +S(k′ − k) = D2 � +g δk′−k,g. If disorder of the second type, +namely strong disorder, however, the atoms deviate signifi- +cantly from their ideal positions in the crystal, and deviations +amogst neighboring atoms are correlated. This is the case of +extended defects, amorphous regions, molten materials, etc, +and this type of disorder causes the loss of long range crys- +talline order. In these paracrystalline structures, not only the +intensity of the diffraction peaks will decrease but, most im- +portantly, their widths will suffer from a nonlinear increase of +their integral breadth, δg, for successive orders of Bragg re- +flections. The complete paracrystalline theory was proposed +by Hosemann [58]. Hosemann included fluctuations of vari- +ance σ that introduce correlations between pairs of atoms, +⟨frifrj⟩, that decrease with separation ultimately causing the +peaks in the structure factor of the material to broaden the +larger the reciprocal lattice. The result is a structure factor +composed by a sum of Lorentzians [59] +Sq(k′ − k) = +� +g +Smax(g) +1 + ℓ2 +hkl(q − k′ + k − g)2 , +(6) +of amplitudes Smax(g) = 4/σ2g2 and breadths for Bragg +reflections, |δg| ≡ 1/ℓhkl = σ2π2(h2 + k2 + l2)/a0, given in +terms of the original lattice parameter a0 and the momentum +transfer, q. Hosemann’s paracrystalline theory allows us then +to interpolate continuously between pristine and random cases +through the fluctuation parameter σ: +• for σ → 0 we have ℓhkl → ∞, ∀h, k, l and we obtain +Sq(k′ − k) = � +g δq,k′−k+g, enforcing the kinematic +constraint of momentum conservation, q = k′ − k + g, +typical of pristine crystals [59]; +• for σ → ∞ we have ℓhkl → 0, ∀h, k, l and we end +up with Sq(k′ − k) = Smax(0) → 1, isotropic, for +arbitrary q, k, k′ and determined solely by the g = 0 +contribution, typical of infinite, aperiodic systems [59]; +• for 0 ≤ σ ≤ ∞ we have ∞ ≥ ℓhkl ≥ 0 and the +structure factor, Sq(k′ −k), will be composed by sharp +Bragg peaks at small g (large ℓhkl) and isotropic discs +for larger g (small ℓhkl), as shown in Fig. 2, relax- +ing the kinematic constraint of momentum conserva- +tion, q ̸≈ k′ − k + g, typical of a paracrystal, liquids, +strongly disordered or amorphous systems [59]. +IV. +DISORDER AND GAP FLUCTUATIONS +We now address how the superconducting state of the effec- +tive interaction derived in Sec. II is affected by the structural +disorder effects introduced in the previous section. We start +from the effective Hamiltonian in Eq. (2) and, within a mean- +field decoupling of the quartic term, write the equation for the +superconducting gap as +∆k = − +� +k′,p′ +Vk,k′f0,p′ ⟨cp′−k′↓ck′↑⟩ , +(7) +where we set p = 0, since we want to describe ampli- +tude fluctuations for the superconducting gap in the Cooper +channel. +For the superconducting state formed by singlet +pairs with finite CM momentum, the system can be repre- +sented by the spin-independent imaginary time Green’s func- +tion G(k, k′, τ) = − +� +Tτck,σ(τ)c† +k′,σ(0) +� +and the anomolous +pair propagators F(k, k′, τ) += +⟨Tτck,σ(τ)ck′σ′(0)⟩ and +F∗(k, k′, τ) = +� +Tτc† +k,σ(τ)c† +k′,σ′(0) +� +for σ ̸= σ′. Within +Nambu’s formalism, we can write the decoupled effective +Hamiltonian from Eq. (2) and the electronic components from +Hel in matrix form and derive in first order perturbation theory +the electronic Green’s function for an inhomogeneous system +with disorder as +G (k, k′, iωn) = G0 (k, k′, iωn) ++ +� +p,p′ +G0 (k, p, iωn) Tp,p′σ3G (p′, k′, iωn) , +where G0 (k, k′, iωn) is the matrix form of the translation- +ally invariant electronic Green’s function in frequency space, +iωn are the fermionic Matsubara frequencies and σ3 is a Pauli +matrix. +The diagonal elements of this matrix are defined +by the bare Green’s function in the superconducting state, +G0(k, iωn), and its off-diagonal terms are represented by the +anomalous propagators F0(k, iωn) which are written as +G0 (k, iωn) = +− (iωn + ξk) +ω2n + ξ2 +k + |∆k|2 , +F0 (k, iωn) = +∆k +ω2n + ξ2 +k + |∆k|2 . +In order to proceed, we shall take a couple of approximations: +first we consider the case of overdoped cuprates, which puts +the system in a high concentration of disorder, thus Tp,p′ = +T f(p, p′), where disorder influences the momentum trans- +fer controled by the phase factor f(p, p′) with strenght T . +Second we assume that for a translationally invariant system +the normal and anomalous Green’s functions can be rewrit- +ten as G0(k, k′, iωn) = G0(k, iωn)δk,k′ and F0(k, k′, iωn) = +F0(k, iωn)δ−k,k′. Following these couple of approximations, +the first order pertubation theory expansion of the interacting +Green’s function is simplified +G (k, k′, iωn) = G0 (k, iωn) δk,k′ ++ T fk,k′G0 (k, iωn) σ3G0 (k′, iωn) . (8) +From the gap equation in Eq. (7) and from the definition of +the anomalous propagator, we write + +6 +∆k = − +� +k′,p′ +Vk,k′f0,p′ ⟨cp′−k′↓ck′↑⟩ += − +� +k′,p′ +Vk,k′f0,p′ +� +1 +β +� +ωn +F (p′ − k′, k′, iωn) +� +,(9) +with β = 1/T being the inverse temperature (in units of +kB = 1). By using the matrix form in Eq. (8), we get the form +of the interacting anomalous propagator, where it is worth not- +ing that the normal and anomalous propagators mix in the +impurity scattering. +Despite the anomalous Green’s func- +tion being invariant for time reversal, the normal one is not, +and since disorder produces the transformation F0(k, iωn) ↔ +G0(k, iωn) we clearly see this is a mechanism that breaks time +reversal invariance. As a consequence, this mechanism breaks +the Cooper pair that leaks into the normal metal surrounding +the puddles. +In order to understand the effects of disorder and finite CM +momentum in the gap equation, we substitute the form of the +anomalous propagator given by the matrix in Eq. (8) inside +Eq. (9) to write the gap equation as ∆k = ∆BCS +k ++ δ∆k, +where +∆BCS +k += − +� +k′ +Vk,k′∆k′ +2Ek′ +tanh +�βEk′ +2 +� +, +(10) +is the BCS limit for the gap equation, arising from the first +term in Eq. (8), with the bare anomolous propagators and +Ek = +� +ξ2 +k + ∆2 +k. Then +δ∆k = T +� +k′,p′ +Vk,k′f0,p′fp′,0 +1 +β +� +ωn +{F0G0 + G0F0}(11) +is the correction to the superconductor gap due to effects of +disorder in the system. The factor [f0,p′fp′,0] can be treated +within a mean over disorder in order to calculate the inter- +ference factor as [f0,p′fp′,0] = |f0,p′|2 → S(p′), where +S(0, p′) is the static structure factor. +Thus, the correction +to the gap equation can be written in terms of the structure +factor and we see that fluctuations associated with small CM +momentum p′ → 0 are absent, since the structure factor +S(p′) → 0 and the gap equation is dominated by the BCS +contribution. On the other hand, fluctuations associated with a +finite center-of-mass momentum dominate over the BCS con- +tribution when p′ ≫ 0 and S(p′) → 1. In a general manner, +the structure factor can be written as a sum of Lorentzians with +peaks in wave vectors of the reciprocal lattice, as discussed in +Sec. III and shown in Fig. 2. +Finally, we proceed by taking the Matsubara summations +over the set of mixed Green’s functions as in Eq. (11) to arrive +at the correction in terms of the disorder strenght T and the +finite CM momentum of the Cooper pairs p′ as +δ∆k = T +� +k′,p′ +Vk,k′S (p′) 1 +2 +� ∆k′,p′ +Ek′−p′ +ξk′ +Ek′ + ∆k′ +Ek′ +ξk′−p′ +Ek′−p′ +� +× +� +� +� +Ek′−p′ tanh +� +βEk′ +2 +� +− Ek′ tanh +� βEk′−p′ +2 +� +E2 +k′−p′ − E2 +k′ +� +� +� . +(12) +It is importance to notice the dependence of the correction +on the structure factor S(p′) controlling momentum transfer. +In the limit of small amount of disorder, the so called first- +type disorder [57], as discussed in Sec. III, pointlike deffects +does not affect the BCS gap, in accordance with Anderson’s +Theorem, as we shall see in the next section. On the other +hand, in the limit of high concentration of puddles, the system +is in the limit of second-type disorder, associated with strain- +induced lattice deformations, and both the amplitude of the +superconducting gap and the critical temperature are affected. +In order to proceed to the numerical analsysis, we perform +an approximation for the structure factor based on the limits +of disorder discussed above. For the first-type disorder, we +choose S(p′) = δ0,p′, since no momentum transfer will be +associated with pairs with finite CM momentum in the dilute +limit. On the other hand, for the second-type disorder, we +write S(p′) = 1, assuming a system with high concentration +of puddles. These two limits for the disorder of the 1st and +2nd types can be understood as a hard cutoff for the CM mo- +mentum distribution within the structure factor and are made +to simplify Eq. (12) to the following numerical analysis. +V. +NUMERICAL ANALYSIS +In order to fully understand the effects of disorder and +CM momentum of the Cooper pairs in the superconduct- +ing gap amplitude we perfom a numerical integration of Eq. +(12). We use the decomposition Vk,k′ = −V0η(k)η(k′) and +∆k = ∆0η(k), where η(k) = cos kx − cos ky is a d−wave +form factor, which gives the amplitude fluctuations of the or- +der parameter with the same symmetry. When stated for com- +parison, we shall also use Vk,k′ = −V0 and ∆k = ∆0 when +considering a s−wave symmetry for the interaction and the +gap. For the calculations in the square lattice, we consider a +two-dimensional electronic dispersion with nearest- and next- +nearest-neighbor hopping elements (t, t′) as +ϵk = −2t (cos kx + cos ky) + 4t′ cos kx cos ky − µ, (13) +where µ is the chemical potential that controls the electronic +density. This type of electronic dispersion is general for 2D +transport in strongly correlated systems and is suitable for the +description of the conduction band associated with the CuO2 +planes of high-Tc cuprates. +In the following calculations, all parameters are defined in +units of 4t and we set µ/4t = −0.45, away from the half- +filled case µ/4t = 0.0 (see Fig. 3), since the mean-field theory + +7 +Figure 3. Fermi surface structure used in calculations. Left: The 3D +plot of Eq. (13) in the first Brillouin zone in yellow and the chemical +potential cut defining the Fermi level in blue. Right: The Fermi level +defined by the cut at µ/4t = −0.45. The vectors k, fixed in the +direction (0, π), and k′, varying across the Fermi surface, are also +shown. +yields incorrect results for a two-dimensional lattice near half- +filling [60] and we avoid particle-hole symmetry [61]. For +this reason, we can take t′ = 0. We also set V0/4t = 1.0, +in the limit where the mean-field theory is still valid. For the +summations over p′, we define p′ = k − k′, where k, k′ are +the momenta of the two paired electrons, which we set |k| = +|k′| = kF as two momenta in the Fermi surface. The CM +momenta are then defined by fixing k in the direction of the +point (0, π) and by varying k′ across the Fermi surface, as +shown in Fig. 3. +We start by analyzing the zero temperature limit T = 0 of +Eq. (12), where the hyperbolic tangents can be simplified. In +Fig. 4 we show how the gap amplitude ∆0 is affect by disor- +der T in the limit of disorder of the 1st type, S(p′) = δ0,p′, +or weak concentration of puddles, and strong concentration, +S(p′) = 1, in the limit of disorder of the 2nd kind. The +gap amplitude is insensitive to disorder in the dilute limit for +s−wave pairing, thus ∆0 = ∆BCS +0 +and the BCS limit is re- +covered, in accordance with Anderson’s theorem. However, +in the opposite limit, the disorder strongly affects the ampli- +tude of the gap for d−wave pairing, introducing fluctuations +and decreasing its absolute value in about 50% in the strong +disorder limit, when compared to the clean case. +It is worth noting that the reduction is not linear as the +strenght of disorder approaches the values of the fixed pairing +potential, T → V0, where the pertubation theory still holds. +This can be traced back to the fact that the gap equation is +a self-consistent equation for the aboslute value of ∆0, even +after the approximations considered. Thus we see that even +in the zero temperature limit, disorder tends to destroy super- +conductivity in a system with high concentration of oxygen +interstititals, as in the overdoped cuprates. +We also investigate the effects of specific finite CM momen- +tum on the amplitude of the gap when T = 0. We choose a set +of momenta {p} and substitute in Eq. (12) the corresponding +structure factor, namely S(ps) = δp′,ps, where ps are the mo- +menta in the set. All ps are multiples of kF of each direction +considered, namely (0, π) and (π, π). In Fig. 5 we display the +Figure 4. T = 0 limit for the amplitude fluctuatios of the supercon- +ducting gap as a function of disorder strenght compared to the clean +system. Dilute limit (red), for disorder of the 1st kind and s−wave +symmetry, and high concentration of puddles for disorder of the 2nd +kind and d−wave symmetry (blue). The black dashed line is a guide +to the eye. Gap values are given in terms of ∆0 in the absence of +disorder T = 0. +evolution of the amplitude of the superconducting order pa- +rameter ∆0 as a function of the CM momentum of the Cooper +pairs p, for fixed disorder strenght T = 0.1. The supercon- +ducting order parameter is modulated, with period determined +by the distance between adjacents Fermi surfaces in each di- +rection, being 3.75|kF| for (0, π) and 5.75|kF| for (π, π). Re- +markably, this is in direct contact with the diffraction pattern +displayed in Fig. 2. However, since we are considering a +hard cutoff for the structure factor in terms of delta functions, +the amplitude of the gap modulation is not altered by the dis- +tance from the origin. We expect that by including a more +realistic model for the structure factor, the amplitudes of the +modulations will decay with p, with its effect stronger in the +(π, π) direction, since larger reciprocal lattice vectors G im- +ply a broader structure factor, thus diminishing the amplitude +of the superconducting gap. Altogether, the interplay between +disorder and finite center-of-mass momentum Cooper pairs is +able to strongly affect the superconducting order parameter. +Now we turn to the finite temperature case T ̸= 0 for the +d−wave symmetric order parameter to understand how disor- +der and CM momenta for the Cooper pairs affects the critical +temperature Tc. In Fig. 6 we show the evolution of the su- +perconducting gap with temperature, for different values of +the disorder strenght T . It is clear that with increasing dis- +order, not only ∆0(0) decreases, as pointed in the zero tem- +perature limit, but we also evidence a decrease in the criti- +cal temperature Tc, defined as the value of temperature that +∆0(T, T ) → 0, with disorder, as shown in the inset. This +means that pair breaking is induced by the scattering of the +finite CM momentum Cooper pairs with the nanosized oxy- +gen puddles of the system and by increasing disorder, Tc is +significantly reduced. +This pair breaking effect is due to the fact that the phase +space required to pair formation is reduced when p increases +in absolute value. In the small scattering momentum transfer + +2 +k +03 +斤-2 +0 +-1 +0 +ky +元 +元 +0 +2 +Kx +元 +2 +-2 +-2 +-2 +0 +2 +kx1.0 +0.9 +△o(T)/△o(0) +0.8 +0.7 +2nd d wave +0.6 +lst s wave +0.0 +0.2 +0.4 +0.6 +0.8 +T8 +Figure 5. Left: Extended Brillouin zones in the upper positive part of +momentum space. The arrows indicate the distance between the cen- +ters of each Fermi surface in terms of the Fermi vector |kF | of each +direction considered. Right: The amplitude of the superconducting +order parameter as a function of different CM momentum vectors +|p|, in the directions (0, π) and (π, π). ∆0(p) is given in units of +the gap at p = 0. +sector, p < |kF|, the gap is almost unnafacted by the presence +of disorder when comprared to the value when p = 0, since +the shape of the Fermi surface intersection of the two paired +electrons suffers little change. However, when p approaches +the maximum absolute value of 2|kF| within the first Brillouin +zone, the phase space for pair formation is greatly reduced and +disorder induces pair breaking, captured by the reduction of +the superconducting order paremeter. The modulation occurs +for p > 2|kF|, since electrons from different Brillouin zones +participate in the scattering and pairing process. Therefore, +these results point to the combined effect of finite center-of- +mass momentum pairs being scattered by structural disorder +induced by the network of oxygen puddles as a mechanism +for the reduction of the superconducting gap and the critical +temperature in the overdoped regime. +VI. +CONCLUSION AND DISCUSSION +In this work we presented an extension of the proposed +model for the formation of networks of puddles and its ef- +fects on the superconductivity in oxygen-doped cuprates [38]. +We show that the presence of puddles, in the overdoped side +of the phase diagram, introduces strong disorder in the sys- +tem that induces the formation of finite center-of-mass mo- +mentum Cooper pairs. We derive an analytical expression for +the amplitude fluctuations in the superconducting gap induced +by the puddles, within a mean-field BCS-like approach, in +terms of the disorder strenght T and the finite CM momenta +p. We numerically solve this expression to show that even +in the zero temperature limit the gap is strongly affected by +disorder-induced CM Cooper pairs. In the limit of strong dis- +order, the gap tends to close and, in the finite temperature case, +Tc tracks the reduction of the superconducting gap, also being +strongly affected by disorder. It is important to emphasize +that we do not account the effect of longer-range Coulomb re- +pulsion, restricting the application of our results to screened +systems [68]. +Figure 6. Temperature dependence of the superconducting order pa- +rameter for different values of disorder strenght (colored bar). Gap +alues are given in terms of the clean case T = 0 and temperature in +terms of T 0 +c also of the clean case. Inset: The critical temperature +dependence normalized to the clean value as a function of disorder +strenght. The black dashed line is a guide to the eye. +The experimental observations of structural scale invari- +ance of dopants detected by scanning micro-x-ray diffraction +[36], the promotion of critical temperature [37], the agglom- +eration of interstitial oxygens in regions of strong local strain +in the crystal structure of cuprate superconductors [42, 43] +and the proposed theoretical reports regarding the presence +of networks of nanoscale superconducting islands in high- +temperature superconductors [62–65] are in close connection +with the results reported here. Eventhough we are showing +that the superconducting state is depleted in the presence of +strong disorder in the overdoped regime, it is clear from the +above mentioned surveys that the importance of these net- +works and its interplay with electronic degrees of freedom +pass across the whole phase diagram of hole-doped cuprates. +In Ref. [38], the present authors show how the complex +networks formed by the oxygen puddles can transionate to a +synchronized phase, controlled by the superfluid density, in a +way that the concentration of dopant atoms controls the emer- +gence of local superconductivity in the underdoped regime +and how the systems evolves to a bulk superconductor as the +concentration of dopants, thus puddles, increases as the sys- +tems approaches the optimally doped regime. It is important +to emphasize that within this framework, the state studied in +this work is described by the bulk superconductor state in the +synchronized phase of the network formed by the oxygen pud- +dles (see Fig. 1), in the sense that we require the network of +puddles to be fully synchronized in order to the band of elec- +trons to interact with the global mode of vibration of the syn- +chronized network. Our approach is based on a mean-field +approximation for the complex network, therefore we point to +the importance of describing different topologies for the orga- +nization of the puddles and how this can affect not only the +transition to the superconducting state [66], but also its pos- +sible interplay with the superconducting fluctuations of pre- +formed Cooper pairs observed in the pseudogap phase above +Tc [67], in terms of local superconductivity. + +8 +1.00 +(0, πt) +(π, ) +6 +0.98 +[ KF +0.96 +4 +5 +K +7 +.75/kFl +0.94 +3 +2 +0.92 +0 +0.90 +-2 +0 +2 +4 +6 +8 +01234567891011 +kx +Ip/ / / kFl1.0 +1.0 +0.8 +0.8 +.0.6 +0.6 +T)/△o( +1.0 +Ao(T, +0.4 +0.4 +.0.8 +0.2 +0.2 +0.6 +0.0 +0.2 +0.4 +0.6 +0.8 +T +0.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.09 +Appendix A: Unitary transformation +In this Appendix section, we show the derivation of the +effective Hamiltonian containing the pairing interaction be- +tween two electrons forming a Cooper pair with finite center- +of-mass momentum. The starting point is the full Hamiltonian +written in momentum space H = Hel+Hp+Hel−p, which is +the summation over the contributions of the electrons, puddles +and electron-puddle interaction, respectively. Introducing an +unitary transformation of the form H′ = e−SHeS, where S +is the transformation matrix introduced in Sec. II, we can ex- +pand the exponentials up to second order in powers of S to +write the transformed Hamiltonian as +H′ = H + [H, S] + 1 +2[[H, S], S], +(A1) +and by treating Hel−p as a perturbation, we can divide the full +Hamiltonian as H = H0 + Hel−p, where H0 contains the +kinetic terms of electrons and puddles, to write +H′ = H0 + Hel−p + [H0, S] + [Hel−p, S] + 1 +2[[H0, S], S]. +Since the goal is to eliminate the interaction, the defining +equation for the transformation matrix comes from the elim- +ination of the first-order term [H0, S] + Hel−p = 0, from +which we can extract the factors x and y for S. In this way, +the transformed Hamiltonian can be written in terms of an ef- +fective interaction that comes from recombining the terms in +the commutators +H′ = H0 + 1 +2[Hel−p, S], +(A2) +thus the problem is reduced to an effective system described +by H = H0 + Heff, where Heff = +1 +2 [Hel−p, S]. +By +performing the calculation over the commutator [H0, S], the +choice of x and y that eliminate the first-order term is given +by +xk,k′,q = +1 +ξk′ − ξk − ωq +, +yk,k′,q = +1 +ξk′ − ξk + ωq +, +and the transformation matrix S is fully defined. Then we +proceed to the calculation of the effective Hamiltonian that +comes from the commutator of the now defined matrix S and +the electron-puddle interaction, which gives a combination of +M(q, Q)M(−q, Q′), where Q = k − k′ and Q′ = k′′ − k′′′ +are two auxiliar variables that accomodate the variety of in- +dices arising from the commutator. Recalling the definition of +the factor M given in the main text, we see that +M(q, Q)M(−q, Q′) = +� +R,R′ +g(Q)g(Q′) +× ei(R−R′)·qe−i(Q·R+Q′·R′), +which can be simplified by taking R = R′ since each R de- +scribes the position of a nanosized puddle and we are assum- +ing the dilute limit of oxygen puddles, as discussed in the +main text, in accordance with STEM and STM measurements +[42, 43]. In this way, the effective Hamiltonian is written as +Heff = +� +k′,k′′′,q,Q,Q′ +V (q, Q, Q′)M(q, Q)M(−q, Q′) +× c† +k′′′+Q′c† +k′+Qck′ck′′′, +(A3) +with V (q, Q, Q′) = ωq/[(ξk′′′ − ξk′′′+Q′)2 − ω2 +q]. Proceed- +ing with the calculation, we note that within BCS theory, the +effective Hamiltonian describes the interaction between elec- +trons with opposite momenta k′ = −k′′′, with zero CM mo- +mentum. However, in our case, the auxiliar variables Q and +Q′ introduces a momentum transfer connected with a finite +CM momentum for the pairs, for each fermionic operator in +the effective Hamiltonian that comes from the commutator +[Hel−p, S]. In this sense, we perform a change of variables +introducing the finite CM momentum k′ + k′′′ = p, in a way +that we can eliminate the dependence on the auxiliar variables. +The new variables introduced are written as k = k′′′ + Q′ and +−k + p′ = k′ + Q, where p and p′ are the CM momenta of +the Cooper pairs. In the limit where the interaction g(k, k′) is +independent of the CM momenta, we can decouple the effec- +tive interaction and end up with the effective Hamiltonian +Heff = +� +k,k′ +� +p,p′ +V (k, k′)f(p, p′)c† +k,↑c† +p−k,↓cp′−k′,↓ck′,↑, +with +V (k, k′) = +ω0 +(ξk′ − ξk)2 − ω2 +0 +|g(k − k′)|2 +f(p, p′) = +� +R +e−i(p′−p)·R +where we assume ωq = ω0, a dispersionless phonon mode for +each puddle. + +10 +[1] J. Bardeen, L. N. Cooper, and J. 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Lett. 108, 017002 (2012) + diff --git a/E9E1T4oBgHgl3EQfEgPe/content/tmp_files/load_file.txt b/E9E1T4oBgHgl3EQfEgPe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..00dd8d358e40a5bccb3b7afa9af331ccab8c24c4 --- /dev/null +++ b/E9E1T4oBgHgl3EQfEgPe/content/tmp_files/load_file.txt @@ -0,0 +1,822 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf,len=821 +page_content='Disorder-induced finite center-of-mass momentum Cooper pairing and its consequences to the critical temperature and superconducting gap of overdoped cuprates Victor Velasco1 and Marcello B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Silva Neto1 1Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, Brazil One of the most studied classes of unconventional high-temperature superconductors is the hole-doped cuprates, where special attention is given to those doped with extra interstitial oxygens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In this context, the formation of spatially inhomogeneous agglomerates of dopant oxygen atoms in the form of nanosized puddles is not only relevant, but also subject of intense recent experimental and theoretical surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Following these efforts, in this work we show the consequences of the presence of networks of oxygen puddles in the supercon- ducting state of overdoped cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Starting from the inhomogeneous disordered background brought by the network of puddles, we show that an effective interaction between electrons can be mediated by the local vibra- tional degrees of freedom of each puddle, but the pairs arising from this interaction have a finite center-of-mass momentum p, thus breaking up the Cooper channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Furthermore, we derive an analytical expression for the amplitude of the superconducting gap ∆k in terms of disorder and finite center-of-mass momentum and show that amplitude fluctuations are induced in the superconducting state by the presence of the puddles, where both the gap and the critical temperature are affect and reduced by disorder and finite momentum pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Finally, we discuss our findings in the context of networks of superconducting oxygen nano-puddles in cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' INTRODUCTION It is a well known fact within the Bardeen-Cooper- Schrieffer (BCS) theory of superconductivity that the two quasi-particles forming the bound states that constitute the superconductor, named Cooper pairs, have momentum k and −k, near the Fermi surface, with oposite spins ↑ and ↓, form- ing a singlet with zero center-of-mass momentum [1], in what is usually called the Cooper channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' However, the ex- istence of a finite-momentum superconducting ground state has recently been raised theoretically [2–7] and supported by several experiments in correlated quantum materials [8– 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Moreover, the possibility of emergent finite-momentum pair states, in the form of pair density waves, in a variety of well-established superconducting compounds, for example transition-metal dichalcogenides and in cuprates [12], points to the importance of understanding the intrinsic characteris- tics of these states and its interplay with other common fea- tures present in these systems, such as disorder [13] and in the presence of magnetic fields [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Although condensed matter models usually start from the notion of a perfect crystal, a plethora of notable effects are only accessible when this notion is no longer true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' One fa- mous example is the problem of the high-Tc superconductiv- ity on cuprates, in which a region of d-wave pairing occurs in the form of a dome-shaped area and as a function of doping in its phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Here, doping, either intentional or acciden- tal, usually takes place, for example, via chemical substitution in La2−xSrxCuO4 [15], or via inclusion of interstitial dopant oxygen atoms (Oi) in Bi2Sr2CaCu2O8+δ [16], La2CuO4+y [17] or YBa2Cu3O6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='5+y [18], which can be treated as point- like scattering centers as well as extended defects that intro- duce disorder and deviate the neighboring atoms from their crystallographic positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' This brings to light a fundamen- tal question regarding the context of the dome-shaped area of high temperature superconductivity in cuprates, on what mechanism is responsible for the reduction in Tc upon over- doping as well as to the subsequent disappearance of super- conductivity at a critical doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Usually, this is ascribed to intrinsic effects, in which pairing correlations diminish with doping, due to screening of local Coulomb interactions [19], but some authours have also addressed the role of disorder in surpressing superconductivity [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Disorder, however, is usually incorporated as random on-site energies in Hubbard- like models that can lead to Anderson localization phenomena [23–25], thus it is important to extend this effects to include also the possiblity of severe structural disorder within finite regions of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' One of the most significant results from the study of disor- der effects in superconductivity is the well known Anderson’s theorem, which states that both the transition temperature, Tc, and the isotropic gap, ∆0, of s−wave superconductors are in- sensitive to the presence of weak disorder at the mean-field level of BCS-like models [26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' One of the requirements of the theorem is that the density of states remains unchanged when compared to the pure metal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' As such, if the in- fluence of disorder is strong enough to deplete the density of states, the theorem no longer holds, and disorder dramat- ically affects superconductivity [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In this case of strong disorder and high concentration of impurity centers, the su- perconducting correlation length is comparable to the disor- der correlation length, and the mean-field equations can lead to self-organized granularity where fluctuations of the local order parameter are present [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' This is likely to be the case for overdoped cuprate superconductors with high concentra- tion of interstitial oxygens that can lead to the formation of nanosized oxygen puddles, regions with agglomeration of Oi, that support superconductivity [17, 18, 31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The case of unconventional high temperature d−wave su- perconductivity in hole-doped cuprates is of experimental and theoretical relevance since its discovery [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Apart from sev- eral different physical characteristics, one of the main differ- ences between these materials and the conventional BCS su- perconductors is that the superconducting gap amplitude is not homogenous when the system undergoes the supercon- ducting transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' This is evidenced by scanning tunneling microscopy (STM) spectra in Bi2Sr2CaCu2O8+δ at different arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='02892v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='supr-con] 7 Jan 2023 2 doping levels, where the inhomogenous gap in the supercon- ducting regime is revealed to be represented by a variety of gap sizes and amplitues occuring in all samples as the con- centration of dopants is varied [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Most remarkably, there is a clear correlation between the position of Oi agglomer- ates and the amplitudes of the gaps, since regions with larger groups of dopants are observed to correspond to regions of larger gap amplitudes [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Paralelly, Oi dopants have been observed to self-organize into nanosize regions, or puddles, as mentioned above, via µXRS in HgBa2CuO4+δ [35], as well as in other cuprate compounds [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Remarkably, it has been observed that spatial variations in the self-organization of the nanosized Oi-rich puddles have a direct effect on su- perconductivity, through variations in the critical temperature [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Therefore, it is of paramount importance a deeper un- derstanding, from a theoretical perspective, of the role of the oxygen puddles in the physics of hole-doped cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In this work, we aim to investigate the effects of how struc- tural disorder caused by the agglomeration of Oi in puddles is responsible for the appearence of finite (nonzero) center-of- mass (CM) momentum Cooper pairs in overdoped cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' This will be done by making use of a previously reported model proposed to describe how superconductivity rises in cuprates, on the underdoped side of the phase diagram, in terms of the phase synchronization of networks of nanosized superconducting puddles, rich in interstitial dopant oxygens [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Following, we extend the puddle model to derive analyt- ical expressions showing how the superconducting gap, and thus the critical temperature, are affected by the presence of Cooper pairs with finite CM momentum and structural dis- order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Finally, we show numerically that both Tc and ∆0 decrease with increasing disorder, thus pointing to a simple physical mechanism to explain the closing of the supercon- ducting dome-shaped area of the phase diagram, as being due to the reduction of the available phase space for Cooper pair- ing due to the development of a nonzero, finite CM momen- tum Cooper pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' This paper is divided as following: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' II we explain the puddle model, which is the base for the calculations pre- sented in this work, and derive the effective interaction be- tween electrons and the network of puddles, giving rise to a finite CM momentum pair state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Following, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' III we de- scribe the effects of structural disorder that the agglomeration of Oi within each puddle causes to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' IV we derive the the self-consistent equation for the amplitude of the superconducting gap in terms of disorder and finite CM momentum Cooper pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Then Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' V is devoted to the nu- merical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Finally, we discuss the implications of our results within the framework of networks of nano-sized puddles and summarize our findings in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' INHOMOGENEOUS OXYGEN PUDDLES The oxygen rich nanopuddles have different elastic proper- ties than their surroundings, and can therefore be considered as elastic insertions in an otherwise homogeneous medium, with its own vibrational mode, forming a network of super- Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Pictorical view of the disordered background introduced by the network of puddles (blue) in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The network consists of puddles of different sizes, defined by the radius of each insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Electrons (black and red) scatter in each puddle and, in the super- conducting state, percolate within the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' conducting nanoscale puddles, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 1, which is the starting point for the model that captured how supercon- ductivity may arise in cuprates due to the phase synchroniza- tion of each nanopuddle [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In terms of the Kuramoto model for sychronization of phase oscillators [39, 40], each nano- sized puddle is assigned to a phase, that in the underdoped regime evolves independently of the others, giving rise to lo- calized patches of superconductivity, as revaled by STM and other techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' With increased concentration of Oi through doping, the superfluid density is responsible for the enhance- ment of the interactions between the puddles and, in terms of the Kuramoto model, to lock their phases in a synchronous way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Following a BCS-like procedure, the order parameter for synchronization is connected to the amplitude of the bulk su- perconductor gap, that is non zero only after the locking of the global phase in the synchronized phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The synchronization and the large frequency of the global network of puddles is also responsible for large values of Tc in the optimally doped cuprates within the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Inspired by these experimental and theoretical findings, we introduce a model Hamiltonian that captures the inter- action between electrons and localized vibrations that arise from the agglomeration of interstitial oxygens in one pud- dle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' This interaction must be local, since each electron will only interact with the quantized vibration whenever it is in the region defined by the puddle (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The minimal model that captures this physical situation can be divided in H = Hel + Hp + Hel−p, with Hel = � k,σ ξkc† k,σck,σ + � k,k′ Tk,k′c† k′,σck,σ, where the first term represents a band of electrons with dis- persion ξk measured relative to the chemical potential, with creation c† k,σ and annihilation ck,σ fermionic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The second term represents the scattering of electrons in each in- 3 homogeneity described by the puddles, with strenght con- troled by the spin-preserving momentum transfer disorder ma- trix Tk,k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The oxygen puddles are described by local phonon modes Hp = � q ℏωqa† qaq, with frequencies ωq and the creation (a† q) and annihilation (aq) bosonic operators, responsible for the description of the localized vibration of each puddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Finally, the interaction term can be described as Hel−p = � r,R,σ g(r − R)c† r,σcr,σ � a† R + aR � , where r and R are the electron and puddle locations, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The puddle is a finite size region in space, thus R de- fines the center of this region that can be modeled as a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The interaction strenght g(r − R) is only relevant whenever the electron is in the region around the puddle, which can be modeled using a Gogny-type short range interaction that is dependent on the radius of the oxygen agglomeration re- gion [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' After performing the transformation to momentum space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' the interaction term is written as Hel−p = � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='q M(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' k − k′)c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='σck′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='σ � a† −q + aq � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (1) where M(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' k − k′) = � R g(k − k′) exp [i(q − [k − k′]) · R] is associated with the fact that the puddles are not present in all sites,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' rather they are inhomogeneously distributed around the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' thus the summation has to be retained only to these regions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' which is relevant for the case of Bi2Sr2CaCu2O8+δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' since locations of dopant oxygens are observed to be consis- tent with the position inferred from local strain analysis of the incommensurate structure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' as imaged by scanning transmis- sion electron microscopy (STEM) [42],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' which means that the crucial oxygen dopants are periodically distributed in corre- lation with local strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' However, not all strained regions are occupied with dopant oxygen atoms, that is the distribution of Oi is inhomogeneous, which justifies our approximation and is consistent with STM measurements [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In the limits of a clean or a totally doped system, this term can be treated ex- actly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The factor g(k − k′) is the Fourier transform of the in- teracting potential between the electrons and the puddles and controls the momentum transfer between the incoming and scattered electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' One can see from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (1) that the presence of a finite den- sity of puddles spread around the systems give rise to a off- diagonal term associated with the momentum transfer k − k′ that comes from the interacting potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In the limit that the summation over M(q, k − k′) can be made exactly, one re- covers the usual definition of an electron-phonon interaction, where the momentum transfer is the momentum of the local phononic mode q, as in the Frohlich [44] and Holstein [45] models, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In order to explore the effects of this kind of interaction in the form of pairing, we introduce an unitary transformation H′ = e−SHeS, with an ansatz for the transformation matrix S = � k,k′,σ,q,Q M(q, k − k′)c† k,σck′,σ � xa† −q + yaq � , where x and y are factors determined a posteriori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' After the transformation (see Appendix A for details),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' we end with an effective interaction written as Heff = � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='k′ � p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='p′ V (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' k′)f(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' p′)c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='↑c† p−k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='↓cp′−k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='↓ck′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='↑ (2) with V (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' k′) = D(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' k′)|g(k − k′)|2 being the potential arising from the interaction between electrons and puddles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' D(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' k′) the phononic propagator associated with the lo- cal phonon modes produced by the vibrating puddles and f(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' p′) = � R e−i(p−p′)·R the phase factor controlling mo- mentum transfer between the interacting electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In the regime where the phononic propagator is negative, given that ξk ≈ ξk′, we have an effective attractive interaction between the electrons mediated by the nanopuddles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Remarkably, this interaction leads to the formation of finite center-of-mass mo- mentum Cooper pairs represented by p and p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Therefore, from the perspective of inhomogeneously distributed puddles bringing disorder to an otherwise clean medium, a bound state between two electrons can be formed with a finite center-of- mass momentum that is associated with the strenght of the interaction between the electrons forming the pair and the ag- glomeration of interstitial dopant oxygens in one nanopuddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' It is important to notice that the states arising from the ef- fective Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (2) are different from other pro- posed pair states with finite CM momentum, as for example the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, where fi- nite center-of-mass momentum Cooper pairs can be stabilized under a finite magnetic field via the Zeeman coupling [46, 47], and the recently proposed current driven FFLO state [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Moreover, it has been shown that, even without the presence of a magnetic field or other external potentials, a finite CM momentum Cooper pair can be stable in a superconducting ground state as pointed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' [7], but the authors do not ex- plore the effects that can give rise to this kind of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Here we start from the fact that nanosized puddles are formed via doping and the responsible for the CM momentum of the pairs is disorder induced by the puddles in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Eventhough we are not considering any specific form for the interaction potential g(k − k′), it is important to com- ment that the only requirement is that it must be a finite size potential in real space, which means that is not a point-like disorder center that is scattering the electrons in the interac- tion term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (1), rather is a region in space defined by the agglomeration of oxygen interstitials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In this case, we can point to potentials like the Woods-Saxon potential [49] that is 4 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Top: Structure factor for disordered media from Hose- mann’s paracrystalline theory [58], given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (6) in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The pristine case corresponds to the ℓ → ∞ limit, where the structure factor is given by delta-peaks at reciprocal lattice vectors and mo- mentum is conserved (here ℓ is a measure of disorder and for this reason should be inversely related to the residual resistivity shift due to structural disorder, ℓ ∝ 1/δρ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Bottom left:: − Bragg diffraction pattern for a structually disordered medium, showing Bragg peaks at the central region and Bragg rings at the outter region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Bottom right − plot of the structure factor as a function of momentum transfer, ∆Q, showing well defined Bragg peaks, for small momentum trans- fer, at the reciprocal lattice vectors, G, while the Bragg peaks be- come ever broader, at larger momentum transfer, eventually merging into rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' used to describe the forces applied on protons and neutrons in the atomic nucleous or the Gogny-type interactions [50–52], which is another kind of nucleon-nucleon potential that has also found applications in astrophysics [53], as possible can- didates to describe the electron-puddle interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' However, a precise and detailed description of such potential would re- quired more knowledege about the formation of the nanosized puddles and its effects on the crystal structure of the host ma- terial, which would affect the electronic degrees of freedom [54], but this is outside the scope of the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' STRUCTURAL DISORDER Before we proceed to the characterization of the supercon- ducting state that arises from the effective Hamiltonian de- rived in the last section, it is important to briefly discuss which kind of disorder is giving the Cooper pairs a finite CM mo- mentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In order to do that, we introduce concepts arising from the study of structural disorder, which is the kind of per- turbation that the agglomeration of Oi causes in the crystalline structure of different cuprate systems, as for example by tilt- ing the CuO6 octahedra in La2CuO4+δ [55] and by altering the distance between the apical oxygen and the planar copper atom in Bi2Sr2CaCu2O8+δ [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Translational invariance is one of the most fundamental properties of pristine crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The concept of a Brillouin zone, that repeats itself by translations of reciprocal lattice vectors, allows us to organize electrons in energy bands, ϵn(k), labeled by a band index, n, and function of a quasi- momentum (wave-vector) quantum number, k, in terms of which periodic Bloch wave-functions, un,k(r), are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' A perfect crystal is characterized by very intense and sharp peaks in the Fraunhofer diffraction pattern of Bragg scatter- ing experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The existence of such sharp peaks follows directly from Heisenberg’s uncertainty principle and their lo- cation is determined by the crystalline-lattice structure factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Simply put, an extended Bloch wave with well defined mo- mentum state, k, that interacts with ions located at arbitrary positions, ri, of the crystal (infinite uncertainty ∆r → ∞), scatters into another extended Bloch wave with momentum state, k′, with zero uncertainty, ∆k → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The entire process carries a phase φ(k′ − k) = 1 √ N � ri friei(k′−k)·ri, (3) where N is the number of lattice sites in the crystal and fri is an atomic form factor that gives the probability that an atom is located at a certain crystallographic position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The scattered intensity is proportional to |φ(k′−k)|2 and is thus determined by the lattice structure factor, S(k′ − k) = 1 N � ri,rj frifrjei(k′−k)·(ri−rj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (4) For a pristine crystal all atoms are at their ideal locations, fri = frj = 1 and thus S(k′ − k) = � g δk′−k,g, where g is a reciprocal lattice vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The Fraunhoffer diffraction pattern in this case thus corresponds to δ−like peaks as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 2 and the kinematic constraint of quasi-momentum conserva- tion, k′ = k + g, (5) forms the basis for Bloch’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In the opposite limit of a random atom gas, however, an extended Bloch wave with well defined momentum state, k, that interacts with ions located at a particular, well defined position, ri, of the crystal (zero un- certainty ∆r → 0), scatters into another extended Bloch wave with momentum state, k′, with infinite uncertainty, ∆k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In this case, frifrj = δri,rj, and S(q) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' There are no kinematic constraints whatsoever relating k and k′ to g and the Fraunhoffer diffraction pattern in this case corresponds to an isotropic disc of even intensity, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Interpolating between the pristine and random limits de- scribed above by increasing disorder is pivotal to the descrip- tion of inherently inhomogeneous systems, such as the one of ramdom oxygen puddles described in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' If disorder is of the first type, namely weak disorder, all atoms deviate only slightly from their ideal positions in the crys- tal, independently of the deviations of their neighbors [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' This is the case of pointlike defects, thermal vibrations or micro-mechanical strains, and this kind of disorder preserves long range crystalline order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In this case the widths of the peaks in the Fraunhoffer diffraction pattern are not affected,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=" Bragg diffraction patterns lα1/po measures the amount of distortions Smax(g) S℃(k'- k) = k'-k-q," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content="0 1＋l2(k'kqg) g0 20 Structure factor Sq(k'-k) 15 10 5 0 0 2 4 Reciprocal vector AQ=k-k-q (units of G) uncertainty in reciprocal lattice breakdown momentum conservation5 and only their intensity is slightly reduced since for uncor- related Gaussian disorder," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' frifrj = D2 < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' where D2 is the Debye-Waller factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The structure factor is given by S(k′ − k) = D2 � g δk′−k,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' If disorder of the second type, namely strong disorder, however, the atoms deviate signifi- cantly from their ideal positions in the crystal, and deviations amogst neighboring atoms are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' This is the case of extended defects, amorphous regions, molten materials, etc, and this type of disorder causes the loss of long range crys- talline order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In these paracrystalline structures, not only the intensity of the diffraction peaks will decrease but, most im- portantly, their widths will suffer from a nonlinear increase of their integral breadth, δg, for successive orders of Bragg re- flections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The complete paracrystalline theory was proposed by Hosemann [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Hosemann included fluctuations of vari- ance σ that introduce correlations between pairs of atoms, ⟨frifrj⟩, that decrease with separation ultimately causing the peaks in the structure factor of the material to broaden the larger the reciprocal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The result is a structure factor composed by a sum of Lorentzians [59] Sq(k′ − k) = � g Smax(g) 1 + ℓ2 hkl(q − k′ + k − g)2 , (6) of amplitudes Smax(g) = 4/σ2g2 and breadths for Bragg reflections, |δg| ≡ 1/ℓhkl = σ2π2(h2 + k2 + l2)/a0, given in terms of the original lattice parameter a0 and the momentum transfer, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Hosemann’s paracrystalline theory allows us then to interpolate continuously between pristine and random cases through the fluctuation parameter σ: for σ → 0 we have ℓhkl → ∞, ∀h, k, l and we obtain Sq(k′ − k) = � g δq,k′−k+g, enforcing the kinematic constraint of momentum conservation, q = k′ − k + g, typical of pristine crystals [59];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' for σ → ∞ we have ℓhkl → 0, ∀h, k, l and we end up with Sq(k′ − k) = Smax(0) → 1, isotropic, for arbitrary q, k, k′ and determined solely by the g = 0 contribution, typical of infinite, aperiodic systems [59];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' for 0 ≤ σ ≤ ∞ we have ∞ ≥ ℓhkl ≥ 0 and the structure factor, Sq(k′ −k), will be composed by sharp Bragg peaks at small g (large ℓhkl) and isotropic discs for larger g (small ℓhkl), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 2, relax- ing the kinematic constraint of momentum conserva- tion, q ̸≈ k′ − k + g, typical of a paracrystal, liquids, strongly disordered or amorphous systems [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' DISORDER AND GAP FLUCTUATIONS We now address how the superconducting state of the effec- tive interaction derived in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' II is affected by the structural disorder effects introduced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' We start from the effective Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (2) and, within a mean- field decoupling of the quartic term, write the equation for the superconducting gap as ∆k = − � k′,p′ Vk,k′f0,p′ ⟨cp′−k′↓ck′↑⟩ , (7) where we set p = 0, since we want to describe ampli- tude fluctuations for the superconducting gap in the Cooper channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' For the superconducting state formed by singlet pairs with finite CM momentum, the system can be repre- sented by the spin-independent imaginary time Green’s func- tion G(k, k′, τ) = − � Tτck,σ(τ)c† k′,σ(0) � and the anomolous pair propagators F(k, k′, τ) = ⟨Tτck,σ(τ)ck′σ′(0)⟩ and F∗(k, k′, τ) = � Tτc† k,σ(τ)c† k′,σ′(0) � for σ ̸= σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Within Nambu’s formalism, we can write the decoupled effective Hamiltonian from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (2) and the electronic components from Hel in matrix form and derive in first order perturbation theory the electronic Green’s function for an inhomogeneous system with disorder as G (k, k′, iωn) = G0 (k, k′, iωn) + � p,p′ G0 (k, p, iωn) Tp,p′σ3G (p′, k′, iωn) , where G0 (k, k′, iωn) is the matrix form of the translation- ally invariant electronic Green’s function in frequency space, iωn are the fermionic Matsubara frequencies and σ3 is a Pauli matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The diagonal elements of this matrix are defined by the bare Green’s function in the superconducting state, G0(k, iωn), and its off-diagonal terms are represented by the anomalous propagators F0(k, iωn) which are written as G0 (k, iωn) = − (iωn + ξk) ω2n + ξ2 k + |∆k|2 , F0 (k, iωn) = ∆k ω2n + ξ2 k + |∆k|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In order to proceed, we shall take a couple of approximations: first we consider the case of overdoped cuprates, which puts the system in a high concentration of disorder, thus Tp,p′ = T f(p, p′), where disorder influences the momentum trans- fer controled by the phase factor f(p, p′) with strenght T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Second we assume that for a translationally invariant system the normal and anomalous Green’s functions can be rewrit- ten as G0(k, k′, iωn) = G0(k, iωn)δk,k′ and F0(k, k′, iωn) = F0(k, iωn)δ−k,k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Following these couple of approximations, the first order pertubation theory expansion of the interacting Green’s function is simplified G (k, k′, iωn) = G0 (k, iωn) δk,k′ + T fk,k′G0 (k, iωn) σ3G0 (k′, iωn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (8) From the gap equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (7) and from the definition of the anomalous propagator, we write 6 ∆k = − � k′,p′ Vk,k′f0,p′ ⟨cp′−k′↓ck′↑⟩ = − � k′,p′ Vk,k′f0,p′ � 1 β � ωn F (p′ − k′, k′, iωn) � ,(9) with β = 1/T being the inverse temperature (in units of kB = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' By using the matrix form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (8), we get the form of the interacting anomalous propagator, where it is worth not- ing that the normal and anomalous propagators mix in the impurity scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Despite the anomalous Green’s func- tion being invariant for time reversal, the normal one is not, and since disorder produces the transformation F0(k, iωn) ↔ G0(k, iωn) we clearly see this is a mechanism that breaks time reversal invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' As a consequence, this mechanism breaks the Cooper pair that leaks into the normal metal surrounding the puddles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In order to understand the effects of disorder and finite CM momentum in the gap equation, we substitute the form of the anomalous propagator given by the matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (8) inside Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (9) to write the gap equation as ∆k = ∆BCS k + δ∆k, where ∆BCS k = − � k′ Vk,k′∆k′ 2Ek′ tanh �βEk′ 2 � , (10) is the BCS limit for the gap equation, arising from the first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (8), with the bare anomolous propagators and Ek = � ξ2 k + ∆2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Then δ∆k = T � k′,p′ Vk,k′f0,p′fp′,0 1 β � ωn {F0G0 + G0F0}(11) is the correction to the superconductor gap due to effects of disorder in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The factor [f0,p′fp′,0] can be treated within a mean over disorder in order to calculate the inter- ference factor as [f0,p′fp′,0] = |f0,p′|2 → S(p′), where S(0, p′) is the static structure factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Thus, the correction to the gap equation can be written in terms of the structure factor and we see that fluctuations associated with small CM momentum p′ → 0 are absent, since the structure factor S(p′) → 0 and the gap equation is dominated by the BCS contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' On the other hand, fluctuations associated with a finite center-of-mass momentum dominate over the BCS con- tribution when p′ ≫ 0 and S(p′) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In a general manner, the structure factor can be written as a sum of Lorentzians with peaks in wave vectors of the reciprocal lattice, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' III and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Finally, we proceed by taking the Matsubara summations over the set of mixed Green’s functions as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (11) to arrive at the correction in terms of the disorder strenght T and the finite CM momentum of the Cooper pairs p′ as δ∆k = T � k′,p′ Vk,k′S (p′) 1 2 � ∆k′,p′ Ek′−p′ ξk′ Ek′ + ∆k′ Ek′ ξk′−p′ Ek′−p′ � × � � � Ek′−p′ tanh � βEk′ 2 � − Ek′ tanh � βEk′−p′ 2 � E2 k′−p′ − E2 k′ � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (12) It is importance to notice the dependence of the correction on the structure factor S(p′) controlling momentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In the limit of small amount of disorder, the so called first- type disorder [57], as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' III, pointlike deffects does not affect the BCS gap, in accordance with Anderson’s Theorem, as we shall see in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' On the other hand, in the limit of high concentration of puddles, the system is in the limit of second-type disorder, associated with strain- induced lattice deformations, and both the amplitude of the superconducting gap and the critical temperature are affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In order to proceed to the numerical analsysis, we perform an approximation for the structure factor based on the limits of disorder discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' For the first-type disorder, we choose S(p′) = δ0,p′, since no momentum transfer will be associated with pairs with finite CM momentum in the dilute limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' On the other hand, for the second-type disorder, we write S(p′) = 1, assuming a system with high concentration of puddles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' These two limits for the disorder of the 1st and 2nd types can be understood as a hard cutoff for the CM mo- mentum distribution within the structure factor and are made to simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (12) to the following numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' NUMERICAL ANALYSIS In order to fully understand the effects of disorder and CM momentum of the Cooper pairs in the superconduct- ing gap amplitude we perfom a numerical integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' We use the decomposition Vk,k′ = −V0η(k)η(k′) and ∆k = ∆0η(k), where η(k) = cos kx − cos ky is a d−wave form factor, which gives the amplitude fluctuations of the or- der parameter with the same symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' When stated for com- parison, we shall also use Vk,k′ = −V0 and ∆k = ∆0 when considering a s−wave symmetry for the interaction and the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' For the calculations in the square lattice, we consider a two-dimensional electronic dispersion with nearest- and next- nearest-neighbor hopping elements (t, t′) as ϵk = −2t (cos kx + cos ky) + 4t′ cos kx cos ky − µ, (13) where µ is the chemical potential that controls the electronic density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' This type of electronic dispersion is general for 2D transport in strongly correlated systems and is suitable for the description of the conduction band associated with the CuO2 planes of high-Tc cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In the following calculations, all parameters are defined in units of 4t and we set µ/4t = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='45, away from the half- filled case µ/4t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 3), since the mean-field theory 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Fermi surface structure used in calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Left: The 3D plot of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (13) in the first Brillouin zone in yellow and the chemical potential cut defining the Fermi level in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Right: The Fermi level defined by the cut at µ/4t = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The vectors k, fixed in the direction (0, π), and k′, varying across the Fermi surface, are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' yields incorrect results for a two-dimensional lattice near half- filling [60] and we avoid particle-hole symmetry [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' For this reason, we can take t′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' We also set V0/4t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='0, in the limit where the mean-field theory is still valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' For the summations over p′, we define p′ = k − k′, where k, k′ are the momenta of the two paired electrons, which we set |k| = |k′| = kF as two momenta in the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The CM momenta are then defined by fixing k in the direction of the point (0, π) and by varying k′ across the Fermi surface, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' We start by analyzing the zero temperature limit T = 0 of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (12), where the hyperbolic tangents can be simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 4 we show how the gap amplitude ∆0 is affect by disor- der T in the limit of disorder of the 1st type, S(p′) = δ0,p′, or weak concentration of puddles, and strong concentration, S(p′) = 1, in the limit of disorder of the 2nd kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The gap amplitude is insensitive to disorder in the dilute limit for s−wave pairing, thus ∆0 = ∆BCS 0 and the BCS limit is re- covered, in accordance with Anderson’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' However, in the opposite limit, the disorder strongly affects the ampli- tude of the gap for d−wave pairing, introducing fluctuations and decreasing its absolute value in about 50% in the strong disorder limit, when compared to the clean case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' It is worth noting that the reduction is not linear as the strenght of disorder approaches the values of the fixed pairing potential, T → V0, where the pertubation theory still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' This can be traced back to the fact that the gap equation is a self-consistent equation for the aboslute value of ∆0, even after the approximations considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Thus we see that even in the zero temperature limit, disorder tends to destroy super- conductivity in a system with high concentration of oxygen interstititals, as in the overdoped cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' We also investigate the effects of specific finite CM momen- tum on the amplitude of the gap when T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' We choose a set of momenta {p} and substitute in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' (12) the corresponding structure factor, namely S(ps) = δp′,ps, where ps are the mo- menta in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' All ps are multiples of kF of each direction considered, namely (0, π) and (π, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 5 we display the Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' T = 0 limit for the amplitude fluctuatios of the supercon- ducting gap as a function of disorder strenght compared to the clean system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Dilute limit (red), for disorder of the 1st kind and s−wave symmetry, and high concentration of puddles for disorder of the 2nd kind and d−wave symmetry (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The black dashed line is a guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Gap values are given in terms of ∆0 in the absence of disorder T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' evolution of the amplitude of the superconducting order pa- rameter ∆0 as a function of the CM momentum of the Cooper pairs p, for fixed disorder strenght T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The supercon- ducting order parameter is modulated, with period determined by the distance between adjacents Fermi surfaces in each di- rection, being 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='75|kF| for (0, π) and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='75|kF| for (π, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Re- markably, this is in direct contact with the diffraction pattern displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' However, since we are considering a hard cutoff for the structure factor in terms of delta functions, the amplitude of the gap modulation is not altered by the dis- tance from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' We expect that by including a more realistic model for the structure factor, the amplitudes of the modulations will decay with p, with its effect stronger in the (π, π) direction, since larger reciprocal lattice vectors G im- ply a broader structure factor, thus diminishing the amplitude of the superconducting gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Altogether, the interplay between disorder and finite center-of-mass momentum Cooper pairs is able to strongly affect the superconducting order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Now we turn to the finite temperature case T ̸= 0 for the d−wave symmetric order parameter to understand how disor- der and CM momenta for the Cooper pairs affects the critical temperature Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 6 we show the evolution of the su- perconducting gap with temperature, for different values of the disorder strenght T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' It is clear that with increasing dis- order, not only ∆0(0) decreases, as pointed in the zero tem- perature limit, but we also evidence a decrease in the criti- cal temperature Tc, defined as the value of temperature that ∆0(T, T ) → 0, with disorder, as shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' This means that pair breaking is induced by the scattering of the finite CM momentum Cooper pairs with the nanosized oxy- gen puddles of the system and by increasing disorder, Tc is significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' This pair breaking effect is due to the fact that the phase space required to pair formation is reduced when p increases in absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In the small scattering momentum transfer 2 k 03 斤-2 0 1 0 ky 元 元 0 2 Kx 元 2 2 2 2 0 2 kx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='9 △o(T)/△o(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='7 2nd d wave 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='6 lst s wave 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='8 T8 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Left: Extended Brillouin zones in the upper positive part of momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The arrows indicate the distance between the cen- ters of each Fermi surface in terms of the Fermi vector |kF | of each direction considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Right: The amplitude of the superconducting order parameter as a function of different CM momentum vectors |p|, in the directions (0, π) and (π, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' ∆0(p) is given in units of the gap at p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' sector, p < |kF|, the gap is almost unnafacted by the presence of disorder when comprared to the value when p = 0, since the shape of the Fermi surface intersection of the two paired electrons suffers little change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' However, when p approaches the maximum absolute value of 2|kF| within the first Brillouin zone, the phase space for pair formation is greatly reduced and disorder induces pair breaking, captured by the reduction of the superconducting order paremeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The modulation occurs for p > 2|kF|, since electrons from different Brillouin zones participate in the scattering and pairing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Therefore, these results point to the combined effect of finite center-of- mass momentum pairs being scattered by structural disorder induced by the network of oxygen puddles as a mechanism for the reduction of the superconducting gap and the critical temperature in the overdoped regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' CONCLUSION AND DISCUSSION In this work we presented an extension of the proposed model for the formation of networks of puddles and its ef- fects on the superconductivity in oxygen-doped cuprates [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' We show that the presence of puddles, in the overdoped side of the phase diagram, introduces strong disorder in the sys- tem that induces the formation of finite center-of-mass mo- mentum Cooper pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' We derive an analytical expression for the amplitude fluctuations in the superconducting gap induced by the puddles, within a mean-field BCS-like approach, in terms of the disorder strenght T and the finite CM momenta p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' We numerically solve this expression to show that even in the zero temperature limit the gap is strongly affected by disorder-induced CM Cooper pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In the limit of strong dis- order, the gap tends to close and, in the finite temperature case, Tc tracks the reduction of the superconducting gap, also being strongly affected by disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' It is important to emphasize that we do not account the effect of longer-range Coulomb re- pulsion, restricting the application of our results to screened systems [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Temperature dependence of the superconducting order pa- rameter for different values of disorder strenght (colored bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Gap alues are given in terms of the clean case T = 0 and temperature in terms of T 0 c also of the clean case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Inset: The critical temperature dependence normalized to the clean value as a function of disorder strenght.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The black dashed line is a guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The experimental observations of structural scale invari- ance of dopants detected by scanning micro-x-ray diffraction [36],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' the promotion of critical temperature [37],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' the agglom- eration of interstitial oxygens in regions of strong local strain in the crystal structure of cuprate superconductors [42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 43] and the proposed theoretical reports regarding the presence of networks of nanoscale superconducting islands in high- temperature superconductors [62–65] are in close connection with the results reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Eventhough we are showing that the superconducting state is depleted in the presence of strong disorder in the overdoped regime, it is clear from the above mentioned surveys that the importance of these net- works and its interplay with electronic degrees of freedom pass across the whole phase diagram of hole-doped cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' [38], the present authors show how the complex networks formed by the oxygen puddles can transionate to a synchronized phase, controlled by the superfluid density, in a way that the concentration of dopant atoms controls the emer- gence of local superconductivity in the underdoped regime and how the systems evolves to a bulk superconductor as the concentration of dopants, thus puddles, increases as the sys- tems approaches the optimally doped regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' It is important to emphasize that within this framework, the state studied in this work is described by the bulk superconductor state in the synchronized phase of the network formed by the oxygen pud- dles (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 1), in the sense that we require the network of puddles to be fully synchronized in order to the band of elec- trons to interact with the global mode of vibration of the syn- chronized network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Our approach is based on a mean-field approximation for the complex network, therefore we point to the importance of describing different topologies for the orga- nization of the puddles and how this can affect not only the transition to the superconducting state [66], but also its pos- sible interplay with the superconducting fluctuations of pre- formed Cooper pairs observed in the pseudogap phase above Tc [67], in terms of local superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='00 (0, πt) (π, ) 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='98 [ KF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='96 4 5 K 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='75/kFl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='94 3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='92 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='90 2 0 2 4 6 8 01234567891011 kx Ip/ / / kFl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='0 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content='09 Appendix A: Unitary transformation In this Appendix section, we show the derivation of the effective Hamiltonian containing the pairing interaction be- tween two electrons forming a Cooper pair with finite center- of-mass momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The starting point is the full Hamiltonian written in momentum space H = Hel+Hp+Hel−p, which is the summation over the contributions of the electrons, puddles and electron-puddle interaction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Introducing an unitary transformation of the form H′ = e−SHeS, where S is the transformation matrix introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' II, we can ex- pand the exponentials up to second order in powers of S to write the transformed Hamiltonian as H′ = H + [H, S] + 1 2[[H, S], S], (A1) and by treating Hel−p as a perturbation, we can divide the full Hamiltonian as H = H0 + Hel−p, where H0 contains the kinetic terms of electrons and puddles, to write H′ = H0 + Hel−p + [H0, S] + [Hel−p, S] + 1 2[[H0, S], S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Since the goal is to eliminate the interaction, the defining equation for the transformation matrix comes from the elim- ination of the first-order term [H0, S] + Hel−p = 0, from which we can extract the factors x and y for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In this way, the transformed Hamiltonian can be written in terms of an ef- fective interaction that comes from recombining the terms in the commutators H′ = H0 + 1 2[Hel−p, S], (A2) thus the problem is reduced to an effective system described by H = H0 + Heff, where Heff = 1 2 [Hel−p, S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' By performing the calculation over the commutator [H0, S], the choice of x and y that eliminate the first-order term is given by xk,k′,q = 1 ξk′ − ξk − ωq , yk,k′,q = 1 ξk′ − ξk + ωq , and the transformation matrix S is fully defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Then we proceed to the calculation of the effective Hamiltonian that comes from the commutator of the now defined matrix S and the electron-puddle interaction, which gives a combination of M(q, Q)M(−q, Q′), where Q = k − k′ and Q′ = k′′ − k′′′ are two auxiliar variables that accomodate the variety of in- dices arising from the commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Recalling the definition of the factor M given in the main text, we see that M(q, Q)M(−q, Q′) = � R,R′ g(Q)g(Q′) × ei(R−R′)·qe−i(Q·R+Q′·R′), which can be simplified by taking R = R′ since each R de- scribes the position of a nanosized puddle and we are assum- ing the dilute limit of oxygen puddles, as discussed in the main text, in accordance with STEM and STM measurements [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In this way, the effective Hamiltonian is written as Heff = � k′,k′′′,q,Q,Q′ V (q, Q, Q′)M(q, Q)M(−q, Q′) × c† k′′′+Q′c† k′+Qck′ck′′′, (A3) with V (q, Q, Q′) = ωq/[(ξk′′′ − ξk′′′+Q′)2 − ω2 q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' Proceed- ing with the calculation, we note that within BCS theory, the effective Hamiltonian describes the interaction between elec- trons with opposite momenta k′ = −k′′′, with zero CM mo- mentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' However, in our case, the auxiliar variables Q and Q′ introduces a momentum transfer connected with a finite CM momentum for the pairs, for each fermionic operator in the effective Hamiltonian that comes from the commutator [Hel−p, S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In this sense, we perform a change of variables introducing the finite CM momentum k′ + k′′′ = p, in a way that we can eliminate the dependence on the auxiliar variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' The new variables introduced are written as k = k′′′ + Q′ and −k + p′ = k′ + Q, where p and p′ are the CM momenta of the Cooper pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} +page_content=' In the limit where the interaction g(k, k′) is independent of the CM momenta, we can decouple the effec- tive interaction and end up with the effective Hamiltonian Heff = � k,k′ � p,p′ V (k, k′)f(p, p′)c† k,↑c† p−k,↓cp′−k′,↓ck′,↑, with V (k, k′) = ω0 (ξk′ − ξk)2 − ω2 0 |g(k − 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfEgPe/content/2301.02892v1.pdf'} diff --git a/ENE0T4oBgHgl3EQfywJf/content/tmp_files/2301.02663v1.pdf.txt b/ENE0T4oBgHgl3EQfywJf/content/tmp_files/2301.02663v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..84b2fe22aad3108290a5a938245cd0fd253f91eb --- /dev/null +++ b/ENE0T4oBgHgl3EQfywJf/content/tmp_files/2301.02663v1.pdf.txt @@ -0,0 +1,645 @@ +arXiv:2301.02663v1 [math.GR] 6 Jan 2023 +ON THE CHARACTERIZATION OF ALTERNATING GROUPS BY CODEGREES +MALLORY DOLORFINO, LUKE MARTIN, ZACHARY SLONIM, YUXUAN SUN, AND YONG YANG +Abstract. Let G be a finite group and Irr(G) the set of all irreducible complex characters of G. Define the +codegree of χ ∈ Irr(G) as cod(χ) := |G:ker(χ)| +χ(1) +and denote by cod(G) := {cod(χ) | χ ∈ Irr(G)} the codegree +set of G. Let An be an alternating group of degree n ≥ 5. In this paper, we show that An is determined up +to isomorphism by cod(An). +1. Introduction +Let G be a finite group and Irr(G) the set of all irreducible complex characters of G. For any χ ∈ Irr(G), +define the codegree of χ as cod(χ) := |G:ker(χ)| +χ(1) +. Then define the codegree set of G as cod(G) := {cod(χ) | +χ ∈ Irr(G)}. The concept of codegrees was originally considered in [8], where the codegree was defined as +cod(χ) := +|G| +χ(1), and it was later modified to its current definition by [22] so that cod(χ) is the same for +G and G/N when N ≤ ker(χ). Several properties of codegrees have been studied, such as the relationship +between the codegrees and element orders, codegrees of p-groups, and groups with few codegrees. +The codegree set of a group is closely related to the character degree set of a group, which is defined as +cd(G) := {χ(1) | χ ∈ Irr(G)}. The relationship between the character degree set and a group’s structure is +an active area of research – many properties of a group’s structure are largely determined by its character +degree set. In 1990, Bertram Huppert made the following conjecture about the relationship between a simple +group H and a finite group G having equal character degree sets. +Huppert’s Conjecture: Let H be a finite nonabelian simple group and G a finite group such that +cd(H) = cd(G). Then G ∼= H × A, where A is an abelian group. +Huppert’s conjecture has been verified for many cases such as the alternating groups, sporadic groups, +and simple groups of Lie type with low rank, but it has yet to be verified for simple groups of Lie type with +high rank. Recently, a similar conjecture related to codegrees has been posed. +Codegree Version of Huppert’s Conjecture: Let H be a finite nonabelian simple group and G a +finite group such that cod(H) = cod(G). Then G ∼= H. +This conjecture appears in the Kourovka Notebook of Unsolved Problems in Group Theory as question +20.79 [18]. It has been verified for PSL(2, q), PSL(3, 4), Alt7, J1, 2B2(22f+1) where f ≥ 1, M11, M12, M22, M23 +and PSL(3, 3) by [1, 3, 13]. The conjecture has also been verified for PSL(3, q) and PSU(3, q) in [19] and +2G2(q) in [14]. Recently, the authors verified the conjecture for all sporadic simple groups in [11]. +In this paper, we provide a general proof verifying this conjecture for all alternating groups of degree +greater than or equal to 5. The methods used may be generalized to simple groups of Lie type, giving +promising results for characterizing all simple groups by their codegree sets. +Theorem 1.1. Let An be an alternating group of degree n ≥ 5 and G a finite group. If cod(G) = cod(An), +then G ∼= An. +Throughout the paper, we follow the notation used in Isaacs’ book [16] and the ATLAS of Finite Groups +[9]. +2000 Mathematics Subject Classification. 20C15, 20D06. +1 + +2. Preliminary Results +We first introduce some lemmas which will be used later. +Lemma 2.1. [21, Lemma 4.2] Let S be a finite nonabelian simple group. Then there exists 1S ̸= χ ∈ Irr(S) +that extends to Aut(S). +Lemma 2.2. [17, Theorem 4.3.34] Let N be a minimal normal subgroup of G such that N = S1 × · · · × St +where Si ∼= S is a nonabelian simple group for each i = 1, . . . , t. If χ ∈ Irr(S) extends to Aut(S), then +χ × · · · × χ ∈ Irr(N) extends to G. +Lemma 2.3. [13, Remark 2.6] Let G be a finite group and S a finite nonabelian simple group with cod(G) = +cod(S). Then G is a perfect group. +Lemma 2.4. [15] Let G be a finite group and S a finite nonabelian simple group such that cod(S) ⊆ cod(G). +Then |S| divides |G|. +Lemma 2.5. Let G be a finite group with N ⊴ G. Then cod(G/N) ⊆ cod(G). +Proof. From [16, Lemma 2.22], we can define Irr(G/N) = {ˆχ(gN) = χ(g) | χ ∈ Irr(G) and N ⊆ ker(χ)}. +Take any ˆχ ∈ Irr(G/N). By definition, we know that ˆχ(1) = χ(1), so the denominators of cod(ˆχ) and cod(χ) +are equal. In addition, ker(ˆχ) ∼= ker(χ)/N, so |ker(χ)| = |N| · |ker(ˆχ)|. Thus |G/N : ker(ˆχ)| = +|G|/|N| +| ker(χ)|/|N| = +|G| +| ker(χ)|, so cod(ˆχ) = cod(χ) and therefore cod(G/N) ⊆ cod(G). +□ +Lemma 2.6. Let G be a finite group with normal subgroups N and M such that N ≤ M. Then, cod(G/M) ⊆ +cod(G/N). +Proof. By the Third Isomorphism Theorem, we know that G/M ∼= (G/N)/(M/N) is a quotient of G/N, +and by Lemma 2.5, cod(G/M) ⊆ cod(G/N). +□ +Lemma 2.7. Let S be a finite nonabelian simple group and G be a nontrivial finite group with cod(G) ⊆ +cod(S). Then, |S| < |G| · |Irr(G)|. +Proof. We know that for each irreducible character χ ∈ Irr(S), χ(1)2 < |S|. Because S is simple, if χ is non- +trivial, then ker(χ) = 1, so cod(χ) = +|S| +χ(1) > +� +|S|. Then, since cod(G) ⊆ cod(S), for each irreducible non- +trivial character ψ ∈ Irr(G), cod(ψ) > +� +|S|. Thus, |G:ker(ψ)| +ψ(1) +> +� +|S| which implies that +|G| +|ker(ψ)|√ +|S| > ψ(1). +So, ψ(1) < +|G| +√ +|S|. +Then � +ψ∈Irr(G) ψ(1)2 < | Irr(G)| |G|2 +|S| , and by character theorems, we’ll have |G| < +| Irr(G)| |G|2 +|S| . Thus |S| < |G| · |Irr(G)|. +□ +3. Main Results +We start with some lemmas which limit the simple groups whose codegree set can be contained in the +codegree set of an alternating group. +Lemma 3.1. Let H be an alternating group of degree m ̸= n, where m, n ≥ 5. Then cod(H) ̸⊆ cod(An). +Proof. Suppose cod(Am) ⊆ cod(An). Then, from Lemma 2.4, |Am| divides |An|, so m < n. Let ax denote +the minimal non-trivial codegree of Ax. +We show that an−1 < an so that cod(Am) ̸⊆ cod(An) follows +immediately. +We know that irreducible representations of the symmetric group Sn are in one-to-one correspondence +with the partitions of n. Let λ be a partition of n and Vλ be the corresponding irreducible representation +of Sn. We note that a partition of n can be visualized by a Young diagram and we let hλ(i, j) be the hook +length of the (i, j)th square of the Young diagram corresponding to λ, i.e. the number of cells (a, b) of λ +such that a = i and b ≥ j or b = j and a ≥ i. By the hook length formula, +n! +dim(Vλ) = � hλ(i, j) := Hλ. +Let Uλ be an irreducible constituent of the restriction of Vλ to An, ResSn +An Vλ. If λ is not self-conjugate +(λ ̸= λ′), then ResSn +An Vλ remains irreducible, so Uλ = ResSn +An Vλ. In this case, +n! +dim(Uλ) = Hλ. If λ is self- +conjugate, then the restriction of Vλ to An splits into two irreducible representations of the same dimension, +so dim(Uλ) = 1 +2dim(Vλ). In this case, +n! +dim(Uλ) = 2Hλ. +2 + +Now, an = min{ +n!/2 +dim(Uλ) | Uλ ∈ Irr(An)} = 1 +2 min({Hλ | λ ̸= λ′} ∪ {2Hλ | λ = λ′}). We want to show that +an−1 < an. First, assume that an = 1 +22Hλ for some λ = λ′. Then we can remove a square from λ to give a +non-self-conjugate partition µ of n − 1. Since Hµ < Hλ < 2Hλ and an−1 ≤ 1 +2Hµ, we know an−1 < an. +Now assume that an = 1 +2Hλ for some λ ̸= λ′. Then if n ≥ 3, we can remove a square from λ to obtain a +non-self-conjugate partition µ of n − 1. Since Hµ < Hλ and an−1 ≤ 1 +2Hµ, an−1 < an. Thus, if m < n, then +am < an, contradicting the assumption that cod(Am) ⊆ cod(An). +□ +Lemma 3.2. Let H be a sporadic simple group or the Tits group. Then if n ≥ 5, cod(H) ̸⊆ cod(An). +Proof. In search of a contradiction, let H be a sporadic simple group or the Tits group such that cod(H) ⊆ +cod(An). From Lemmas 2.7 and 2.4, we deduce a tight restriction on the order of H. Namely, |H| = |An|/k +where 1 ≤ k < |Irr(H)| is an integer. Now, for each sporadic (or Tits) group H, we can computationally +check (using Julia [6]) which alternating groups An satisfy both |H| divides |An| and |An| +|H| < |Irr(H)|. We +find only one possible exception: An = A10 and H = J2 where |A10| +|J2| = 3 < 21 = |Irr(J2)|. In this case, we +check that cod(J2) ̸⊆ cod(A10) using the ATLAS [9]. +□ +Lemma 3.3. Let H be a classical simple group of Lie type. Then cod(H) ̸⊆ cod(An) for all n ≥ 5. +Proof. There are 6 families of classical simple groups of Lie type. +These are PSL(m + 1, q), Ω(2m + +1, q), PSp(2m, q), O+(2m, q), PSU(m + 1, q), and O−(2m, q).l We prove the lemma in each case. Let k(G) +denote the number of conjugacy classes of G, we reproduce [12, Table 2] for reference. +Table 1. Class Numbers for Classical Groups +G +k(G) ≤ +Comments +SL(n, q) +2.5qn−1 +SU(n, q) +8.26qn−1 +Sp(2n, q) +10.8qn +q odd +Sp(2n, q) +15.2qn +q even +SO(2n + 1, q) +7.1qn +q odd +Ω(2n + 1, q) +7.3qn +q odd +SO±(2n, q) +7.5qn +q odd +Ω±(2n, q) +6.8qn +q odd +O±(2n, q) +9.5qn +q odd +SO±(2n, q) +14qn +q even +O±(2n, q) +15qn +q even +(1) Let H = PSL(m + 1, q) where q = pk and m ≥ 1. From the order formula found in [7], qm(m+1)/2 +divides |PSL(m + 1, q)|. From Legendre’s formula, we know that for any prime p, |n!|p ≤ p +n +p−1 . +If q = pk, then we have |n!|q ≤ q +n +k(p−1) and thus |An|q ≤ q +n +k(p−1) . By Lemma 2.4, |PSL(m + 1, q)| +divides |An|, so qm(m+1)/2 divides |An|. Thus m(m+1) +2 +≤ +n +k(p−1), giving n ≥ m(m+1)k(p−1) +2 +. Therefore, +|An| ≥ +���A m(m+1)k(p−1) +2 +���. +Now, we note that k(PSL(m + 1, q)) ≤ k(SL(m + 1, q)) since PSL(m + 1, q) is a quotient of +SL(m + 1, q). Then from Table 1, we have that |Irr(PSL(m + 1, q))| = k(PSL(m + 1, q)) ≤ k(SL(m + +1, q)) ≤ 2.5qm. +Applying Lemma 2.7 gives |An| < |PSL(m + 1, q)| · |Irr(PSL(m + 1, q))|. Hence +|A m(m+1)k(p−1) +2 +| < |PSL(m + 1, q)| · 2.5qm. Now we show that if we consider the left and right sides +as functions of m with constants p and k, then asymptotically, the value of |A m(m+1)k(p−1) +2 +| grows +faster than that of |PSL(m + 1, q)| · 2.5qm. We know that the left function behaves asymptotically +as (m2)!, and using the order formula for PSL(m + 1, q), we know that the right function behaves +asymptotically as qf(m), where f(m) is a polynomial with degree 2. Thus the left function grows +faster than the right function since x! >> cx for any constant c when x is large. Similarly, we can +prove this result considering the two sides as functions of p and k. +3 + +Then, we search for the maximum possible value of m which satisfies the inequality given the +smallest possible values of p and k, which are 2 and 1, respectively. We find that m ≤ 6 and, using +a similar process for p and k, that p ≤ 17 and k ≤ 63. Now, we have limited our search to a finite +number of groups which we can check in the same way as for the sporadic groups. From this, we +find a small list of exceptions, listed in Table 2: +Table 2. Exceptions satisfying |PSL(m + 1, q)| divides |An| and +|An| < |PSL(m + 1, q)| · 2.5qm +m +q +n +1 +4 +5 +1 +4 +6 +1 +8 +7 +1 +9 +6 +1 +9 +7 +1 +5 +5 +1 +5 +6 +1 +7 +7 +2 +4 +8 +2 +4 +9 +3 +2 +8 +3 +2 +9 +Now, all of these exceptions can be found in the ATLAS, and it is routine to check that none of +these groups satisfy cod(PSL(m + 1, q)) ⊆ cod(An) unless PSL(m + 1, q) ∼= An. Thus, if PSL(m + +1, q) ̸∼= An, then cod(PSL(m + 1, q) ̸⊆ cod(An). +(2) Let H = Ω(2m+1, q) where q = pk is odd and m ≥ 2. Note that when q = 2k is even, Ω(2m+1, q) ∼= +PSp(2m, q), which we deal with in the next case. From [7], qm2 divides |Ω(2m + 1, q)|. Thus, using +Table 1 similarly to above, |Am2k(p−1)| < |Ω(2m+ 1, q)|·7.3qm. As above, we computationally check +that we get a contradiction if m > 2, p > 3, or k > 1, so m = 2, p = 3, and k = 1 is the only +possibility. We get the list of exceptions listed in Table 3 after checking divisibility. +Table 3. Exceptions satisfying |Ω(2m + 1, q)| divides |An| and +|An| < |Ω(2m + 1, q)| · 7.3qm +m +q +n +2 +3 +9 +Again, we check the ATLAS and find that cod(Ω(5, 3)) ̸⊆ cod(A9). +(3) Let H = PSp(2m, q) where q = pk and m ≥ 3. From [7], qm2 divides |PSp(2m, q)|. Since PSp(2m, q) +is a quotient of Sp(2m, q), we have k(PSp(2m, q)) ≤ k(Sp(2m, q)). From Table 1, |Am2k(p−1)| < +|PSp(2m, q)| · 15.2qm. We computationally check that we get a contradiction if m > 4, p > 2, or +k > 2, so m = 3 or 4, p = 2, and k = 1 or 2 are the only possibilities. We get no exceptions after +checking divisibility. +(4) Let H = O+(2m, q) where q = pk and m ≥ 4. From [7], qm(m−1) divides |O+(2m, q)|. Using Table +1, we have that |Am(m−1)k(p−1)| < |O+(2m, q)| · 15qm. As above, we computationally check that we +get a contradiction if m > 4, p > 2, or k > 1 so m = 4, p = 2, and k = 1 is the only possibility, and +we get no possible exceptions after checking divisibility. +(5) Let H = PSU(m+1, q) where q = pk and m ≥ 2. From [7], qm(m+1)/2 divides |PSU(m+1, q)|. Since +PSU(m + 1, q) is a quotient of SU(m + 1, q), we have k(PSU(m + 1, q)) ≤ k(SU(m + 1, q)). From +Table 1, |A m(m+1)k(p−1) +2 +| < |PSU(m + 1, q)| · 8.26qm. Again, we computationally check that we get a +4 + +contradiction if m > 6, p > 7, or k > 42 so m ≤ 6, p ≤ 7, and k ≤ 42 are the only possibilities. We +get Table 4 after checking divisibility: +Table 4. Exceptions satisfying |PSU(m + 1, q)| divides |An| and +|An| < |PSU(m + 1, q)| · 8.26qm +m +q +n +2 +3 +9 +3 +2 +9 +We check the ATLAS to find that cod(PSU(3, 3)) ̸⊆ cod(A9), and we note that PSU(4, 2) ∼= +Ω(5, 3), which we have already ruled out. +(6) Let H = O−(2m, q) where q = pk and m ≥ 4. From [7], qm(m−1) divides |O−(2m, q)|. Thus, using +Table 1 similarly to above, |Am(m−1)k(p−1)| < |O−(2m, q)| · 15qm. Again, we computationally check +that we get a contradiction if m > 5, p > 3, or k > 3 so m ≤ 5, p ≤ 3, and k ≤ 3 are the only +possibilities, and we get no possible exceptions after checking divisibility. +□ +Lemma 3.4. Let H be an exceptional simple group of Lie type. Then if n ≥ 5, cod(H) ̸⊆ cod(An). +Proof. There are 10 familes of exceptional simple groups of Lie type (other than the Tits group). These are +E6(q), E7(q), E8(q), F4(q), G2(q),2 E6(q),3 D4(q),2 B2(q),2 F4(q), and 2G2(q). We prove the lemma in each +case. First, we reproduce [12, Table 1] for reference. +Table 5. Class Numbers for Exceptional Groups +G +k(G) ≤ +Comments +2B2(q) +q + 3 +q = 22m+1 +2G2(q) +q + 8 +q = 32m+1 +G2(q) +q2 + 2q + 9 +2F4(q) +q2 + 4q + 17 +q = 22m+1 +3D4(q) +q4 + q3 + q2 + q + 6 +F4(q) +q4 + 2q3 + 7q2 + 15q + 31 +E6(q) +q6 + q5 + 2q4 + 2q3 + 15q2 + 21q + 60 +2E6(q) +q6 + q5 + 2q4 + 4q3 + 18q2 + 26q + 62 +E7(q) +q7 + q6 + 2q5 + 7q4 + 17q3 + 35q2 + 71q + 103 +E8(q) +q8 + q7 + 2q6 + 3q5 + 10q4 + 16q3 + 40q2 + 67q + 112 +(1) Let H ∼= E6(q) where q = pk. From the order formula found in [7], q36 divides |E6(q)|. From [5], +we know that for any prime p, |n!|p ≤ p +n +p−1 . If q = pk, then we have |n!|q ≤ q +n +k(p−1) and thus +|An|q ≤ q +n +k(p−1) where |An|p is the p-part of An. By Lemma 2.4, |E6(q)| divides |An| so q36 divides +|An|. Thus 36 ≤ +n +k(p−1) and n ≥ 36k(p − 1). Therefore, |An| ≥ |A36k(p−1)|. +Now, we note from Table 5 that |Irr(E6(q))| = k(E6(q)) ≤ q6 + q5 + 2q4 + 2q3 + 15q2 + 21q + 60. +Applying Lemma 2.7 gives |An| < |E6(q)|·|Irr(E6(q))|. Hence, |A36k(p−1)| < |E6(q)|·(q6 +q5 +2q4 + +2q3 + 15q2 + 21q + 60). As with the classical Lie type groups, we can computationally find an upper +bound on p and k since the left side grows faster in terms of p and k than the right side. In this +case, we find that no values of p and k satisfy the inequality, since substituting p = 2 and k = 1 +gives |A36| > |E6(2)| · (26 + 25 + 2 · 24 + 2 · 23 + 15 · 22 + 21 · 2 + 60). Thus, there are no possible +values for q and n such that cod(E6(q)) ⊆ cod(An). +(2) Let H ∼= E7(q) where q = pk. From [7], q63 divides |E7(q)|. From Table 5, |A63k(p−1)| < |E7(q)| · +(q7 +q7 +2q5 +7q4 +17q3 +35q2+71q +103). We computationally check that we get a contradiction +for p = 2, k = 1, so there are no possible exceptions. +5 + +(3) Let H ∼= E8(q) where q = pk. From [7], q120 divides |E8(q)|. Thus, using Table 5 as above, we have +|A120k(p−1)| < |E8(q)|·(q8+q7+2q6+3q5+10q4+16q3+40q2+67q+112). Now, we computationally +check that we get a contradiction for p = 2, k = 1, so there are no possible exceptions. +(4) Let H ∼= F4(q) where q = pk. From [7], q24 divides |F4(q)|. From Table 5, |A24k(p−1)| < |F4(q)|·(q4 + +2q3 + 7q2 + 15q + 31). Again, we computationally check that we get a contradiction for p = 2, k = 1, +so there are no possible exceptions. +(5) Let H ∼= G2(q) where q = pk. From [7], q6 divides |G2(q)|. Thus, using Table 5 as above, |A6k(p−1)| < +|G2(q)| · (q2 + 2q + 9). Now, we find that p = 2, k = 1 satisfies the inequality, but any other values +of p and k do not. However, we note that G2(2) is not simple, so we instead consider its derived +subgroup G2(2)′ (which still satisfies the above inequality). We check for exceptions where |G2(2)′| +divides |An| and |An| < |G2(2)′| · (22 + 2 · 2 + 9), but there are none. +(6) Let H ∼= 2E6(q) where q = pk. +From [7], q36 divides |2E6(q)|. +Using Table 5, |A36k(p−1)| < +|2E6(q)| · (q6 + q5 + 2q4 + 4q3 + 18q2 + 26q + 62). Again, we computationally check that we get a +contradiction for p = 2, k = 1, so there are no possible exceptions. +(7) Let H ∼= 3D4(q) where q = pk. From [7], q12 divides |3D4(q)|. Thus, using Table 5 similarly to +above, |A12k(p−1)| < |3D4(q)| · (q4 + q3 + q2 + q + 6). Now, we find that p = 2, k = 1 satisfies the +inequality, but any other values of p and k do not. As for the sporadic groups, we check for possible +exceptions where |3D4(2)| divides |An| and |An| < |3D4(2)| · (24 + 23 + 22 + 2 + 2), but there are +none. +(8) Let H ∼= 2B2(q) where q = 22m+1 and m ≥ 1. From [7], q2 divides |2B2(q)|. From Table 5, we +have that |A2(2m+1)| < |2B2(q)| · (q + 3). In this case, we computationally check that we get a +contradiction if m > 4, so m must be less than 5. However, checking the divisibility condition, we +get no exceptions. +(9) Let H ∼= 2F4(q) where q = 22m+1 and m ≥ 1. From [7], q12 divides |2F4(q)|. Thus, using Table +5 as above, |A12(2m+1)| < |2F4(q)| · (q2 + 4q + 17). Now, we computationally check that we get a +contradiction for m = 1, so there are no exceptions +(10) Let H ∼= 2G2(q) where q = 32m+1 and m ≥ 1. +From [7], q3 divides |2G2(q)|. +From Table 5, +|A3(2m+1)·2| < |2G2(q)| · (q + 8). Again, we computationally check that we get a contradiction for +m = 1, so there are no exceptions. +□ +Theorem 3.5. Let G be a finite group such that cod(G) = cod(An) where n ≥ 5. Let N be a maximal +subgroup of G. Then, G/N ∼= An. +Proof. By Lemma 2.3, G is perfect. Thus G/N is a nonabelian simple group. By Lemma 2.6, we have +cod(G/N) ⊆ cod(G) = cod(An). By Lemmas 3.1, 3.2, 3.3, and 3.4, G/N cannot be an alternating group +of degree m ̸= n, a sporadic simple group or the Tits group, a classical simple group of Lie type, or an +exceptional simple group of Lie type. Thus, G/N ∼= An. +□ +Now we present the proof of Theorem 1.1. +Proof. Let G be a minimal counterexample and N be a maximal normal subgroup of G. By Lemma 2.3, G +is perfect, and by Theorem 3.5, G/N ∼= An. In particular, N ̸= 1 as G ̸∼= An. +Step 1: N is a minimal normal subgroup of G. +Suppose L is a non-trivial normal subgroup of G with L < N. Then by Lemma 2.6, we have cod(G/N) ⊆ +cod(G/L) ⊆ cod(G). +However, cod(G/N) = cod(An) = cod(G) so equality must be obtained in each +inclusion. Thus, cod(G/L) = cod(An) which implies that G/L ∼= An since G is a minimal counterexample. +This is a contradiction since we also have G/N ∼= An, but L < N. +Step 2: N is the only non-trivial, proper normal subgroup of G. +Otherwise we assume M is another proper nontrivial normal subgroup of G. If N is included in M, then +M = N or M = G since G/N is simple, a contradiction. Then N ∩ M = 1 and G = N × M. Since M is also +a maximal normal subgroup of G, we have N ∼= M ∼= An. Choose ψ1 ∈ Irr(N) and ψ2 ∈ Irr(M) such that +cod(ψ1) = cod(ψ2) = max(cod(An)). Set χ = ψ1 · ψ2 ∈ Irr(G). Then cod(χ) = (max(cod(An)))2 /∈ cod(G), +a contradiction. +Step 3: For each non-trivial χ ∈ Irr(G|N) := Irr(G) − Irr(G/N), χ is faithful. +6 + +We construct Irr(G/N) as the same as Lemma 2.5. Then it follows by the definition of Irr(G|N) that if +χ ∈ Irr(G|N), N ̸≤ ker(χ). Thus since N is the unique nontrivial, proper, normal subgroup of G, ker(χ) = G +or ker(χ) = 1. Therefore, ker(χ) = 1 for all nontrivial χ ∈ Irr(G|N). +Step 4: N is an elementary abelian group. +Suppose that N is not abelian. Since N is a minimal normal subgroup, by [10, Theorem 4.3A (iii)], +N = Sn where S is a nonabelian simple group and n ∈ Z+. +By Lemmas 2.1 and 2.2, there is a non- +trivial character χ ∈ Irr(N) which extends to some ψ ∈ Irr(G). Now, ker(ψ) = 1 by Step 3, so cod(ψ) = +|G|/ψ(1) = |G/N| · |N|/χ(1). However, by assumption, we have that cod(G) = cod(An) = cod(G/N). Thus, +cod(ψ) ∈ cod(G) = cod(G/N), so cod(ψ) = |G/N|/φ(1) for some φ ∈ Irr(G/N). Hence, |G/N| is divisible by +cod(ψ) which contradicts the fact that cod(ψ) = |G/N| · |N|/χ(1), as χ(1) ̸= |N|. Thus N must be abelian. +Now to show that N is elementary abelian, let a prime p divide |N|. Then N has a p-Sylow subgroup +K, and K is the unique p-Sylow subgroup of N since N is abelian, so K is characteristic in N. Thus, +K is a normal subgroup of G, so K = N as N is minimal. +Thus |N| = pn. Now, take the subgroup +N p = {np | n ∈ N} of N, which is proper by Cauchy’s theorem. Since N p is characteristic in N, it must +be normal in G, so N p is trivial by the uniqueness of N. Thus every element of N has order p, and N is +elementary abelian. +Step 5: CG(N) = N. +First note that since N is normal, CG(N) ⊴ G. Additionally, since N is abelian by Step 4, N ≤ CG(N). +By the maximality of N, we must have CG(N) = N or CG(N) = G. If CG(N) = N, we are done. +If not, then CG(N) = G, so N must be in the center of G. Then since N is the unique minimal normal +subgroup of G by Step 2, we must have that |N| is prime. If not, there always exists a proper non-trivial +subgroup K of N, and K is normal since it is contained in Z(G), contradicting the minimality of N. Moreover, +since G is perfect, we have that Z(G) = N, and N is isomorphic to a subgroup of the Schur multiplier of +G/N [16, Corollary 11.20]. +Now, we note that it is well-known that for n > 7, the Schur multiplier of An is Z2, so G ∼= 2.An. +From [20], 2.An always has a character degree of order 2⌊(n−2)/2⌋. Let χ be such an irreducible character of +2.An with χ(1) = 2⌊(n−2)/2⌋. Recall that by Step 2, there is only one non-trivial proper normal subgroup of +G ∼= 2.An. In particular N ∼= Z2 is the only non-trivial proper normal subgroup of G. Thus |ker(χ)| = 1 +or 2. Then we have cod(χ) = |2.An:ker(χ)| +χ(1) +. If |ker(χ)| = 1, then cod(χ) = +n! +2⌊(n−2)/2⌋ , and if |ker(χ)| = 2, +then cod(χ) = +n!/2 +2⌊(n−2)/2⌋ = +n! +2⌊n/2⌋ . In either case, for any prime p ̸= 2, | cod(χ)|p = |n!|p = |An|p. Since +cod(G) = cod(An), we know that cod(χ) ∈ cod(An). Therefore, there is a character degree of An which is a +power of 2. +However, from [20], we know that for n > 7, An only has a character degree equal to a power of 2 when +n = 2d + 1 for some positive integer d. In this case, 2d = n − 1 ∈ cd(An) so we need |An| +n−1 = +|2.An| +2⌊(n−2)/2⌋ or +|2.An| +2⌊n/2⌋ . Hence, +1 +n−1 = +2 +2⌊(n−2)/2⌋ = +1 +2⌊(n−2)/2⌋−1 or +1 +2⌊n/2⌋−1 so n − 1 = 2⌊(n−2)/2⌋−1 or 2⌊n/2⌋−1. However, the +only integer solution to either of these equations occurs when n = 9 and 9 − 1 = 8 = 23 = 2⌊9/2⌋−1. In this +case, we check the ATLAS [9] to find that the codegree sets of A9 and 2.A9 do not have the same order. +This is a contradiction, so CG(N) = N. +Step 6: Let λ be a non-trivial character in Irr(N) and ϑ ∈ Irr(IG(λ)|λ), the set of irreducible constituents +of λIG(λ), where IG(λ) is the inertia group of λ ∈ G. Then |IG(λ)| +ϑ(1) +∈ cod(G). Also, ϑ(1) divides |IG(λ)/N|, +and |N| divides |G/N|. Lastly, IG(λ) < G, i.e. λ is not G-invariant. +Let λ be a non-trivial character in Irr(N) and ϑ ∈ Irr(IG(λ)|λ). Let χ be an irreducible constituent of +ϑG. By [16, Corollary 5.4], we know χ ∈ Irr(G), and by [16, Definition 5.1], we have χ(1) = +|G| +|IG(λ)| · ϑ(1). +Moreover, we know tat ker(χ) = 1 by Step 2, and thus cod(χ) = +|G| +χ(1) = |IG(λ)| +ϑ(1) , so |IG(λ)| +ϑ(1) +∈ cod(G). Now, +since N is abelian, λ(1) = 1, so we have ϑ(1) = ϑ(1)/λ(1) which divides |IG(λ)| +|N| , so |N| divides +|IG(λ)| +ϑ(1) . +Moreover, we know that cod(G) = cod(G/N), and all elements in cod(G/N) divide |G/N|, so |N| divides +|G/N|. +Next, we want to show IG(λ) is a proper subgroup of G. To reach a contradiction, assume IG(λ) = G. +Then ker(λ) ⊴ G. From Step 2, we know ker(λ) = 1, and from Step 4, we know N is a cyclic group of prime +7 + +order. Thus by the Normalizer-Centralizer theorem, we have G/N = NG(N)/CG(N) ≤ Aut(N) so G/N is +abelian, a contradiction. +Step 7: Final contradiction. +From Step 4, N is an elementary abelian group of order pm for some prime p and integer m ≥ 1. By +the Normalizer-Centralizer theorem, An ∼= G/N = NG(N)/CG(N) ≤ Aut(N) and m > 1. Note that in +general, Aut(N) = GL(m, p). By Step 6, |N| divides |G/N|, so we know that |N| = pm divides |An| and +G/N ∼= An ≲ GL(m, p). We prove by contradiction that this cannot occur. +First, we claim that if pm divides |An| and An ≲ (GL(m, p), then p must equal 2. To show this, we note +that for p > 2, by [5], we have that if pm divides |An|, then m < n +2 . However, Theorem 1.1 of [24] shows that +if n > 6, the minimal faithful degree of a modular representation of An over a field of characteristic p is at +least n − 2. Since embedding An as a subgroup of GL(m, p) is equivalent to giving a faithful representation +of degree m over a field of characteristic p, we have that m ≥ n − 2. This is a contradiction since n +2 > n − 2 +implies n < 4. Therefore, p = 2. +Now, let p = 2. +As above, from [5], we obtain |n!|2 ≤ 2n−1. +Thus, if 2m divides |An|, then m ≤ +|An|2 ≤ 2n−2. Now, Theorem 1.1 of [23] shows that if n > 8, then the minimal faithful degree of a modular +representation of An over a field of characteristic 2 is at least n − 2. Therefore, we must have m ≥ n − 2, so +m = |An|2 = 2n−2 is the only option. +Let λ ∈ Irr(N), ϑ ∈ Irr(IG(λ)|λ), and T := IG(λ). Then 1 < |G : T | < |N| = 2n−2 for |G : T | is +the number of all conjugates of λ. By Step 5, we know that +|T | +ϑ(1) ∈ cod(G) and moreover that |N| divides +|T | +ϑ(1). Since |N|2 = |An|2 and cod(G) = cod(An), we know that +��� |T | +ϑ(1) +��� +2 = |N|2. Thus +��� |T/N| +ϑ(1) +��� +2 = 1 so the +2-parts of |T/N| and ϑ(1) are equal. Thus for every ϑ ∈ Irr(T | λ), we have |ϑ(1)|2 = |T/N|2. However, +|T/N| = � +ϑ∈Irr(T |λ) ϑ(1)2. Hence, if |ϑ(1)|2 = 2k ≥ 2 for every ϑ ∈ Irr(T | λ), we would have |T/N|2 = 22k +contradicting the fact that |ϑ(1)|2 = |T/N|2. Therefore, |T/N|2 = 1. Thus, since |G/N|2 ≥ |N|2 = 2n−2, we +have |G : T |2 = |G/N : T/N|2 ≥ 2n−2, so |G : T | ≥ 2n−2 = |N|, which is a contradiction. +We have one final exception to consider: n = 8, p = 2, and m = 4, 5 or 6. In this case, A8 ∼= GL(4, 2) and +26 divides |A8|. Now, cod(A8) = {1, 26·32·5, 25·32·5, 24·32·7, 26·3·5, 24·32·5, 26·32, 26·7, 23·32·5, 32·5·7, 25·32} +from [13]. We will look at each possibility for m in turn. +First, let m = 4. Then we have G/N ∼= A8 ∼= GL(4, 2), N = (Z2)4 so G is an extension of GL(4, 2) by N. +Suppose first that this extension is split and G is a semidirect product. This semidirect product is defined +by a homomorphism φ : GL(4, 2) → Aut((Z2)4) ∼= GL(4, 2). However, since GL(4, 2) is simple, ker(φ) = 1 or +GL(4, 2). In the first case, we have the trivial direct product, so there are at least two copies of GL(4, 2) as +normal subgroups of G, which contradicts Step 2. In the second case, φ is some automorphism of GL(4, 2). +Here, we can check using GAP that any such φ creates a semidirect product GL(4, 2) ⋊φ (Z2)4 which does +not have the same codegree set as A8. Now, suppose that the extension is non-split. Then, [4] gives that +there is a unique non-split extension 24.GL(4, 2). However, we find using GAP that it doesn’t have the same +codegree set as A8. +Second, let m = 5. +As above, |G : T | < |N| = 25 and +|T | +ϑ(1) ∈ cod(G) such that 25 divides +|T | +ϑ(1). +Further, | |T/N| +ϑ(1) |2 ≤ 2 so |T/N|2 ≤ 4 and |G/N : T/N|2 ≥ 16. Thus, we have 16 divides |G/N : T/N| and +|G/N : T/N| < 32. But we check the index of all subgroups of G/N ∼= A8 using GAP and find that none of +them satisfy these two properties. +Finally, let m = 6. Now, |N|2 = |A8|2. For this case the same argument as above for general An holds, +and we reach a contradiction. Thus we find that every |N| = pm produces a contradiction, so N = 1 and +G ∼= An. +□ +4. Acknowledgements +This research was conducted under NSF-REU grant DMS-1757233, DMS-2150205 and NSA grant H98230- +21-1-0333, H98230-22-1-0022 by Dolorfino, Martin, Slonim, and Sun during the Summer of 2022 under the +supervision of Yang. The authors gratefully acknowledge the financial support of NSF and NSA, and also +thank Texas State University for providing a great working environment and support. Yang was also partially +supported by grants from the Simons Foundation (#499532, #918096, to YY). The authors would also like +to thank Prof. Richard Stanley for his help. +8 + +References +[1] N. Ahanjideh, Nondivisibility among irreducible character co-degrees. Bull. Aust. Math. Soc., 105 (2022), 68-74. +[2] K. Aziziheris, F. Shafiei, F. Shirjian, Simple groups with few irreducible character degrees. J. Algebra Appl., 20 (2021), +2150139. +[3] A. Bahri, Z. Akhlaghi, B. Khosravi, An analogue of Huppert’s conjecture for character codegrees. Bull. Aust. Math. Soc., +104 (2021), no. 2, 278-286. +[4] A. B. M. Basheer and J. Moori, Fischer Matrices of Dempwolff Group 25.GL(5, 2). Int. J. Group Theory, 1 (2012), 43-63. +[5] C. Bessenrodt, H. P. Tong-Viet, J. Zhang, Huppert’s conjecture for alternating groups. J. Algebra, 470 (2017), 353-378. +[6] J. Bezanson, S. Karpinski, V. B. Shah, A. Edelman, Julia: A fast dynamic language for technical computing. ArXiv +Preprint, ArXiv:1209.5145. +[7] R. W. Carter, Simple Groups of Lie Type. Wiley, 1989. +[8] D. Chillag and M. Herzog, On character degrees quotients. Arch. Math., 55 (1990), 25-29. +[9] J. H. Conway et. al, Atlas of Finite Groups. Oxford Clarendon Press, 1985. +[10] J. D. Dixon and B. Mortimer, Permutation Groups. Spring, 1996. +[11] M. Dolorfino, L. Martin, Z. Slonim, Y. Sun, Y. Yang, On the characterization of sporadic simple groups by codegrees. +submitted. +[12] J. Fulman and R. Guralnick, Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with appli- +cations to derangements. Trans. Amer. Math. Soc., 364 (2012), 3023-3070. +[13] M. Gintz, M. Kortje, M. laurence, Y. Liu, Z. Wang, Y. Yang, On the characterization of some nonabelian simple groups +with few codegrees. Comm. Algebra, 50 (2022), 3932-3939. +[14] H. Guan, X. Zhang, Y. Yang, Recognizing Ree groups +2G2(q) using the codegree set. Bull. Aust. Math. Soc., +https://www.doi.org/10.1017/S0004972722001022. +[15] N. N. Hung, Group pseudo-algebras of finite simple groups. In progress. +[16] I. M. Isaacs, Character Theory of Finite Groups. New York Academic Press, 1976. +[17] G. James and A. Kerber, The Representation Theory of the Symmetric Group. Addison-Wesley Publishing Company, 1981. +[18] E. I. Khukrho and V. D. Mazurov, Unsolved Problems in Group Theory. The Kourovka Notebook. No. 20. Russian Academy +of Sciences, 2022. +[19] Y. Liu and Y. Yang, Huppert’s analogue conjecture for PSL(3, q) and PSU(3, q). Results Math., 78 (2023), No. 7. +[20] G. Malle and A.E. Zalesskii, Prime power degree representations of quasi-simple groups. Arch. Math., 77 (2001), 461-468. +[21] A. Moret´o, Complex group algebra of finite groups: Brauer’s problem 1. Adv. Math., 208 (2007), 236-248. +[22] G. Qian, Y. Wang, H. Wei, Co-degrees of irreducible characters in finite groups. J. Algebra, 312 (2007), 946-955. +[23] A. Wagner, The faithful linear representations of least degree of Sn and An over a field of characteristic 2. Math. Z., 151 +(1976), 127-138. +[24] A. Wagner, The faithful linear representations of least degree of Sn and An over a field of odd characteristics. Math. Z., +154 (1977), 104-113. +Mallory Dolorfino, Kalamazoo College, Kalamazoo, Michigan, USA, mallory.dolorfino19@kzoo.edu +Luke Martin, Gonzaga University, Spokane, Washington, USA, lwmartin2019@gmail.com +Zachary Slonim, University of California, Berkeley, Berkeley, California, USA, zachslonim@berkeley.edu +Yuxuan Sun, Haverford College, Haverford, Pennsylvania, USA, ysun1@haverford.edu +Yong Yang, Texas State University, San Marcos, Texas, USA, yang@txstate.edu +9 + diff --git a/ENE0T4oBgHgl3EQfywJf/content/tmp_files/load_file.txt b/ENE0T4oBgHgl3EQfywJf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8df7360b1d8fb79a485b9989e6463ed0b9b29883 --- /dev/null +++ b/ENE0T4oBgHgl3EQfywJf/content/tmp_files/load_file.txt @@ -0,0 +1,641 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf,len=640 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='02663v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='GR] 6 Jan 2023 ON THE CHARACTERIZATION OF ALTERNATING GROUPS BY CODEGREES MALLORY DOLORFINO, LUKE MARTIN, ZACHARY SLONIM, YUXUAN SUN, AND YONG YANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let G be a finite group and Irr(G) the set of all irreducible complex characters of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Define the codegree of χ ∈ Irr(G) as cod(χ) := |G:ker(χ)| χ(1) and denote by cod(G) := {cod(χ) | χ ∈ Irr(G)} the codegree set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let An be an alternating group of degree n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this paper, we show that An is determined up to isomorphism by cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Introduction Let G be a finite group and Irr(G) the set of all irreducible complex characters of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' For any χ ∈ Irr(G), define the codegree of χ as cod(χ) := |G:ker(χ)| χ(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then define the codegree set of G as cod(G) := {cod(χ) | χ ∈ Irr(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' The concept of codegrees was originally considered in [8], where the codegree was defined as cod(χ) := |G| χ(1), and it was later modified to its current definition by [22] so that cod(χ) is the same for G and G/N when N ≤ ker(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Several properties of codegrees have been studied, such as the relationship between the codegrees and element orders, codegrees of p-groups, and groups with few codegrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' The codegree set of a group is closely related to the character degree set of a group, which is defined as cd(G) := {χ(1) | χ ∈ Irr(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' The relationship between the character degree set and a group’s structure is an active area of research – many properties of a group’s structure are largely determined by its character degree set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In 1990, Bertram Huppert made the following conjecture about the relationship between a simple group H and a finite group G having equal character degree sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Huppert’s Conjecture: Let H be a finite nonabelian simple group and G a finite group such that cd(H) = cd(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then G ∼= H × A, where A is an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Huppert’s conjecture has been verified for many cases such as the alternating groups, sporadic groups, and simple groups of Lie type with low rank, but it has yet to be verified for simple groups of Lie type with high rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Recently, a similar conjecture related to codegrees has been posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Codegree Version of Huppert’s Conjecture: Let H be a finite nonabelian simple group and G a finite group such that cod(H) = cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then G ∼= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' This conjecture appears in the Kourovka Notebook of Unsolved Problems in Group Theory as question 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='79 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' It has been verified for PSL(2, q), PSL(3, 4), Alt7, J1, 2B2(22f+1) where f ≥ 1, M11, M12, M22, M23 and PSL(3, 3) by [1, 3, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' The conjecture has also been verified for PSL(3, q) and PSU(3, q) in [19] and 2G2(q) in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Recently, the authors verified the conjecture for all sporadic simple groups in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this paper, we provide a general proof verifying this conjecture for all alternating groups of degree greater than or equal to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' The methods used may be generalized to simple groups of Lie type, giving promising results for characterizing all simple groups by their codegree sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let An be an alternating group of degree n ≥ 5 and G a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If cod(G) = cod(An), then G ∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Throughout the paper, we follow the notation used in Isaacs’ book [16] and the ATLAS of Finite Groups [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 20C15, 20D06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Preliminary Results We first introduce some lemmas which will be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' [21, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='2] Let S be a finite nonabelian simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then there exists 1S ̸= χ ∈ Irr(S) that extends to Aut(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' [17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='34] Let N be a minimal normal subgroup of G such that N = S1 × · · · × St where Si ∼= S is a nonabelian simple group for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' , t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If χ ∈ Irr(S) extends to Aut(S), then χ × · · · × χ ∈ Irr(N) extends to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' [13, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='6] Let G be a finite group and S a finite nonabelian simple group with cod(G) = cod(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then G is a perfect group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' [15] Let G be a finite group and S a finite nonabelian simple group such that cod(S) ⊆ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then |S| divides |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let G be a finite group with N ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then cod(G/N) ⊆ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='22], we can define Irr(G/N) = {ˆχ(gN) = χ(g) | χ ∈ Irr(G) and N ⊆ ker(χ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Take any ˆχ ∈ Irr(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By definition, we know that ˆχ(1) = χ(1), so the denominators of cod(ˆχ) and cod(χ) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In addition, ker(ˆχ) ∼= ker(χ)/N, so |ker(χ)| = |N| · |ker(ˆχ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus |G/N : ker(ˆχ)| = |G|/|N| | ker(χ)|/|N| = |G| | ker(χ)|, so cod(ˆχ) = cod(χ) and therefore cod(G/N) ⊆ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let G be a finite group with normal subgroups N and M such that N ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then, cod(G/M) ⊆ cod(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By the Third Isomorphism Theorem, we know that G/M ∼= (G/N)/(M/N) is a quotient of G/N, and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5, cod(G/M) ⊆ cod(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let S be a finite nonabelian simple group and G be a nontrivial finite group with cod(G) ⊆ cod(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then, |S| < |G| · |Irr(G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We know that for each irreducible character χ ∈ Irr(S), χ(1)2 < |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Because S is simple, if χ is non- trivial, then ker(χ) = 1, so cod(χ) = |S| χ(1) > � |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then, since cod(G) ⊆ cod(S), for each irreducible non- trivial character ψ ∈ Irr(G), cod(ψ) > � |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, |G:ker(ψ)| ψ(1) > � |S| which implies that |G| |ker(ψ)|√ |S| > ψ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' So, ψ(1) < |G| √ |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then � ψ∈Irr(G) ψ(1)2 < | Irr(G)| |G|2 |S| , and by character theorems, we’ll have |G| < | Irr(G)| |G|2 |S| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus |S| < |G| · |Irr(G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Main Results We start with some lemmas which limit the simple groups whose codegree set can be contained in the codegree set of an alternating group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let H be an alternating group of degree m ̸= n, where m, n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then cod(H) ̸⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Suppose cod(Am) ⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then, from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4, |Am| divides |An|, so m < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let ax denote the minimal non-trivial codegree of Ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We show that an−1 < an so that cod(Am) ̸⊆ cod(An) follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We know that irreducible representations of the symmetric group Sn are in one-to-one correspondence with the partitions of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let λ be a partition of n and Vλ be the corresponding irreducible representation of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We note that a partition of n can be visualized by a Young diagram and we let hλ(i, j) be the hook length of the (i, j)th square of the Young diagram corresponding to λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' the number of cells (a, b) of λ such that a = i and b ≥ j or b = j and a ≥ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By the hook length formula, n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' dim(Vλ) = � hλ(i, j) := Hλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let Uλ be an irreducible constituent of the restriction of Vλ to An, ResSn An Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If λ is not self-conjugate (λ ̸= λ′), then ResSn An Vλ remains irreducible, so Uλ = ResSn An Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' dim(Uλ) = Hλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If λ is self- conjugate, then the restriction of Vλ to An splits into two irreducible representations of the same dimension, so dim(Uλ) = 1 2dim(Vλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' dim(Uλ) = 2Hλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 2 Now, an = min{ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='/2 dim(Uλ) | Uλ ∈ Irr(An)} = 1 2 min({Hλ | λ ̸= λ′} ∪ {2Hλ | λ = λ′}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We want to show that an−1 < an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' First, assume that an = 1 22Hλ for some λ = λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then we can remove a square from λ to give a non-self-conjugate partition µ of n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since Hµ < Hλ < 2Hλ and an−1 ≤ 1 2Hµ, we know an−1 < an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now assume that an = 1 2Hλ for some λ ̸= λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then if n ≥ 3, we can remove a square from λ to obtain a non-self-conjugate partition µ of n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since Hµ < Hλ and an−1 ≤ 1 2Hµ, an−1 < an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, if m < n, then am < an, contradicting the assumption that cod(Am) ⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let H be a sporadic simple group or the Tits group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then if n ≥ 5, cod(H) ̸⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In search of a contradiction, let H be a sporadic simple group or the Tits group such that cod(H) ⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4, we deduce a tight restriction on the order of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Namely, |H| = |An|/k where 1 ≤ k < |Irr(H)| is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, for each sporadic (or Tits) group H, we can computationally check (using Julia [6]) which alternating groups An satisfy both |H| divides |An| and |An| |H| < |Irr(H)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We find only one possible exception: An = A10 and H = J2 where |A10| |J2| = 3 < 21 = |Irr(J2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, we check that cod(J2) ̸⊆ cod(A10) using the ATLAS [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let H be a classical simple group of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then cod(H) ̸⊆ cod(An) for all n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' There are 6 families of classical simple groups of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' These are PSL(m + 1, q), Ω(2m + 1, q), PSp(2m, q), O+(2m, q), PSU(m + 1, q), and O−(2m, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='l We prove the lemma in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let k(G) denote the number of conjugacy classes of G, we reproduce [12, Table 2] for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Class Numbers for Classical Groups G k(G) ≤ Comments SL(n, q) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5qn−1 SU(n, q) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='26qn−1 Sp(2n, q) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='8qn q odd Sp(2n, q) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='2qn q even SO(2n + 1, q) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1qn q odd Ω(2n + 1, q) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3qn q odd SO±(2n, q) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5qn q odd Ω±(2n, q) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='8qn q odd O±(2n, q) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5qn q odd SO±(2n, q) 14qn q even O±(2n, q) 15qn q even (1) Let H = PSL(m + 1, q) where q = pk and m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From the order formula found in [7], qm(m+1)/2 divides |PSL(m + 1, q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Legendre’s formula, we know that for any prime p, |n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='|p ≤ p n p−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If q = pk, then we have |n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='|q ≤ q n k(p−1) and thus |An|q ≤ q n k(p−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4, |PSL(m + 1, q)| divides |An|, so qm(m+1)/2 divides |An|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus m(m+1) 2 ≤ n k(p−1), giving n ≥ m(m+1)k(p−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Therefore, |An| ≥ ���A m(m+1)k(p−1) 2 ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we note that k(PSL(m + 1, q)) ≤ k(SL(m + 1, q)) since PSL(m + 1, q) is a quotient of SL(m + 1, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then from Table 1, we have that |Irr(PSL(m + 1, q))| = k(PSL(m + 1, q)) ≤ k(SL(m + 1, q)) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='7 gives |An| < |PSL(m + 1, q)| · |Irr(PSL(m + 1, q))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Hence |A m(m+1)k(p−1) 2 | < |PSL(m + 1, q)| · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now we show that if we consider the left and right sides as functions of m with constants p and k, then asymptotically, the value of |A m(m+1)k(p−1) 2 | grows faster than that of |PSL(m + 1, q)| · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We know that the left function behaves asymptotically as (m2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=', and using the order formula for PSL(m + 1, q), we know that the right function behaves asymptotically as qf(m), where f(m) is a polynomial with degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus the left function grows faster than the right function since x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' >> cx for any constant c when x is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Similarly, we can prove this result considering the two sides as functions of p and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 3 Then, we search for the maximum possible value of m which satisfies the inequality given the smallest possible values of p and k, which are 2 and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We find that m ≤ 6 and, using a similar process for p and k, that p ≤ 17 and k ≤ 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we have limited our search to a finite number of groups which we can check in the same way as for the sporadic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From this, we find a small list of exceptions, listed in Table 2: Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Exceptions satisfying |PSL(m + 1, q)| divides |An| and |An| < |PSL(m + 1, q)| · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5qm m q n 1 4 5 1 4 6 1 8 7 1 9 6 1 9 7 1 5 5 1 5 6 1 7 7 2 4 8 2 4 9 3 2 8 3 2 9 Now, all of these exceptions can be found in the ATLAS, and it is routine to check that none of these groups satisfy cod(PSL(m + 1, q)) ⊆ cod(An) unless PSL(m + 1, q) ∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, if PSL(m + 1, q) ̸∼= An, then cod(PSL(m + 1, q) ̸⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (2) Let H = Ω(2m+1, q) where q = pk is odd and m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Note that when q = 2k is even, Ω(2m+1, q) ∼= PSp(2m, q), which we deal with in the next case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], qm2 divides |Ω(2m + 1, q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, using Table 1 similarly to above, |Am2k(p−1)| < |Ω(2m+ 1, q)|·7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' As above, we computationally check that we get a contradiction if m > 2, p > 3, or k > 1, so m = 2, p = 3, and k = 1 is the only possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We get the list of exceptions listed in Table 3 after checking divisibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Exceptions satisfying |Ω(2m + 1, q)| divides |An| and |An| < |Ω(2m + 1, q)| · 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3qm m q n 2 3 9 Again, we check the ATLAS and find that cod(Ω(5, 3)) ̸⊆ cod(A9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (3) Let H = PSp(2m, q) where q = pk and m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], qm2 divides |PSp(2m, q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since PSp(2m, q) is a quotient of Sp(2m, q), we have k(PSp(2m, q)) ≤ k(Sp(2m, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Table 1, |Am2k(p−1)| < |PSp(2m, q)| · 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='2qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We computationally check that we get a contradiction if m > 4, p > 2, or k > 2, so m = 3 or 4, p = 2, and k = 1 or 2 are the only possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We get no exceptions after checking divisibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (4) Let H = O+(2m, q) where q = pk and m ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], qm(m−1) divides |O+(2m, q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Using Table 1, we have that |Am(m−1)k(p−1)| < |O+(2m, q)| · 15qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' As above, we computationally check that we get a contradiction if m > 4, p > 2, or k > 1 so m = 4, p = 2, and k = 1 is the only possibility, and we get no possible exceptions after checking divisibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (5) Let H = PSU(m+1, q) where q = pk and m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], qm(m+1)/2 divides |PSU(m+1, q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since PSU(m + 1, q) is a quotient of SU(m + 1, q), we have k(PSU(m + 1, q)) ≤ k(SU(m + 1, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Table 1, |A m(m+1)k(p−1) 2 | < |PSU(m + 1, q)| · 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='26qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Again, we computationally check that we get a 4 contradiction if m > 6, p > 7, or k > 42 so m ≤ 6, p ≤ 7, and k ≤ 42 are the only possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We get Table 4 after checking divisibility: Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Exceptions satisfying |PSU(m + 1, q)| divides |An| and |An| < |PSU(m + 1, q)| · 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='26qm m q n 2 3 9 3 2 9 We check the ATLAS to find that cod(PSU(3, 3)) ̸⊆ cod(A9), and we note that PSU(4, 2) ∼= Ω(5, 3), which we have already ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (6) Let H = O−(2m, q) where q = pk and m ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], qm(m−1) divides |O−(2m, q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, using Table 1 similarly to above, |Am(m−1)k(p−1)| < |O−(2m, q)| · 15qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Again, we computationally check that we get a contradiction if m > 5, p > 3, or k > 3 so m ≤ 5, p ≤ 3, and k ≤ 3 are the only possibilities, and we get no possible exceptions after checking divisibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let H be an exceptional simple group of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then if n ≥ 5, cod(H) ̸⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' There are 10 familes of exceptional simple groups of Lie type (other than the Tits group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' These are E6(q), E7(q), E8(q), F4(q), G2(q),2 E6(q),3 D4(q),2 B2(q),2 F4(q), and 2G2(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We prove the lemma in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' First, we reproduce [12, Table 1] for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Class Numbers for Exceptional Groups G k(G) ≤ Comments 2B2(q) q + 3 q = 22m+1 2G2(q) q + 8 q = 32m+1 G2(q) q2 + 2q + 9 2F4(q) q2 + 4q + 17 q = 22m+1 3D4(q) q4 + q3 + q2 + q + 6 F4(q) q4 + 2q3 + 7q2 + 15q + 31 E6(q) q6 + q5 + 2q4 + 2q3 + 15q2 + 21q + 60 2E6(q) q6 + q5 + 2q4 + 4q3 + 18q2 + 26q + 62 E7(q) q7 + q6 + 2q5 + 7q4 + 17q3 + 35q2 + 71q + 103 E8(q) q8 + q7 + 2q6 + 3q5 + 10q4 + 16q3 + 40q2 + 67q + 112 (1) Let H ∼= E6(q) where q = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From the order formula found in [7], q36 divides |E6(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [5], we know that for any prime p, |n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='|p ≤ p n p−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If q = pk, then we have |n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='|q ≤ q n k(p−1) and thus |An|q ≤ q n k(p−1) where |An|p is the p-part of An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4, |E6(q)| divides |An| so q36 divides |An|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus 36 ≤ n k(p−1) and n ≥ 36k(p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Therefore, |An| ≥ |A36k(p−1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we note from Table 5 that |Irr(E6(q))| = k(E6(q)) ≤ q6 + q5 + 2q4 + 2q3 + 15q2 + 21q + 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='7 gives |An| < |E6(q)|·|Irr(E6(q))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Hence, |A36k(p−1)| < |E6(q)|·(q6 +q5 +2q4 + 2q3 + 15q2 + 21q + 60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' As with the classical Lie type groups, we can computationally find an upper bound on p and k since the left side grows faster in terms of p and k than the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, we find that no values of p and k satisfy the inequality, since substituting p = 2 and k = 1 gives |A36| > |E6(2)| · (26 + 25 + 2 · 24 + 2 · 23 + 15 · 22 + 21 · 2 + 60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, there are no possible values for q and n such that cod(E6(q)) ⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (2) Let H ∼= E7(q) where q = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q63 divides |E7(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Table 5, |A63k(p−1)| < |E7(q)| · (q7 +q7 +2q5 +7q4 +17q3 +35q2+71q +103).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We computationally check that we get a contradiction for p = 2, k = 1, so there are no possible exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 5 (3) Let H ∼= E8(q) where q = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q120 divides |E8(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, using Table 5 as above, we have |A120k(p−1)| < |E8(q)|·(q8+q7+2q6+3q5+10q4+16q3+40q2+67q+112).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we computationally check that we get a contradiction for p = 2, k = 1, so there are no possible exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (4) Let H ∼= F4(q) where q = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q24 divides |F4(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Table 5, |A24k(p−1)| < |F4(q)|·(q4 + 2q3 + 7q2 + 15q + 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Again, we computationally check that we get a contradiction for p = 2, k = 1, so there are no possible exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (5) Let H ∼= G2(q) where q = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q6 divides |G2(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, using Table 5 as above, |A6k(p−1)| < |G2(q)| · (q2 + 2q + 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we find that p = 2, k = 1 satisfies the inequality, but any other values of p and k do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, we note that G2(2) is not simple, so we instead consider its derived subgroup G2(2)′ (which still satisfies the above inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We check for exceptions where |G2(2)′| divides |An| and |An| < |G2(2)′| · (22 + 2 · 2 + 9), but there are none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (6) Let H ∼= 2E6(q) where q = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q36 divides |2E6(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Using Table 5, |A36k(p−1)| < |2E6(q)| · (q6 + q5 + 2q4 + 4q3 + 18q2 + 26q + 62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Again, we computationally check that we get a contradiction for p = 2, k = 1, so there are no possible exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (7) Let H ∼= 3D4(q) where q = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q12 divides |3D4(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, using Table 5 similarly to above, |A12k(p−1)| < |3D4(q)| · (q4 + q3 + q2 + q + 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we find that p = 2, k = 1 satisfies the inequality, but any other values of p and k do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' As for the sporadic groups, we check for possible exceptions where |3D4(2)| divides |An| and |An| < |3D4(2)| · (24 + 23 + 22 + 2 + 2), but there are none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (8) Let H ∼= 2B2(q) where q = 22m+1 and m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q2 divides |2B2(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Table 5, we have that |A2(2m+1)| < |2B2(q)| · (q + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, we computationally check that we get a contradiction if m > 4, so m must be less than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, checking the divisibility condition, we get no exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (9) Let H ∼= 2F4(q) where q = 22m+1 and m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q12 divides |2F4(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, using Table 5 as above, |A12(2m+1)| < |2F4(q)| · (q2 + 4q + 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we computationally check that we get a contradiction for m = 1, so there are no exceptions (10) Let H ∼= 2G2(q) where q = 32m+1 and m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q3 divides |2G2(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Table 5, |A3(2m+1)·2| < |2G2(q)| · (q + 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Again, we computationally check that we get a contradiction for m = 1, so there are no exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let G be a finite group such that cod(G) = cod(An) where n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let N be a maximal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then, G/N ∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3, G is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus G/N is a nonabelian simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='6, we have cod(G/N) ⊆ cod(G) = cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4, G/N cannot be an alternating group of degree m ̸= n, a sporadic simple group or the Tits group, a classical simple group of Lie type, or an exceptional simple group of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, G/N ∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ Now we present the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let G be a minimal counterexample and N be a maximal normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3, G is perfect, and by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5, G/N ∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In particular, N ̸= 1 as G ̸∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Step 1: N is a minimal normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Suppose L is a non-trivial normal subgroup of G with L < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='6, we have cod(G/N) ⊆ cod(G/L) ⊆ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, cod(G/N) = cod(An) = cod(G) so equality must be obtained in each inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, cod(G/L) = cod(An) which implies that G/L ∼= An since G is a minimal counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' This is a contradiction since we also have G/N ∼= An, but L < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Step 2: N is the only non-trivial, proper normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Otherwise we assume M is another proper nontrivial normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If N is included in M, then M = N or M = G since G/N is simple, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then N ∩ M = 1 and G = N × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since M is also a maximal normal subgroup of G, we have N ∼= M ∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Choose ψ1 ∈ Irr(N) and ψ2 ∈ Irr(M) such that cod(ψ1) = cod(ψ2) = max(cod(An)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Set χ = ψ1 · ψ2 ∈ Irr(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then cod(χ) = (max(cod(An)))2 /∈ cod(G), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Step 3: For each non-trivial χ ∈ Irr(G|N) := Irr(G) − Irr(G/N), χ is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 6 We construct Irr(G/N) as the same as Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then it follows by the definition of Irr(G|N) that if χ ∈ Irr(G|N), N ̸≤ ker(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus since N is the unique nontrivial, proper, normal subgroup of G, ker(χ) = G or ker(χ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Therefore, ker(χ) = 1 for all nontrivial χ ∈ Irr(G|N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Step 4: N is an elementary abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Suppose that N is not abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since N is a minimal normal subgroup, by [10, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3A (iii)], N = Sn where S is a nonabelian simple group and n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='2, there is a non- trivial character χ ∈ Irr(N) which extends to some ψ ∈ Irr(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, ker(ψ) = 1 by Step 3, so cod(ψ) = |G|/ψ(1) = |G/N| · |N|/χ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, by assumption, we have that cod(G) = cod(An) = cod(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, cod(ψ) ∈ cod(G) = cod(G/N), so cod(ψ) = |G/N|/φ(1) for some φ ∈ Irr(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Hence, |G/N| is divisible by cod(ψ) which contradicts the fact that cod(ψ) = |G/N| · |N|/χ(1), as χ(1) ̸= |N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus N must be abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now to show that N is elementary abelian, let a prime p divide |N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then N has a p-Sylow subgroup K, and K is the unique p-Sylow subgroup of N since N is abelian, so K is characteristic in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, K is a normal subgroup of G, so K = N as N is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus |N| = pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, take the subgroup N p = {np | n ∈ N} of N, which is proper by Cauchy’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since N p is characteristic in N, it must be normal in G, so N p is trivial by the uniqueness of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus every element of N has order p, and N is elementary abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Step 5: CG(N) = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' First note that since N is normal, CG(N) ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Additionally, since N is abelian by Step 4, N ≤ CG(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By the maximality of N, we must have CG(N) = N or CG(N) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If CG(N) = N, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If not, then CG(N) = G, so N must be in the center of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then since N is the unique minimal normal subgroup of G by Step 2, we must have that |N| is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If not, there always exists a proper non-trivial subgroup K of N, and K is normal since it is contained in Z(G), contradicting the minimality of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Moreover, since G is perfect, we have that Z(G) = N, and N is isomorphic to a subgroup of the Schur multiplier of G/N [16, Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we note that it is well-known that for n > 7, the Schur multiplier of An is Z2, so G ∼= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [20], 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='An always has a character degree of order 2⌊(n−2)/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let χ be such an irreducible character of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='An with χ(1) = 2⌊(n−2)/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Recall that by Step 2, there is only one non-trivial proper normal subgroup of G ∼= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In particular N ∼= Z2 is the only non-trivial proper normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus |ker(χ)| = 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then we have cod(χ) = |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='An:ker(χ)| χ(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If |ker(χ)| = 1, then cod(χ) = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 2⌊(n−2)/2⌋ , and if |ker(χ)| = 2, then cod(χ) = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='/2 2⌊(n−2)/2⌋ = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 2⌊n/2⌋ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In either case, for any prime p ̸= 2, | cod(χ)|p = |n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='|p = |An|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since cod(G) = cod(An), we know that cod(χ) ∈ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Therefore, there is a character degree of An which is a power of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, from [20], we know that for n > 7, An only has a character degree equal to a power of 2 when n = 2d + 1 for some positive integer d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, 2d = n − 1 ∈ cd(An) so we need |An| n−1 = |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='An| 2⌊(n−2)/2⌋ or |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='An| 2⌊n/2⌋ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Hence, 1 n−1 = 2 2⌊(n−2)/2⌋ = 1 2⌊(n−2)/2⌋−1 or 1 2⌊n/2⌋−1 so n − 1 = 2⌊(n−2)/2⌋−1 or 2⌊n/2⌋−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, the only integer solution to either of these equations occurs when n = 9 and 9 − 1 = 8 = 23 = 2⌊9/2⌋−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, we check the ATLAS [9] to find that the codegree sets of A9 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='A9 do not have the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' This is a contradiction, so CG(N) = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Step 6: Let λ be a non-trivial character in Irr(N) and ϑ ∈ Irr(IG(λ)|λ), the set of irreducible constituents of λIG(λ), where IG(λ) is the inertia group of λ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then |IG(λ)| ϑ(1) ∈ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Also, ϑ(1) divides |IG(λ)/N|, and |N| divides |G/N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Lastly, IG(λ) < G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' λ is not G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let λ be a non-trivial character in Irr(N) and ϑ ∈ Irr(IG(λ)|λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let χ be an irreducible constituent of ϑG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By [16, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4], we know χ ∈ Irr(G), and by [16, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1], we have χ(1) = |G| |IG(λ)| · ϑ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Moreover, we know tat ker(χ) = 1 by Step 2, and thus cod(χ) = |G| χ(1) = |IG(λ)| ϑ(1) , so |IG(λ)| ϑ(1) ∈ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, since N is abelian, λ(1) = 1, so we have ϑ(1) = ϑ(1)/λ(1) which divides |IG(λ)| |N| , so |N| divides |IG(λ)| ϑ(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Moreover, we know that cod(G) = cod(G/N), and all elements in cod(G/N) divide |G/N|, so |N| divides |G/N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Next, we want to show IG(λ) is a proper subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' To reach a contradiction, assume IG(λ) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then ker(λ) ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Step 2, we know ker(λ) = 1, and from Step 4, we know N is a cyclic group of prime 7 order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus by the Normalizer-Centralizer theorem, we have G/N = NG(N)/CG(N) ≤ Aut(N) so G/N is abelian, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Step 7: Final contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Step 4, N is an elementary abelian group of order pm for some prime p and integer m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By the Normalizer-Centralizer theorem, An ∼= G/N = NG(N)/CG(N) ≤ Aut(N) and m > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Note that in general, Aut(N) = GL(m, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Step 6, |N| divides |G/N|, so we know that |N| = pm divides |An| and G/N ∼= An ≲ GL(m, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We prove by contradiction that this cannot occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' First, we claim that if pm divides |An| and An ≲ (GL(m, p), then p must equal 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' To show this, we note that for p > 2, by [5], we have that if pm divides |An|, then m < n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1 of [24] shows that if n > 6, the minimal faithful degree of a modular representation of An over a field of characteristic p is at least n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since embedding An as a subgroup of GL(m, p) is equivalent to giving a faithful representation of degree m over a field of characteristic p, we have that m ≥ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' This is a contradiction since n 2 > n − 2 implies n < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Therefore, p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, let p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' As above, from [5], we obtain |n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='|2 ≤ 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, if 2m divides |An|, then m ≤ |An|2 ≤ 2n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1 of [23] shows that if n > 8, then the minimal faithful degree of a modular representation of An over a field of characteristic 2 is at least n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Therefore, we must have m ≥ n − 2, so m = |An|2 = 2n−2 is the only option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let λ ∈ Irr(N), ϑ ∈ Irr(IG(λ)|λ), and T := IG(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then 1 < |G : T | < |N| = 2n−2 for |G : T | is the number of all conjugates of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Step 5, we know that |T | ϑ(1) ∈ cod(G) and moreover that |N| divides |T | ϑ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since |N|2 = |An|2 and cod(G) = cod(An), we know that ��� |T | ϑ(1) ��� 2 = |N|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus ��� |T/N| ϑ(1) ��� 2 = 1 so the 2-parts of |T/N| and ϑ(1) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus for every ϑ ∈ Irr(T | λ), we have |ϑ(1)|2 = |T/N|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, |T/N| = � ϑ∈Irr(T |λ) ϑ(1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Hence, if |ϑ(1)|2 = 2k ≥ 2 for every ϑ ∈ Irr(T | λ), we would have |T/N|2 = 22k contradicting the fact that |ϑ(1)|2 = |T/N|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Therefore, |T/N|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, since |G/N|2 ≥ |N|2 = 2n−2, we have |G : T |2 = |G/N : T/N|2 ≥ 2n−2, so |G : T | ≥ 2n−2 = |N|, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We have one final exception to consider: n = 8, p = 2, and m = 4, 5 or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, A8 ∼= GL(4, 2) and 26 divides |A8|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, cod(A8) = {1, 26·32·5, 25·32·5, 24·32·7, 26·3·5, 24·32·5, 26·32, 26·7, 23·32·5, 32·5·7, 25·32} from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We will look at each possibility for m in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' First, let m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then we have G/N ∼= A8 ∼= GL(4, 2), N = (Z2)4 so G is an extension of GL(4, 2) by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Suppose first that this extension is split and G is a semidirect product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' This semidirect product is defined by a homomorphism φ : GL(4, 2) → Aut((Z2)4) ∼= GL(4, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, since GL(4, 2) is simple, ker(φ) = 1 or GL(4, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In the first case, we have the trivial direct product, so there are at least two copies of GL(4, 2) as normal subgroups of G, which contradicts Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In the second case, φ is some automorphism of GL(4, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Here, we can check using GAP that any such φ creates a semidirect product GL(4, 2) ⋊φ (Z2)4 which does not have the same codegree set as A8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, suppose that the extension is non-split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then, [4] gives that there is a unique non-split extension 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='GL(4, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, we find using GAP that it doesn’t have the same codegree set as A8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Second, let m = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' As above, |G : T | < |N| = 25 and |T | ϑ(1) ∈ cod(G) such that 25 divides |T | ϑ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Further, | |T/N| ϑ(1) |2 ≤ 2 so |T/N|2 ≤ 4 and |G/N : T/N|2 ≥ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, we have 16 divides |G/N : T/N| and |G/N : T/N| < 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' But we check the index of all subgroups of G/N ∼= A8 using GAP and find that none of them satisfy these two properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Finally, let m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, |N|2 = |A8|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' For this case the same argument as above for general An holds, and we reach a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus we find that every |N| = pm produces a contradiction, so N = 1 and G ∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Acknowledgements This research was conducted under NSF-REU grant DMS-1757233, DMS-2150205 and NSA grant H98230- 21-1-0333, H98230-22-1-0022 by Dolorfino, Martin, Slonim, and Sun during the Summer of 2022 under the supervision of Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' The authors gratefully acknowledge the financial support of NSF and NSA, and also thank Texas State University for providing a great working environment and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Yang was also partially supported by grants from the Simons Foundation (#499532, #918096, to YY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' The authors would also like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Richard Stanley for his help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 8 References [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Ahanjideh, Nondivisibility among irreducible character co-degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Mallory Dolorfino, Kalamazoo College, Kalamazoo, Michigan, USA, mallory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='dolorfino19@kzoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='edu Luke Martin, Gonzaga University, Spokane, Washington, USA, lwmartin2019@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='com Zachary Slonim, University of California, Berkeley, Berkeley, California, USA, zachslonim@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='edu Yuxuan Sun, Haverford College, Haverford, Pennsylvania, USA, ysun1@haverford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='edu Yong Yang, Texas State University, San Marcos, Texas, USA, yang@txstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='edu 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} diff --git a/EtAyT4oBgHgl3EQfSfdv/content/tmp_files/2301.00087v1.pdf.txt b/EtAyT4oBgHgl3EQfSfdv/content/tmp_files/2301.00087v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2b9f6d19f564a6681cd32053ba5e1bfe84027665 --- /dev/null +++ b/EtAyT4oBgHgl3EQfSfdv/content/tmp_files/2301.00087v1.pdf.txt @@ -0,0 +1,2084 @@ +Mechanical feedback linearization of single-input +mechanical control systems +Marcin Nowicki1 and Witold Respondek2,3 +1Poznan University of Technology, Institute of Automatic Control +and Robotics, Piotrowo 3a, 61-138 Pozna´n, Poland +2Lodz University of Technology, Institute of Automatic Control, B. +Stefanowskiego 18, 90-537 Lodz, Poland +3INSA de Rouen Normandie, Laboratoire de Math´ematiques, +76801 Saint-Etienne-du-Rouvray, France +January 3, 2023 +Abstract +We present a new type of feedback linearization that is tailored for me- +chanical control systems. We call it a mechanical feedback linearization. +Its basic feature is preservation of the mechanical structure of the system. +For mechanical systems with a scalar control, we formulate necessary and +sufficient conditions that are verifiable using differentiations and algebraic +operations only. We illustrate our results with several examples. +1 +Introduction +An N-dimensional control-affine system with a scalar control +˙z = F(z) + G(z)u, +(Σ) +where z ∈ Z, an open subset of RN, and u ∈ R, is said to be (locally) feedback +linearizable (F-linearizable) if there exist a (local) diffeomorphism Φ : Z → RN +and an invertible feedback of the form u = α(z) + β(z)˜u such that the control +system (Σ), in the new coordinates ˜z = Φ(z) and with the new control ˜u, is a +controllable linear system of the form ˙˜z = A˜z + b˜u. A geometric solution to the +problem of feedback linearization (inspired by [1], and developed independently +in [2] and [3]) provides powerful techniques for designing a closed-loop control +system that have been used in numerous engineering applications. +From a +theoretical point of view, that result identifies a class of nonlinear systems that +can be considered as linear ones in a well-chosen coordinates and with respect +to a well-modified control. +1 +arXiv:2301.00087v1 [math.OC] 31 Dec 2022 + +In this paper, we state and study the following fundamental question: if a +nonlinear control system (Σ) is mechanical and feedback linearizable, are those +two structures compatible? That is, can we feedback linearize the system pre- +serving its mechanical structure? For mechanical control systems, it is natural +to consider mechanical feedback equivalence (in particular, to a linear form) +under mechanical transformations (coordinates changes and feedback) that pre- +serve the mechanical structure of the system. In our recent paper [4], we showed +that even in the simplest underactuated case of 2 degrees of freedom, the struc- +tures (linear and mechanical) may not conform trivially. In the present paper, +we treat the single-input case in its full generality. +There are several motivations for preserving the mechanical structure when +feedback linearizing the system. First, our formulation of the problem of me- +chanical linearization preserves configurations and velocities. We reckon that it +is essential that new configurations (of the linearized system) are functions of the +original configurations only, as well as new velocities are true physical velocities +(in contrast to pseudo-velocities). Therefore, we do not lose the physical inter- +pretation of the system. This could be useful, e.g. for mechanical systems with +constraints on configurations, which are transformed into linear constraints on +configurations. Second, the configuration trajectories are preserved too, which +could be useful in e.g. the motion planning problem (the most natural way to +state the problem for mechanical systems is to follow configuration trajectories). +Third, it is worth mentioning that mechanical feedback linearizability guaran- +tees the linearizing outputs to be functions of configurations only. This may +be of constructional importance because one needs only configuration sensors, +not those of velocities. The next argument is the fact that the resultant linear +mechanical system allows us to employ dedicated techniques for mechanical sys- +tems. An example of such technique is the natural frequency method of tuning +a linear feedback. Finally, when applying mechanical feedback linearization, the +physical interpretation of the external action (force, torque, etc.) is preserved +but is lost for general feedback linearization. +This work is a mechanical counterpart of the classical results on feedback +linearization of control systems [1], [2], [3], see also monographs [6], [7]. Our +intention is to formulate conditions for mechanical linearization (shortly, MF- +linearization) in a possibly similar manner (e.g. using involutivity of certain +distributions). +For a geometric approach to mechanical control systems see [5], [8], [9], [10]. +For mathematical preliminaries concerning the Lie derivative, the Lie bracket, +distributions, etc., see [6], [7]. For linearization of mechanical control systems +along controlled trajectories see [11]. For mechanical state-space linearization of +mechanical control systems see [12] and [13]. Compare also [14], for a pioneering +work on (partial) feedback linearization of mechanical systems. +Although the state-space of mechanical control system is the tangent bundle +TQ of the configuration space Q, we formulate our conditions using objects on +Q only. The key here is a geometric approach to mechanical systems [5] and +considering the Euler-Lagrange equations as the geodesic equation under an +influence of external forces. +2 + +The outline of the paper is as follows. In Section 2, we state the problem. In +Section 3, we develop further the problem of mechanical feedback linearization +and formulate the main result, separately, for mechanical systems with n ≥ 3 +in Theorem 1, and with n = 2 in Theorem 2. +In Section 4, we provide an +application of our results to MF-linearization of several mechanical systems. +Section 6 contains technical results used in proofs that could be of independent +interest. +1.1 +Notation +Throughout the Einstein summation convention is assumed, i.e. any expression +containing a repeated index (upper and lower) implies the summation over that +index up to n, e.g. ωiXi = �n +i=1 ωiXi. +AT +transpose of a matrix (of a vector) A, +In +n × n identity matrix, +Q +configuration manifold, +X(Q) +the set of smooth vector fields on a manifold Q, +TxQ +tangent space at x ∈ Q, +TQ = � +x∈Q TxQ +tangent bundle of Q, +x = (x1, . . . , xn) +a local coordinate system on Q, +φ +a diffeomorphism of Q, and Φ a diffeomorphism of TQ, +Dφ = ∂φ +∂x +the Jacobian matrix of a diffeomorphism φ, +∂˜xi +∂xj := ∂φi +∂xj +the (i, j)-element of the Jacobian matrix Dφ, +∂xj +∂˜xi +the (j, i)-element of the inverse of the Jacobian matrix Dφ, +LXα +Lie derivative of a function α defined as LXα = ∂α +∂xi Xi, +[X, Y ] = ∂Y +∂x X − ∂X +∂x Y = adXY +Lie bracket of vector fields, +∂ +∂xi +the i-th unity vector field, and dxi the i-th unity covector field, in a co- +ordinate system x = (x1, . . . , xn), +Ei = span +� +adj +eg, 0 ≤ j ≤ i +� +distribution on Q spanned by adj +eg, +∇ +covariant derivative, and ∇2 second covariant derivative, +Γi +jk +Christoffel symbols of the second kind of ∇, +2 +Problem statement +Consider an n-dimensional configuration space Q (an open subset of Rn or, in +general, an n-dimensional manifold) equipped with a symmetric affine connec- +tion ∇. The operator of the affine connection ∇ allows to define intrinsically the +acceleration as the covariant derivative ∇ ˙x(t) ˙x(t), see e.g. [5,8,17]. The covari- +ant derivative ∇ : X(Q) × X(Q) → X(Q) of an arbitrary vector field Y = Y i ∂ +∂xi +with respect to X = Xi ∂ +∂xi in coordinates reads +∇XY = +�∂Y i +∂xj Xj + Γi +jkXjY k +� ∂ +∂xi . +(1) +3 + +A mechanical control system (MS) is a 4-tuple (Q, ∇, g, e), where g and e are, +respectively, controlled and uncontrolled vector fields on Q. A curve x(t) : I → +Q, I ⊂ R, is a trajectory of (MS) if it satisfies the following equation +∇ ˙x(t) ˙x(t) = e (x(t)) + g (x(t)) u, +(2) +which can be viewed as an equation that balances accelerations of the system, +where the left-hand side represents geometric accelerations (i.e. accelerations +caused by the geometry of the system) and the right-hand side represents ac- +celerations caused by external actions on the system (controlled or not). Notice +that (2) is a second-order differential equation on Q (indeed, using (1) we con- +clude that ∇ ˙x ˙x depends on ¨x, see [5] for details) and can be rewritten as a +system of first-order differential equations on TQ, which we also call a mechan- +ical control system (MS): +˙xi = yi +˙yi = −Γi +jk(x)yjyk + ei(x) + gi(x)u, +(MS) +for 1 ≤ i ≤ n, where (x, y) = +� +x1, . . . , xn, y1, . . . , yn� +are local coordinates +on the tangent bundle TQ of the configuration manifold Q, and Γi +jk(x) are +Christoffel symbols of the affine connection ∇ that correspond to the Cori- +olis and centrifugal forces. +The vector fields e(x) = (e1(x), . . . , en(x))T and +g(x) = (g1(x), . . . , gn(x))T correspond to, respectively, uncontrolled and con- +trolled actions on the system. Throughout all objects are assumed to be smooth +and the word smooth means C∞-smooth. +Our obvious inspirations are Lagrangian mechanical control systems without +dissipative forces. For the correspondence between (MS) and the Lagrangian +equations of dynamics see [5], [8], [9] and our recent papers [13], [16]. However, +we will consider throughout a more general class of mechanical control systems +allowing for any symmetric (not necessarily a metric) connection and any (not +necessarily potential) vector field e(x). +Consider the group of mechanical feedback transformations GMF generated +by the following transformations: +(i) changes of coordinates in TQ given by Φ : TQ → T ˜Q +(x, y) �→ (˜x, ˜y) = Φ(x, y) = +� +φ(x), ∂φ +∂x(x)y +� +, +(3) +called a mechanical diffeomorphism, where φ : Q → ˜Q is a diffeomorphism +and ∂φ +∂x its Jacobian matrix, +(ii) mechanical feedback transformations, denoted (α, β, γ), of the form +u = γjk(x)yjyk + α(x) + β(x)˜u, +(4) +where γjk, α, β are smooth functions on Q satisfying +γjk = γkj, β(·) ̸= 0. The matrix γ = (γjk) represents a (0, 2)−tensor field. +4 + +Even if the diffeomorphism φ is possibly local on Q, the action of ∂φ +∂x(x) is always +global on fibers TxQ. +Definition 1. The system (MS) is MF-linearizable if there exist mechanical +feedback transformations (Φ, α, β, γ) ∈ GMF bringing (MS) into a linear con- +trollable mechanical system of the form +˙˜xi = ˜yi +˙˜yi = Ei +j ˜xj + bi˜u, +(LMS) +where (˜x, ˜y) are coordinates on TRn = Rn × Rn, the matrix E = (Ei +j) is an +n × n real-valued matrix, the vector field b = bi ∂ +∂˜xi is constant, and the pair +(E, b) is controllable (see [15]). +Represent (MS) as ˙z = F(z) + G(z)u, where z = (x, y) ∈ TQ, F = yi ∂ +∂xi + +� +−Γi +jk(x)yjyk + ei(x) +� +∂ +∂yi , and G = gi(x) ∂ +∂yi . The problem that we formulate +and solve in the paper is whether (MS) is MF-linearizable? That is, do there +exist Φ = (˜x, ˜y) = (φ, ∂φ +∂xy) and (α, β, γ) such that +∂Φ +∂z (z) +� +F + G(yT γy + α) +� +(z) = +� +˜y +E˜x +� +, +∂Φ +∂z (z) (Gβ) (z) = +�0 +b +� +? +Note that MF-linearizability is stronger than the classical feedback lineariz- +ability since, for the latter, Φ : TQ → R2n can be any diffeomorphism (need not +be of mechanical form (3)) and yT γ(x)y +α(x) can be replaced by any function +α(x, y) on TQ and β(x) by any invertible function β(x, y) on TQ. +If we neglect the mechanical structure of ˙z = F(z) + G(z)u, and consider +it as a general control system, we can ask if the system is F-linearizable. The +well-known answer [2,3] asserts that, locally, this is the case if and only if the +distributions Di = span +� +adj +F G, 0 ≤ j ≤ i +� +are involutive and of constant rank +for i = 0, ..., 2n − 1 and D2n−1 = TQ. The natural question arises whether, for +F-linearizable (MS), the general feedback transformations (Φ(z), α(z), β(z)) are +mechanical (i.e. of the form (3) and (4)) or whether they can be replaced by +mechanical ones. +Example 1: +Consider the mechanical system +˙x1 = y1 +˙x2 = y2 +˙y1 = −x2(y1)2 + x2 +˙y2 = u, +(5) +on R4. This system is locally F-linearizable. Indeed, the local diffeomorphism +˜z = Φ(z), where z = (x1, x2, y1, y2), ˜z = (˜x1, ˜x2, ˜y1, ˜y2), given by +˜x1 = x1 +˜x2 = x2 − x2(y1)2 +˜y1 = y1 +˜y2 = +� +(y1)2 − 1 +� � +2(x2)2y1 − y2� +, +5 + +together with the feedback u = 2(x2)3 +6(x2 −(x2)2)(y1)2 + +˜u +(y1)2−1, render the +original system linear and controllable +˙˜x1 = ˜y1 +˙˜x2 = ˜y2 +˙˜y1 = ˜x2 +˙˜y2 = ˜u. +Therefore, the system is F-linearizable. Note, however, that neither the change +of coordinates nor the feedback is mechanical (˜x2 depends on velocities, and +the function β depends on velocities as well) so the mechanical structure is +not preserved. Our question is whether this system can be linearized by other +transformations that preserve the mechanical structure, i.e. +can it be MF- +linearized? +The group of mechanical transformations GMF = {(Φ, α, β, γ)} preserves +trajectories, that is, maps the trajectories of (MS) into those of its MF-equivalent +system ( � +MS). Indeed, if z (t, z0, u(t)) is a trajectory of (MS) (passing through +z0 = (x0, y0) and corresponding to a control u(t)), then ˜z (t, ˜z0, ˜u(t)) = Φ (z (t, z0, u(t))) +is a trajectory of ( � +MS) passing through ˜z0 = Φ(z0) = (φ(x0), ∂φ +∂x(x0)y0) and +corresponding to ˜u(t), where u(t) = y(t)T γ (x(t)) y(t) + α (x(t)) + β (x(t)) ˜u(t). +Moreover, via φ : Q → ˜Q, it establishes a correspondence between configuration +trajectories in Q and ˜Q, i.e. ˜x (t, ˜z0, ˜u(t)) = φ (x(t, z0, u(t))), making the fol- +lowing diagram commutative (notice, however, that π (z(t, z0, u)) = x(t, z0, u) +depends on z0 = (x0, y0), i.e. an initial configuration x0 and initial velocity y0): +z(t, z0, u) +˜z(t, ˜z0, ˜u) +x(t, z0, u) +˜x(t, ˜z0, ˜u) +(Φ,α,β,γ) +π +π +(φ,α,β,γ) +where π : TQ → Q, π(z) = π(x, y) = x, is the canonical projection which +assigns to the pair (x, y) the point x at which the velocity y is attached. +3 +Mechanical feedback linearization +Our main result uses two basic ingredients: the covariant derivative of the con- +nection ∇, see (1), and the involutivity of suitable distributions. We will also +need the second covariant derivative of a vector field Z in the directions (X, Y ), +which is a mapping +∇2 : X(Q) × X(Q) × X(Q) → X(Q) +∇2 +X,Y Z = ∇X∇Y Z − ∇∇XY Z. +(6) +For properties of the second covariant derivative see Lemma 1 in Appendix. +In order to formulate the result, we associate with (MS) the following se- +quence of nested distributions E0 ⊂ E1 ⊂ E2 ⊂ . . . ⊂ Ei ⊂ . . . ⊂ TQ, where +E0 = span {g} , +Ei = span +� +adj +eg, 0 ≤ j ≤ i +� +. +6 + +Remark 1. To analyze the behavior of the distributions Ei under mechanical +feedback transformations (α, β, γ) notice, first, that Ei are invariant under γ +since γ does not act on them. If the distributions Ei are involutive, then they +are invariant under feedback transformations of the form (α, β, 0), i.e. for γ = 0 +they remain unchanged if we replaced e and g by, respectively, e + gα and βg, +cf. [6], [7]. +Now, we formulate our main result for MF-linearization. +First, we state +a theorem for (MS) with n ≥ 3 degrees of freedom. The remaining case of +n = 2 degrees of freedom is treated in Theorem 2. For an explanation of that +distinction, see the comment before Theorem 2 and Remark 3 for a comparison +of both results. +By a local MF-linearization around x0 ∈ Q we mean that it holds on +� +x∈O TxQ, where O is a neighborhood of x0; recall that all transformations +are global on tangent spaces TxQ. +Theorem 1. Assume n ≥ 3. A mechanical control system (MS) is, locally +around x0, MF-linearizable to a controllable (LMS) if and only if +(MF1) rank En−1 = n, +(MF2) Ei is involutive and of constant rank, for 0 ≤ i ≤ n − 2, +(MF3) ∇adieg g ∈ E0 +for 0 ≤ i ≤ n − 1, +(MF4) ∇2 +adkeg,adj +eg e ∈ E1 +for 0 ≤ k, j ≤ n − 1, +Remark 2. Notice that (MF1)-(MF2) are the classical conditions (see [2,3,6, +7]) that assure F-linearization of the system ˙x = e(x)+g(x)u on Q via ˜x = φ(x) +and u = α(x)+β(x)˜u. The remaining two, (MF3)-(MF4), can be interpreted as +compatibility conditions that guarantee vanishing the Christoffel symbols Γi +jk in +the linearizing coordinates ˜x = φ(x), except for those that can be compensated +by feedback u = γjk(x)yjyk + ˜u. +Proof. In the proof we will use two Lemmata 1 and 2, given in Appendix, that +are of independent interest. +Necessity. For (LMS), we have Γi +jk = 0, e = Ex and g = b. It follows that +adi +eg = (−1)iEib and therefore, using the definitions of ∇, given by (1), and of +∇2, given by (6), we calculate +∇adiegadj +eg = 0, +∇2 +adkeg,adj +ege = 0, +(7) +which implies that (MF1)-(MF4) hold for (LMS) (in particular, (MF1) holds +because (LMS) is assumed controllable). To prove necessity of (MF1)-(MF4), +we will show that they are MF-invariant. +All conditions (MF1)-(MF4) are +expressed in a geometrical way, therefore they are invariant under diffeomor- +phisms. The conditions (MF1) and (MF2) are mechanical feedback invariant, +see Remark 1. It remains to show that (MF3) and (MF4) are invariant under the +7 + +mechanical feedback u = γjk(x)yjyk+α(x)+β(x)˜u. For the closed-loop system, +denoted by ”∼”, the Christoffel symbols ˜Γi +jk of ˜∇, ˜e, and ˜g are, respectively, +given by +˜Γi +jk = Γi +jk − giγjk, +˜e = e + gα, +˜g = gβ. +(8) +For any X, Y ∈ X(Q), we have ˜∇XY = ∇XY − γ(X, Y )g = ∇XY +mod E0, +where γ(X, Y ) = γjkXjY k ∈ C∞(Q), therefore +˜∇adi +˜e˜g˜g = ∇adi +˜e˜g˜g − γ(adi +˜e˜g, ˜g)g = ∇adi +˜e˜g˜g +mod E0. +By ∇X˜g = ∇X (gβ) = ∇Xg + (LXβ) g, it follows that instead of calculating +∇adi +˜e˜g˜g it is enough to calculate ∇adi +˜e˜gg, since the second term (LXβ) g ∈ E0. +For i=0, we have ∇˜gg = ∇(gβ)g = β∇gg ∈ E0. It is easy to show that for any +1 ≤ j ≤ n − 1, we have +adj +˜e˜g = βadj +eg + dj−1, +(9) +where dj−1 ∈ Ej−1. Assume ∇adl +˜e˜gg ∈ E0, for 0 ≤ l ≤ i − 1. Then, by formula +(9), ∇adi +˜e˜gg = β∇adiegg + ∇di−1g ∈ E0, because the first term is in E0 by (MF3) +and the second by the induction assumption. We have thus proved necessity of +(MF3). +To show necessity of (MF4), using Lemma 1, calculate +˜∇2 +X,Y Z = ˜∇X ˜∇Y Z − ˜∇ ˜∇XY Z += ˜∇X (∇Y Z − γ(Y, Z)g) − ˜∇(∇XY −γ(X,Y )g)Z += ∇2 +X,Y Z − γ(Y, Z)∇Xg + γ(X, Y )∇gZ +mod E0. +(10) +By the above formula, we get +˜∇2 +adk +˜e ˜g,adj +˜e˜g˜e =∇2 +adk +˜e ˜g,adj +˜e˜g˜e − γ(adj +˜e˜g, ˜e)∇adk +˜e ˜gg ++ γ(adk +˜e˜g, adj +˜e˜g)∇g˜e +mod E0. +The second term, on the right hand side, is in E0 (by (MF3) and its invariance), +while the third term is a function multiplying +∇g˜e = ∇g (e + gα) = ∇ge + α∇gg + Lgα g ∈ E1, +since for (LMS) we have ∇ge = −adeg = −Eb ∈ E1. +The first term ∇2 +adk +˜e ˜g,adj +˜e˜g˜e is, by (9) and Lemma 1(i), a linear combination +with smooth coefficients of ∇2 +adieg,adleg˜e, with 0 ≤ i ≤ k and 0 ≤ l ≤ j. Thus +we calculate ∇2 +adi +eg,adl +eg˜e = ∇2 +adieg,adlege+∇2 +adieg,adleg(gα). The first term vanishes +since (7) holds for (LMS). We calculate the second term using Lemma 1(iii), +and we have ∇2 +adi +eg,adl +eg(gα) = α∇2 +adieg,adlegg + Ladi +egα∇adlegg + Ladlegα∇adi +egg + +(∇2 +adi +eg,adl +egα)g ∈ E0 because the first three terms vanish, due to (7), and +8 + +the last one is in E0. Summarizing the above calculations, we conclude that +˜∇2 +adk +˜e ˜g,adj +˜e˜g˜e ∈ E1 = ˜E1, which proves necessity of (MF4). +Sufficiency. We will transform the system (MS), satisfying (MF1)-(MF4), +into (LMS) in two steps. In the first step, we will normalize the vector fields +e and g and show that condition (MF4) implies zeroing some of the Christoffel +symbols Γi +jk, which exhibit a triangular form in the normalizing coordinates. In +the second step, we compensate the remaining Christoffel symbols. +By conditions (MF1)-(MF2), there exists a function h satisfying Ladj +egh = 0, +for 0 ≤ j ≤ n − 2, and Ladn−1 +e +gh ̸= 0, and thus (˜x, ˜y) = (φ(x), ∂φ +∂x(x)y) is a +local mechanical diffeomorphism, where φ(x) = (Ln−1 +e +h, . . . , Leh, h)T that can +be completed by a feedback transformation (α, β, 0) that map, respectively, βg +into ˜g = (1, 0, . . . , 0)T , e + gα into ˜e = (0, ˜x1, . . . , ˜xn−1)T , and Γi +jk into ˜Γi +jk, see +the classical results of feedback linearization [2], [6], [7]. Then, (Φ, α, β, γ) ∈ +GMF , where (˜x, ˜y) = Φ(x, y) = +� +φ(x), ∂φ +∂x(x)y +� +with φ, α, β just defined and +γjk = ˜Γ1 +jk(˜x), brings (MS) into (we drop ”tildas” for readability) +˙x1 = y1 +˙xi = yi +˙y1 = u +˙yi = −Γi +jkyjyk + xi−1, +2 ≤ i ≤ n, +(11) +to which Lemma 2 applies. +We will show that the Christoffel symbols Γi +jk of (11) satisfy +Γi +kj = 0 +for 1 ≤ k ≤ n − 1, 1 ≤ j ≤ i ≤ n, +Γi +nj = +� +0 +for 1 ≤ j < i ≤ n +λ(xn) +for 2 ≤ j = i ≤ n. +(12) +For system (11), we have adk−1 +e +g = (−1)k−1 +∂ +∂xk and, in particular, g = +∂ +∂x1 . +Calculate ∇adk−1 +e +gg = (−1)k−1∇ +∂ +∂xk gi ∂ +∂xi = (−1)k−1∇ +∂ +∂xk +∂ +∂x1 = (−1)k−1Γi +k1 +∂ +∂xi . +It follows that Γi +k1 = Γi +1k = 0, for 2 ≤ i ≤ n by (MF3), and for i = 1 by the +above form. +Rewrite (MF4) as ∇2 +adk−1 +e +g,adj−1 +e +ge = 0 mod E1, for 1 ≤ j, k ≤ n, and apply +it successively for j = 1, . . . , n and for all 1 ≤ k ≤ n. For j = 1, first calculate +∇ge = ∇ +∂ +∂x1 e = +∂ +∂x2 + Γi +1ses ∂ +∂xi = +∂ +∂x2 +and then +∇adk−1 +e +g (∇ge) = (−1)k−1∇ +∂ +∂xk +∂ +∂x2 = (−1)k−1Γi +k2 +∂ +∂xi . +On the other hand, ∇adk−1 +e +gg = (−1)k−1∇ +∂ +∂xk +∂ +∂x1 = (−1)k−1Γ1 +k1 +∂ +∂x1 = 0 and +hence ∇∇adk−1 +e +gge = 0. Thus, by (6), +∇2 +adk−1 +e +g,ge = ∇adk−1 +e +g (∇ge) − ∇∇adk−1 +e +gge = += (−1)k−1Γi +k2 +∂ +∂xi = 0 +mod E1, +9 + +implying that Γi +k2 = Γi +2k = 0 for any 3 ≤ i ≤ n. +For j = 2, calculate +∇adege = −∇ +∂ +∂x2 e = − ∂ +∂x3 + Γi +2ses ∂ +∂xi = − ∂ +∂x3 − d +where d = d1(x) ∂ +∂x1 + d2(x) ∂ +∂x2 ∈ E1, and then +∇adk−1 +e +g (∇adege) = (−1)k∇ +∂ +∂xk +� ∂ +∂x3 + d +� += += (−1)k � +Γi +k3 + Γi +k1d1 + Γi +k2d2� ∂ +∂xi = += (−1)kΓi +k3 +∂ +∂xi +mod E1. +On the other hand, +∇adk−1 +e +gadeg = (−1)k∇ +∂ +∂xk +∂ +∂x2 = (−1)kΓi +k2 +∂ +∂xi = += (−1)k +� +Γ1 +k2 +∂ +∂x1 + Γ2 +k2 +∂ +∂x2 +� +and ∇∇adk−1 +e +gadege = (−1)kΓ2 +k2 +∂ +∂x3 +mod E1. It follows that, modulo E1, +∇2 +adk−1 +e +g,adege = (−1)k +� n +� +i=4 +Γi +k3 +∂ +∂xi + (Γ3 +k3 − Γ2 +k2) ∂ +∂x3 +� +, +and, using (MF4), we conclude Γi +k3 = Γi +3k = 0 for any 4 ≤ i ≤ n and Γ3 +k3 = Γ2 +k2. +Following the same line (with a more tedious calculation), one can prove the +general induction step. Namely, assuming, for a fixed j, +Γj +kj = Γj−1 +kj−1 +Γi +ks = Γi +sk = 0 +s + 1 ≤ i ≤ n, +1 ≤ s ≤ j, +(13) +one shows by calculating ∇2 +adk−1 +e +g,adj−1 +e +ge, with the help of (24) of Lemma 2, +that +Γj+1 +kj+1 = Γj +kj +Γi +kj+1 = 0 +for j + 2 ≤ i ≤ n +and thus, by the induction assumption and symmetry of the Christoffel symbols, +Γi +ks = Γi +sk = 0 +s + 1 ≤ i ≤ n, +1 ≤ s ≤ j + 1. +(14) +It follows that for each 1 ≤ k ≤ n the matrices consisting of Christoffel symbols +(Γi +kj), for 2 ≤ i, j ≤ n are upper triangular. By the induction argument, (13) +holds for all 2 ≤ j ≤ n and implies, for any 1 ≤ k ≤ n − 1, +Γ2 +k2 = . . . = Γn−1 +kn−1 = Γn +kn = 0. +10 + +since Γn +kn = Γn +nk = 0 (as n > k). On the other hand, for k = n, (13) implies +Γ2 +n2 = . . . = Γn−1 +nn−1 = Γn +nn = λ(x) +for a function λ(x). +Therefore for each 1 ≤ k ≤ n the matrices (Γi +kj), for +2 ≤ i, j ≤ n, are strictly upper triangular, and the last one, for k = n, is upper +triangular with all diagonal elements equal to each other, which we denote by +λ(x). The matrices read +� +Γi +kj +� += +� +� +� +� +� +� +� +� +� +0 +Γ2 +k3 +Γ2 +k4 +. . . +Γ2 +kn−2 +Γ2 +kn−1 +Γ2 +kn +0 +0 +Γ3 +k4 +. . . +Γ3 +kn−2 +Γ3 +kn−1 +Γ3 +kn +... +0 +0 +0 +. . . +0 +Γn−2 +kn−1 +Γn−2 +kn +0 +0 +0 +. . . +0 +0 +Γn−1 +kn +0 +0 +0 +. . . +0 +0 +0 +� +� +� +� +� +� +� +� +� +, +for 1 ≤ k ≤ n − 1, and +� +Γi +nj +� += +� +� +� +� +� +� +� +� +� +λ +Γ2 +n3 +Γ2 +n4 +. . . +Γ2 +nn−2 +Γ2 +nn−1 +Γ2 +nn +0 +λ +Γ3 +n4 +. . . +Γ3 +nn−2 +Γ3 +nn−1 +Γ3 +nn +... +0 +0 +0 +. . . +λ +Γn−2 +nn−1 +Γn−2 +nn +0 +0 +0 +. . . +0 +λ +Γn−1 +nn +0 +0 +0 +. . . +0 +0 +λ +� +� +� +� +� +� +� +� +� +, +and are thus of the desired triangular structure (12) and it remains to prove +that λ = λ(xn). Note that in the above matrices we skip the first row Γ1 +kj and +the first column Γi +k1. This is due to the fact that Γ1 +kj = 0 (which can always be +achieved by a suitable feedback transformation) and Γi +k1 = 0 by (MF3). +Notice +that we have En−2 = span +� ∂ +∂x1 , . . . , +∂ +∂xn−1 +� +and thus applying (24) of Lemma 2, +for j = n and any 1 ≤ k ≤ n, we conclude (set Γn +kn+1 = 0) +(−1)n+k−2∇2 +adk−1 +e +g,adn−1 +e +ge = ∇2 +∂ +∂xk , +∂ +∂xn e += +�∂Γn +ns +∂xk es + Γn +nk+1 + Γn +kn+1 − Γn−1 +kn ++ (Γd +nsΓn +kd − Γd +knΓn +ds)es +� ∂ +∂xn +mod En−2 += +� ∂λ +∂xk en + Γn +nk+1 − Γn−1 +kn +� ∂ +∂xn +mod En−2, +(15) +since, due to the triangular structure (14), Γn +ns = 0 except for s = n giving +Γn +nn = λ and, moreover, the equality Γd +nsΓn +kd−Γd +knΓn +ds = 0 holds. Indeed, in the +latter, Γn +kd = 0 except d = k = n giving Γn +nsΓn +nn − Γn +nnΓn +ns = 0 and Γn +ds = 0 +except for d = s = n giving Γn +nnΓn +kn − Γn +knΓn +nn = 0. +11 + +For (15) we will apply (MF4) in three cases. First, if 1 ≤ k ≤ n − 2, then, +modulo En−2, we have +� ∂λ +∂xk en + Γn +nk+1 − Γn−1 +kn +� +∂ +∂xn = +� ∂λ +∂xk xn−1 +� +∂ +∂xn = 0, +since all Γn +nk+1 = 0 and all Γn−1 +kn += 0 by (14) and k ≤ n − 2. +Second, for +k = n − 1, we have modulo En−2, +� +∂λ +∂xn−1 en + Γn +nn − Γn−1 +n−1n +� ∂ +∂xn = +� +∂λ +∂xn−1 en + λ − λ +� +∂ +∂xn += +� +∂λ +∂xn−1 xn−1 +� +∂ +∂xn = 0. +Therefore +∂λ +∂xk = 0, for 1 ≤ k ≤ n − 1, implying that λ is a function of the last +variable xn only, i.e. λ = λ(xn), which gives the system in the desired form (12) +Third, for k = n, we have modulo En−2, +� ∂λ +∂xn en+Γn +nn+1−Γn−1 +nn +� ∂ +∂xn = +� ∂λ +∂xn xn−1− Γn−1 +nn +� ∂ +∂xn = 0, +implying that Γn−1 +nn += Leλ, since ∂λ(xn) +∂xn xn−1 = Leλ. +Now, transform system (11), satisfying (12), via the local mechanical diffeo- +morphism Φ : TQ → T ¯Q +¯x = φ(x) +¯y = Dφ(x)y, +where +φ(x) = +� +Ln−1 +e +h, . . . , Leh, h +�T , +(16) +with h(xn) = +� xn +0 +Λ(s2)ds2, where Λ(s2) = exp +�� s2 +0 λ(s1)ds1 +� +. +Denote by ¯Γi +jk, ¯e, ¯g the objects of the system expressed in coordinates ¯x = +φ(x). Applying feedback ¯u = −¯Γ1 +jk¯yj ¯yk + Ln +e h + uLgLn−1 +e +h, the transformed +system becomes +˙¯x1 = ¯y1 +˙¯xi = ¯yi +˙¯y1 = ¯u +˙¯yi = −¯Γi +jk¯yj ¯yk + ¯xi−1, +2 ≤ i ≤ n, +(17) +whose vector fields are ¯e = ¯xi−1 ∂ +∂¯xi , where x0 = 0, and ¯g = +∂ +∂¯x1 . Transformed +system (17) is still of the form (11) and at the moment we ignore how Γi +jk have +been changed into ¯Γi +jk. Below we will prove that all ¯Γi +jk vanish. To this end, +we first calculate explicitly the time-evolution of the pair (¯xn, ¯yn) +˙¯xn = d +dth(xn) = Λ(xn) ˙xn = Λ(xn)yn = ¯yn +˙¯yn = d +dt (Λ(xn)yn) = Λ(xn)λ(xn) ˙xnyn + Λ(xn) ˙yn += Λ(xn)λ(xn)ynyn + Λ(xn) ˙yn += Λ(xn)λ(xn)ynyn + Λ(xn) +� +−Γn +nn(xn)ynyn + xn−1� += Λ(xn)xn−1 = ¯xn−1, +12 + +since ¯xn−1 = Leh = Λ(xn)xn−1. It follows that ¯Γn +jk = 0, for all 1 ≤ k, j ≤ n. +For transformed system (17), we rewrite (24) by adding ”bars” as +∇2 +adk−1 +¯e +¯g,adj−1 +¯e +¯g¯e = (−1)j+k +�∂¯Γi +js +∂¯xk ¯es + ¯Γi +jk+1 ++ ¯Γi +kj+1 + (¯Γd +js¯Γi +kd − ¯Γd +kj ¯Γi +ds)¯es − ¯Γi−1 +kj +� ∂ +∂¯xi +(18) +and by (MF4), we have +∇2 +adk−1 +¯e +¯g,adj−1 +¯e +¯g¯e = (−1)j+k¯an +kj(¯x) ∂ +∂¯xn = 0 +mod En−2, +where ¯an +kj(¯x) = +∂¯Γn +js +∂¯xk ¯es + ¯Γn +jk+1 + ¯Γn +kj+1 + (¯Γd +js¯Γn +kd − ¯Γd +kj ¯Γn +ds)¯es − ¯Γn−1 +kj , which +implies (since ¯Γn +kj = 0, for 1 ≤ j, k ≤ n) that ¯an +kj(¯x) = ¯Γn−1 +kj += 0. Now assume +¯Γi +kj = 0 for a certain 1 ≤ i ≤ n − 1 and any 1 ≤ j, k ≤ n. Then (18) and (MF4) +imply ¯Γi−1 +kj += 0. Therefore we have proved that all Christoffel symbols of (17) +vanish and thus the system is a linear controllable (LMS), since the vector field +¯e = ¯xi−1 ∂ +∂¯xi is linear and ¯g = +∂ +∂¯x1 is constant. +The above theorem does not work for systems with 2 degrees of freedom, +i.e. for n=2, as that case is too restrictive for involutivity, see Remark 3 below. +Therefore we state the following theorem for MF-linearization of (MS) with 2 +degrees of freedom. +Theorem 2. A mechanical system (MS) with 2 degrees of freedom is, locally +around x0, MF-linearizable to a controllable linear (LMS) if and only if it +satisfies in a neighborhood of x0 +(MF1)’ g and adeg are independent at x0, +(MF3)’ ∇g g ∈ E0 and ∇adeg g ∈ E0, +(MF5)’ ∇2 +g,adeg adeg − ∇2 +adeg,g adeg ∈ E0. +Remark 3. If n = 2, then E0 is of rank 1, thus involutive and (MF2) is trivially +satisfied, and so is (MF4) because E1 = TQ (cf. Theorem 1). Therefore (MF2)’ +and (MF4)’ are absent and replaced by (MF5)’ that guarantees that we can +compensate the Christoffel symbols (as do (MF3)-(MF4) for n ≥ 3). +Proof. Necessity. Note that (MF1)’ is equivalent to (MF1) and (MF3)’ is (MF3) +of Theorem 1. Although Theorem 1 applies to n ≥ 3, the necessity part of its +proof remains valid for any n ≥ 2 so it shows necessity of (MF1)’-(MF3)’. +Therefore we need to show necessity of (MF5)’. For a controllable (LMS) we +have Γi +jk = 0, g = b and adeg = −Eb are independent, and +∇adiegadj +eg = 0, ∇2 +adj +eg,adkegadi +eg = 0, +� +adj +eg, adk +eg +� += 0, +(19) +13 + +for 0 ≤ i, j, k ≤ 1. We will use formula (10) to show that (MF5)’ is invariant +under mechanical feedback. Denote ˜∇, ˜e, ˜g, γ as in (8). Then we calculate +˜∇2 +˜g,ad˜e˜gad˜e˜g =∇2 +˜g,ad˜e˜gad˜e˜g − γ(ad˜e˜g, ad˜e˜g)∇˜g˜g ++ γ(˜g, ad˜e˜g)∇˜gad˜e˜g +mod E0, +˜∇2 +ad˜e˜g,˜gad˜e˜g =∇2 +ad˜e˜g,˜gad˜e˜g − γ(g, ad˜e˜g)∇ad˜e˜g˜g ++ γ(ad˜e˜g, ˜g)∇˜gad˜e˜g +mod E0. +The second terms of the right hand side of both equations are in E0 due to the +feedback invariance of (MF3)’, while the third terms are equal since γ(X, Y ) = +γ(Y, X) is symmetric. Therefore we conclude +˜∇2 +˜g,ad˜e˜gad˜e˜g − ˜∇2 +ad˜e˜g,˜gad˜e˜g += ∇2 +˜g,ad˜e˜gad˜e˜g − ∇2 +ad˜e˜g,˜gad˜e˜g +mod E0. +Denoting ad˜e˜g = βadeg + d0g (see (9)) and by Lemma 1 (i), we have +∇2 +˜g,ad˜e˜gad˜e˜g = ∇2 +βg,βadeg+d0gad˜e˜g += β2∇2 +g,adegad˜e˜g + βd0∇2 +g,gad˜e˜g +∇2 +ad˜e˜g,˜gad˜e˜g = ∇2 +βadeg+d0g,βgad˜e˜g += β2∇2 +adeg,gad˜e˜g + βd0∇2 +g,gad˜e˜g, +where the last terms on the right are equal, implying +∇2 +˜g,ad˜e˜gad˜e˜g − ∇2 +ad˜e˜g,˜gad˜e˜g += β2 � +∇2 +g,adegad˜e˜g − β2∇2 +adeg,gad˜e˜g +� +and it remains to prove that ∇2 +g,adegad˜e˜g − ∇2 +adeg,gad˜e˜g ∈ E0, which we show +using Lemma 1(iii), where X, Y stand for either g or adeg. Denote ∇Xβ = LXβ +and ∇2 +X,Y β = LXLY β − L∇XY β (see Lemma 1) and calculate +∇2 +X,Y ad˜e˜g = ∇2 +X,Y +� +βadeg + d0g +� += β∇2 +X,Y adeg ++ LXβ∇Y adeg + LY β∇Xadeg + +� +∇2 +X,Y β +� +adeg ++ d0∇2 +X,Y g + LXd0∇Y g + LY d0∇Xg + +� +∇2 +X,Y d0� +g += +� +∇2 +X,Y β +� +adeg +mod E0, +since all ∇2 +X,Y X = 0 and ∇XY = 0 , see (19). Therefore we have +∇2 +g,adegad˜e˜g − ∇2 +adeg,gad˜e˜g += +� +∇2 +g,adegβ − ∇2 +adeg,gβ +� +adeg +mod E0. +Finally, we calculate +∇2 +g,adegβ − ∇2 +adeg,gβ = LgLadegβ − L∇gadegβ +− +� +LadegLgβ − L∇adeggβ +� += L[g,adeg]β = 0, +14 + +which shows necessity of (MF5)’. +Sufficiency. +By (MF1)’, rank E1 = 2, and E0 = span {g} is of constant +rank 1 and thus always involutive, hence the system is, locally around x0 (since +g(x0) ̸= 0), MF-equivalent to (cf. (11)) +˙x1 = y1 +˙x2 = y2 +˙y1 = u +˙y2 = −Γ2 +jkyjyk + x2. +We have g = +∂ +∂x1 , adeg = − ∂ +∂x2 and now we calculate +∇gg = Γ2 +11 +∂ +∂x2 +∇adegg = −Γ2 +12 +∂ +∂x2 , +which by (MF3)’ are in E0 = span +� ∂ +∂x1 +� +, implying Γ2 +11 = Γ2 +12 = Γ2 +21 = 0. It +follows ∇gg = ∇adegg = ∇gadeg = 0, and ∇adegadeg = Γ2 +22 +∂ +∂x2 and thus +∇2 +g,adeg adeg − ∇2 +adeg,g adeg = ∇g∇adegadeg +− ∇∇gadegadeg − ∇adeg∇gadeg − ∇∇adeggadeg += ∇g∇adegadeg = ∇ +∂ +∂x1 Γ2 +22 +∂ +∂x2 = ∂Γ2 +22 +∂x1 +∂ +∂x2 +implying, by (MF5)’, ∂Γ2 +22 +∂x1 = 0, i.e. Γ2 +22(x2) = λ(x2). +Now, we transform the system via the local mechanical diffeomorphism Φ : +TQ → T ¯Q (compare to (16)) +¯x = φ(x) +¯y = Dφ(x)y, +where +φ(x) = (Leh, h)T , +with h(x2) = +� x2 +0 +Λ(s2)ds2 and Λ(s2) = exp +�� s2 +0 λ(s1)ds1 +� +. +We calculate the evolution of the pair (¯x(t), ¯y(t)) of transformed coordinates, +using +d +dth +� +x2(t) +� += Λ +� +x2(t) +� +˙x2(t) and +d +dtΛ +� +x2(t) +� += λ +� +x2(t) +� +Λ +� +x2(t) +� +˙x2(t); +first we get +˙¯x2 = d +dth(x2) = Λ(x2)y2 = ¯y2 +˙¯y2 = Λ(x2)λ(x2)y2y2 + Λ(x2) ˙y2 = Λ(x2)λ(x2)y2y2 ++ Λ(x2) +� +−λ(x2)y2y2 + x2� += Λ(x2)x1 = ¯x1 +and then +˙¯x1 = Λ(x2)y1 + d +dtΛ +� +x2(t) +� +x1y2 = ¯y1 +˙¯y1 = −¯Γ1 +jk¯yj ¯yk + L2 +eh + uLgLeh, +where we denote by ¯Γ1 +jk the Christoffel symbols in the ˙¯y1-equation of the trans- +formed system. Applying the feedback ¯u = −¯Γ1 +jk¯yj ¯yk + L2 +eh + uLgLeh, we get +a controllable linear mechanical system in the canonical form ˙¯x1 = ¯y1, ˙¯y1 = +¯u, ˙¯x2 = ¯y2, ˙¯y2 = ¯x1. +15 + +4 +Examples +Example 1 (cont.): For system (5), we have g = +∂ +∂x2 and adeg = − ∂ +∂x1 are in- +dependent. We check MF-linearizability using Theorem 2. A simple calculation +shows that ∇gg = ∇adegg = 0 ∈ E0, but ∇2 +g,adeg adeg−∇2 +adeg,g adeg = +∂ +∂x1 /∈ E0, +therefore the system is not MF-linearizable. +Thus (5) is an example of a system that is F-linearizable but not MF- +linearizable. For such systems the choice is: either to F-linearize for the price +of loosing the mechanical structure or to keep the mechanical structure but to +get rid of the linearization. +Example 2: Consider the equation of dynamics of the Inertia Wheel Pen- +dulum [18] with constant parameters m0, md, J2: +˙x1 = y1, +˙x2 = y2, +˙y1 = e1 + g1u, +˙y2 = e2 + g2u, +e1 = m0 +md sin x1, e2 = − m0 +md sin x1, g1 = − 1 +md , g2 = md+J2 +J2md . +We will verify whether the conditions of Theorem 2 are satisfied. First, we +calculate adeg = ( m0 +m2 +d cos x1) ∂ +∂x1 − ( m0 +m2 +d cos x1) ∂ +∂x2 . It can be checked that g and +adeg are independent for x1 ̸= ± π +2 , which corresponds to the horizontal position +of the pendulum, therefore (MF1)’ is satisfied everywhere except for x1 = ± π +2 . +Next, we verify condition (MF2)’ by calculating ∇gg = ∇adegg = 0 ∈ E0. +Finally, a direct calculation shows +∇2 +g,adeg adeg = ∇2 +adeg,g adeg = += (m2 +0 +m5 +d +cos2 x1) ∂ +∂x1 − (m2 +0 +m5 +d +cos2 x1) ∂ +∂x2 , +thus ∇2 +g,adeg adeg − ∇2 +adeg,g adeg = 0 ∈ E0 satisfies (MF5)’. The system is MF- +linearizable. A linearizing function is h(x) = md+J2 +J2 +x1 + x2 (all others giving +MF-linearization are of the form σ h(x), where σ ∈ R\ {0}). Due to the proof of +Theorem 2, the linearizing diffeomorphism is (˜x, ˜y) = Φ(x, y) = (φ(x), Dφ(x)y) +with φ(x) = (h, Leh)T . The system in new coordinates reads +˙˜x1 = md + J2 +J2 +y1 + y2 = ˜y1 +˙˜y1 = md + J2 +J2 +�m0 +md +sin x1 − 1 +md +u +� +− m0 +md +sin x1 + md + J2 +m2J2 +u += m0 +J2 +sin x1 = Leh = ˜x2 +(20) +˙˜x1 = m0 +J2 +cos x1y1 = ˜y2 +˙˜y2 = −m0 +J2 +sin x1y1y1 + +m2 +0 +2mdJ2 +sin(2x1) − +m0 +mdJ2 +cos x1u = ˜u. +Example 3: We will study MF-linearizability of the TORA3 system (see +Figure 1), which is based on the TORA system (Translational Oscillator with +16 + +Figure 1: The TORA3 system +Rotational Actuator) studied in the literature, e.g. [19] (however we add gravita- +tional effects). It consists of a two dimensional spring-mass system, with masses +m1, m2 and spring constants k1, k2, respectively. A pendulum of length l3, mass +m3, and moment of inertia J3 is added to the second body. The displacements +of the bodies are denoted by x1 and x2, respectively, and the angle of the pen- +dulum by x3. The gravitational constant is a and u is a torque applied to the +pendulum. The kinetic energy is +T =1 +2m1( ˙x1)2 + 1 +2(m2 + m3)( ˙x2)2 ++ 1 +2(J3 + m3l2 +3)( ˙x3)2 + m3l3 cos x3 ˙x2 ˙x3, +and the mass matrix depends on the configurations. The potential energy is +V = 1 +2k1(x1)2 + 1 +2k2(x2 − x1)2 − m3l3a cos x3. The equations of the dynamics +read +m1¨x1 + k1x1 − k2 +� +x2 − x1� += 0 +(m2 + m3)¨x2 + m3l3 cos x3¨x3 − m3l3 sin x3( ˙x3)2 ++k2 +� +x2 − x1� += 0 +m3l3 cos x3¨x2 + (m3l2 +3 + J3)¨x3 + m3l3a sin x3 = u, +which can be rewritten on TQ as +˙x1 = y1 +˙y1 = η1 +˙x2 = y2 +˙y2 = −¯Γ2 +33y3y3 + η2 + τ 2u +˙x3 = y3 +˙y3 = −¯Γ3 +33y3y3 + η3 + τ 3u +(21) +where ¯Γ2 +33 = +−ν0 sin x3 +ν1+ν2 sin2 x3 , ¯Γ3 +33 = ν2 sin x3 cos x3 +ν1+ν2 sin2 x3 , η1 = − k1 +m1 x1 + k2 +m3 +� +x2 − x1� +, η2 = +1 +2 ν2a sin 2x3−ν3(x2−x1) +ν1+ν2 sin2 x3 +, +η3 = +ν4(x2−x1) cos x3−ν5 sin x3 +ν1+ν2 sin2 x3 +, τ 2 = −m3l3 cos x3 +ν1+ν2 sin2 x3 , τ 3 = +m2+m3 +ν1+ν2 sin2 x3 , with constant +parameters: +ν0 = m3l3(m3l2 +3 + J3), ν1 = m2m3l2 +3 + J3(m2 + m3), ν2 = m2 +3l2 +3, ν3 = +k2 +� +m3l2 +3 + J3 +� +, ν4 = m3l3k2 ν5 = m3l3a(m2 + m3). +17 + +-- +m1 +m2 +ki +k2 +W +-- +W +aTo simplify calculations we apply to the system a preliminary mechanical +feedback1 u = +1 +τ 3 +�¯Γ3 +33y3y3 − η3 + ¯u +� +which yields +˙x1 = y1 +˙x2 = y2 +˙x3 = y3 +˙y1 = −µ1x1 + µ2x2 +˙y2 = µ3 sin x3y3y3 + µ4(x1 − x2) − µ3 cos x3u +˙y3 = ¯u, +(22) +with µ1 = k1+k2 +m1 , µ2 = k2 +m1 , µ3 = +m3l3 +m2+m3 , µ4 = +k2 +m2+m3 . +Since conditions (MF1)-(MF4) of Theorem 1 are MF-invariant, we will check +them for system (22). To summarize: +Γ2 +33 = −µ3 sin x3, +and +Γi +jk = 0 +otherwise, +e = +� +−µ1x1 + µ2x2� ∂ +∂x1 + µ4 +� +x1 − x2� ∂ +∂x2 +g = −µ3 cos x3 ∂ +∂x2 + +∂ +∂x3 = g2 ∂ +∂x2 + +∂ +∂x3 . +We have (notice that calculations are performed on Q only) +adeg = +� +µ2µ3 cos x3� ∂ +∂x1 − +� +µ3µ4 cos x3� ∂ +∂x2 , +ad2 +eg = µ3 cos x3 +� +(µ1µ2 + µ2µ4) ∂ +∂x1 − +� +µ2µ4 + µ2 +4 +� ∂ +∂x2 +� +, +therefore rank E2 = 3 for x3 ̸= ± π +2 , and (MF1) is satisfied. Now +[g, adeg] = − +� +µ2µ3 sin x3� ∂ +∂x1 + +� +µ3µ4 sin x3� ∂ +∂x2 ∈ E1 +and (MF2) is satisfied. Then, for any vector field v = vi(x) ∂ +∂xi , +∇vg = +�∂g2 +∂x3 + Γ2 +33 +� +v3 ∂ +∂x2 = 0, +thus (MC3) is satisfied if we replace v by, in particular, g, adeg, ad2 +eg. Finally, +for (MF4), we calculate +∇2 +g,ge = +� +µ2µ3 sin x3� ∂ +∂x1 − +� +µ3µ4 sin x3� ∂ +∂x2 ∈ E1, +∇2 +adkeg,adj +ege = 0 +otherwise, +thus, the system is MF-linearizable. Now, choose h = +µ4 +µ2 x1 + x2 + µ3 sin x3 +(whose differential dh annihilates g and adeg), thus we take a linearizing diffeo- +morphism (˜x, ˜y) = +� +φ(x), ∂φ +∂x(x)y +� +, with φ(x) = +� +h, Leh, L2 +eh +�T . The linearized +1This preliminary feedback is not necessary and it is possible to check the conditions and +to linearize the system without it, since our method and conditions are feedback invariant. +18 + +system is in the form of (LMS) and reads +˙˜x1 = µ4 +µ2 +y1 + y2 + µ3 cos x3y3 = ˜y1 +˙˜y1 = µ4 +µ2 +˙y1+ ˙y2+µ3(cos x3 ˙y3−sin x3y3y3)= µ4(µ2 − µ1) +µ2 +x1 = ˜x2 +˙˜x2 = µ4(µ2 − µ1) +µ2 +y1 = ˜y2 +˙˜y2 = µ4(µ2 − µ1) +µ2 +˙y1 = µ4(µ2 − µ1) +µ2 +� +µ2x2 − µ1x1� += ˜x3 +˙˜x3 = µ1µ4(µ1 − µ2) +µ2 +y1 + µ4(µ2 − µ1)y2 = ˜y3 +˙˜y3 = (µ2 − µ1)µ3µ4 sin x3y3y3 − (µ1 − µ2)(µ2 +1 + µ2µ4)µ4 +µ2 +x1 ++ (µ1 − µ2)(µ1 + µ4)µ4x2 + (µ1 − µ2)µ3µ4 cos x3u = ˜u. +5 +Conclusions +In this paper, we consider MF-linearization of mechanical control systems (MS) +with scalar control. We formulate the problem as a particular case of feedback +linearization preserving the mechanical structure of (MS) so that the trans- +formed system is both linear and mechanical. As we showed in [4] and confirmed +in this paper, even in the simplest case, the class of MF-linearizable systems is +substantially smaller than that of general F-linearizable ones. Therefore, a nat- +ural question arises, namely to compare the conditions presented in this paper +with those for F-linearization. +The answer lies in the interplay between the +distributions Ei = span +� +adj +eg, 0 ≤ j ≤ i +� +and the ”usual” for F-linearization +Di = span +� +adj +F G, 0 ≤ j ≤ i +� +. We will address this problem in the future. +6 +Appendix +The following lemma can be proved by a direct calculation. +Lemma 1. The second covariant derivative ∇2 +X,Y Z satisfies the following prop- +erties: +(i) linearity over C∞(Q) in X and Y : +∇2 +(α1X1+α2X2),Y Z = α1∇2 +X1,Y Z + α2∇2 +X2,Y Z +∇2 +X,(α1Y1+α2Y2)Z = α1∇2 +X,Y1Z + α2∇2 +X,Y1Z +(ii) linearity over R in Z: +∇2 +X,Y (a1Z1 + a2Z2) = a1∇2 +X,Y Z1 + a2∇2 +X,Y Z2 +19 + +(iii) the product rule: +∇2 +X,Y (βZ) =β∇2 +X,Y Z + LXβ∇Y Z ++ LY β∇XZ + +� +∇2 +X,Y β +� +Z, +where ∇2 +X,Y β = LXLY β−L∇XY β ∈ C∞(Q), Xi, Yi, Zi ∈ X(Q), αi, β ∈ C∞(Q), +and ai ∈ R. +The following lemma is crucial for the proof of Theorem 1. +Lemma 2. For the system +˙x1 = y1 +˙xi = yi +˙y1 = u +˙yi = −Γi +jkyjyk + xi−1, +2 ≤ i ≤ n, +(23) +we have for any 1 ≤ k, j ≤ n, +∇2 +adk−1 +e +g,adj−1 +e +ge = (−1)j+k +�∂Γi +js +∂xk es + Γi +jk+1 + Γi +kj+1 − Γi−1 +kj ++ (Γd +jsΓi +kd − Γd +kjΓi +ds)es +� ∂ +∂xi . +(24) +Proof. For system (23) we calculate ∇2 +adk−1 +e +g,adj−1 +e +ge = (−1)j+k∇2 +∂ +∂xk , +∂ +∂xj e = +∇ +∂ +∂xk ∇ +∂ +∂xj e − ∇∇ +∂ +∂xk +∂ +∂xj e, +where ∇ +∂ +∂xj e = +� +∂ed +∂xj + Γd +jses� +∂ +∂xd , and +∇ +∂ +∂xk +� +∇ +∂ +∂xj e +� += ∇ +∂ +∂xk +� ∂ed +∂xj +� +∂ +∂xd + ∇ +∂ +∂xk +� +Γd +jses� +∂ +∂xd += ∂ed +∂xj ∇ +∂ +∂xk +∂ +∂xd + L +∂ +∂xk +� ∂ed +∂xj +� +∂ +∂xd + +� +Γd +jses� +∇ +∂ +∂xk +∂ +∂xd ++ +� +L +∂ +∂xk +� +Γd +js +� +es + L +∂ +∂xk (es) Γd +js +� +∂ +∂xd += ∂ed +∂xj Γi +kd +∂ +∂xi + Γd +jsesΓi +kd +∂ +∂xi + +� +∂Γi +js +∂xk es + ∂es +∂xk Γi +js +� +∂ +∂xi += +� +∂Γi +js +∂xk es + Γi +jk+1 + Γi +kj+1 + Γd +jsΓi +kdes +� +∂ +∂xi , +since +∂ed +∂xj = 1, if d = j + 1, and zero otherwise, and thus +∂ed +∂xj Γi +kd = Γi +kj+1 +(analogously for the other derivatives). +Now, using ∇ +∂ +∂xk +∂ +∂xj = Γd +kj +∂ +∂xd , we +calculate +∇∇ +∂ +∂xk +∂ +∂xj e = ∇Γd +kj +∂ +∂xd e = Γd +kj +� ∂ei +∂xd + Γi +dses +� ∂ +∂xi += +� +Γi−1 +kj ++ Γd +kjΓi +dses� ∂ +∂xi , +so we have +20 + +∇2 +∂ +∂xk , +∂ +∂xj e = ∇ +∂ +∂xk +� +∇ +∂ +∂xj e +� +− ∇∇ +∂ +∂xk +∂ +∂xj e += +�∂Γi +js +∂xk es + Γi +jk+1 + Γi +kj+1 − Γi−1 +kj ++ (Γd +jsΓi +kd − Γd +kjΓi +ds)es +� ∂ +∂xi . +which yields (24). +References +[1] R. 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Spong, ”Feedback linearization of robot ma- +nipulators and Riemannian curvature”, Journal of Robotic Systems, vol. +12, no. 8, pp. 541-552, 1995. +[15] P. C. Hughes and R. E. Skelton, ”Controllability and observability of linear +matrix-second-order systems”, Journal of Applied Mechanics, vol. 47, no. +2, pp. 415-420, 1980. +[16] M. Nowicki and W. Respondek, ”A mechanical feedback classification of +linear mechanical control systems”, Applied Sciences, vol. 11, no. 22, pp. +10669, 2021. +[17] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Graduate +Texts in Mathematics, Springer, New York, 1997. +[18] M. W. Spong, P. Corke and R. Lozano, ”Nonlinear control of the reaction +wheel pendulum”, Automatica, vol. 37, no. 11, pp. 1845-1851, 2001. +[19] C. Wan, D. Bernstein, V. Coppola, ”Global Stabilization of the Oscillating +Eccentric Rotor”, Nonlinear Dynamics, 10: 49–62, 1995. +22 + diff --git a/EtAyT4oBgHgl3EQfSfdv/content/tmp_files/load_file.txt b/EtAyT4oBgHgl3EQfSfdv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..93927716f8310bf53d84852d040da1d7debf7fba --- /dev/null +++ b/EtAyT4oBgHgl3EQfSfdv/content/tmp_files/load_file.txt @@ -0,0 +1,561 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf,len=560 +page_content='Mechanical feedback linearization of single-input mechanical control systems Marcin Nowicki1 and Witold Respondek2,3 1Poznan University of Technology, Institute of Automatic Control and Robotics, Piotrowo 3a, 61-138 Pozna´n, Poland 2Lodz University of Technology, Institute of Automatic Control, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Stefanowskiego 18, 90-537 Lodz, Poland 3INSA de Rouen Normandie, Laboratoire de Math´ematiques, 76801 Saint-Etienne-du-Rouvray, France January 3, 2023 Abstract We present a new type of feedback linearization that is tailored for me- chanical control systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We call it a mechanical feedback linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Its basic feature is preservation of the mechanical structure of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For mechanical systems with a scalar control, we formulate necessary and sufficient conditions that are verifiable using differentiations and algebraic operations only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We illustrate our results with several examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 1 Introduction An N-dimensional control-affine system with a scalar control ˙z = F(z) + G(z)u, (Σ) where z ∈ Z, an open subset of RN, and u ∈ R, is said to be (locally) feedback linearizable (F-linearizable) if there exist a (local) diffeomorphism Φ : Z → RN and an invertible feedback of the form u = α(z) + β(z)˜u such that the control system (Σ), in the new coordinates ˜z = Φ(z) and with the new control ˜u, is a controllable linear system of the form ˙˜z = A˜z + b˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' A geometric solution to the problem of feedback linearization (inspired by [1], and developed independently in [2] and [3]) provides powerful techniques for designing a closed-loop control system that have been used in numerous engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' From a theoretical point of view, that result identifies a class of nonlinear systems that can be considered as linear ones in a well-chosen coordinates and with respect to a well-modified control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='00087v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='OC] 31 Dec 2022 In this paper, we state and study the following fundamental question: if a nonlinear control system (Σ) is mechanical and feedback linearizable, are those two structures compatible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' That is, can we feedback linearize the system pre- serving its mechanical structure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For mechanical control systems, it is natural to consider mechanical feedback equivalence (in particular, to a linear form) under mechanical transformations (coordinates changes and feedback) that pre- serve the mechanical structure of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In our recent paper [4], we showed that even in the simplest underactuated case of 2 degrees of freedom, the struc- tures (linear and mechanical) may not conform trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In the present paper, we treat the single-input case in its full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' There are several motivations for preserving the mechanical structure when feedback linearizing the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' First, our formulation of the problem of me- chanical linearization preserves configurations and velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We reckon that it is essential that new configurations (of the linearized system) are functions of the original configurations only, as well as new velocities are true physical velocities (in contrast to pseudo-velocities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore, we do not lose the physical inter- pretation of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' This could be useful, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' for mechanical systems with constraints on configurations, which are transformed into linear constraints on configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Second, the configuration trajectories are preserved too, which could be useful in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' the motion planning problem (the most natural way to state the problem for mechanical systems is to follow configuration trajectories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Third, it is worth mentioning that mechanical feedback linearizability guaran- tees the linearizing outputs to be functions of configurations only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' This may be of constructional importance because one needs only configuration sensors, not those of velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The next argument is the fact that the resultant linear mechanical system allows us to employ dedicated techniques for mechanical sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' An example of such technique is the natural frequency method of tuning a linear feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Finally, when applying mechanical feedback linearization, the physical interpretation of the external action (force, torque, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=') is preserved but is lost for general feedback linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' This work is a mechanical counterpart of the classical results on feedback linearization of control systems [1], [2], [3], see also monographs [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Our intention is to formulate conditions for mechanical linearization (shortly, MF- linearization) in a possibly similar manner (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' using involutivity of certain distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For a geometric approach to mechanical control systems see [5], [8], [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For mathematical preliminaries concerning the Lie derivative, the Lie bracket, distributions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=', see [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For linearization of mechanical control systems along controlled trajectories see [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For mechanical state-space linearization of mechanical control systems see [12] and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Compare also [14], for a pioneering work on (partial) feedback linearization of mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Although the state-space of mechanical control system is the tangent bundle TQ of the configuration space Q, we formulate our conditions using objects on Q only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The key here is a geometric approach to mechanical systems [5] and considering the Euler-Lagrange equations as the geodesic equation under an influence of external forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 2 The outline of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In Section 2, we state the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In Section 3, we develop further the problem of mechanical feedback linearization and formulate the main result, separately, for mechanical systems with n ≥ 3 in Theorem 1, and with n = 2 in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In Section 4, we provide an application of our results to MF-linearization of several mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Section 6 contains technical results used in proofs that could be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='1 Notation Throughout the Einstein summation convention is assumed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' any expression containing a repeated index (upper and lower) implies the summation over that index up to n, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ωiXi = �n i=1 ωiXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' AT transpose of a matrix (of a vector) A, In n × n identity matrix, Q configuration manifold, X(Q) the set of smooth vector fields on a manifold Q, TxQ tangent space at x ∈ Q, TQ = � x∈Q TxQ tangent bundle of Q, x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' xn) a local coordinate system on Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' φ a diffeomorphism of Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' and Φ a diffeomorphism of TQ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Dφ = ∂φ ∂x the Jacobian matrix of a diffeomorphism φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ∂˜xi ∂xj := ∂φi ∂xj the (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' j)-element of the Jacobian matrix Dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ∂xj ∂˜xi the (j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' i)-element of the inverse of the Jacobian matrix Dφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' LXα Lie derivative of a function α defined as LXα = ∂α ∂xi Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' [X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Y ] = ∂Y ∂x X − ∂X ∂x Y = adXY Lie bracket of vector fields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ∂ ∂xi the i-th unity vector field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' and dxi the i-th unity covector field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' in a co- ordinate system x = (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , xn), Ei = span � adj eg, 0 ≤ j ≤ i � distribution on Q spanned by adj eg, ∇ covariant derivative, and ∇2 second covariant derivative, Γi jk Christoffel symbols of the second kind of ∇, 2 Problem statement Consider an n-dimensional configuration space Q (an open subset of Rn or, in general, an n-dimensional manifold) equipped with a symmetric affine connec- tion ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The operator of the affine connection ∇ allows to define intrinsically the acceleration as the covariant derivative ∇ ˙x(t) ˙x(t), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' [5,8,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The covari- ant derivative ∇ : X(Q) × X(Q) → X(Q) of an arbitrary vector field Y = Y i ∂ ∂xi with respect to X = Xi ∂ ∂xi in coordinates reads ∇XY = �∂Y i ∂xj Xj + Γi jkXjY k � ∂ ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (1) 3 A mechanical control system (MS) is a 4-tuple (Q, ∇, g, e), where g and e are, respectively, controlled and uncontrolled vector fields on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' A curve x(t) : I → Q, I ⊂ R, is a trajectory of (MS) if it satisfies the following equation ∇ ˙x(t) ˙x(t) = e (x(t)) + g (x(t)) u, (2) which can be viewed as an equation that balances accelerations of the system, where the left-hand side represents geometric accelerations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' accelerations caused by the geometry of the system) and the right-hand side represents ac- celerations caused by external actions on the system (controlled or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Notice that (2) is a second-order differential equation on Q (indeed, using (1) we con- clude that ∇ ˙x ˙x depends on ¨x, see [5] for details) and can be rewritten as a system of first-order differential equations on TQ, which we also call a mechan- ical control system (MS): ˙xi = yi ˙yi = −Γi jk(x)yjyk + ei(x) + gi(x)u, (MS) for 1 ≤ i ≤ n, where (x, y) = � x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , xn, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , yn� are local coordinates on the tangent bundle TQ of the configuration manifold Q, and Γi jk(x) are Christoffel symbols of the affine connection ∇ that correspond to the Cori- olis and centrifugal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The vector fields e(x) = (e1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , en(x))T and g(x) = (g1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , gn(x))T correspond to, respectively, uncontrolled and con- trolled actions on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Throughout all objects are assumed to be smooth and the word smooth means C∞-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Our obvious inspirations are Lagrangian mechanical control systems without dissipative forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For the correspondence between (MS) and the Lagrangian equations of dynamics see [5], [8], [9] and our recent papers [13], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' However, we will consider throughout a more general class of mechanical control systems allowing for any symmetric (not necessarily a metric) connection and any (not necessarily potential) vector field e(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Consider the group of mechanical feedback transformations GMF generated by the following transformations: (i) changes of coordinates in TQ given by Φ : TQ → T ˜Q (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' y) �→ (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ˜y) = Φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' y) = � φ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ∂φ ∂x(x)y � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (3) called a mechanical diffeomorphism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' where φ : Q → ˜Q is a diffeomorphism and ∂φ ∂x its Jacobian matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (ii) mechanical feedback transformations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' denoted (α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' of the form u = γjk(x)yjyk + α(x) + β(x)˜u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (4) where γjk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' β are smooth functions on Q satisfying γjk = γkj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' β(·) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The matrix γ = (γjk) represents a (0, 2)−tensor field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 4 Even if the diffeomorphism φ is possibly local on Q, the action of ∂φ ∂x(x) is always global on fibers TxQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The system (MS) is MF-linearizable if there exist mechanical feedback transformations (Φ, α, β, γ) ∈ GMF bringing (MS) into a linear con- trollable mechanical system of the form ˙˜xi = ˜yi ˙˜yi = Ei j ˜xj + bi˜u, (LMS) where (˜x, ˜y) are coordinates on TRn = Rn × Rn, the matrix E = (Ei j) is an n × n real-valued matrix, the vector field b = bi ∂ ∂˜xi is constant, and the pair (E, b) is controllable (see [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Represent (MS) as ˙z = F(z) + G(z)u, where z = (x, y) ∈ TQ, F = yi ∂ ∂xi + � −Γi jk(x)yjyk + ei(x) � ∂ ∂yi , and G = gi(x) ∂ ∂yi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The problem that we formulate and solve in the paper is whether (MS) is MF-linearizable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' That is, do there exist Φ = (˜x, ˜y) = (φ, ∂φ ∂xy) and (α, β, γ) such that ∂Φ ∂z (z) � F + G(yT γy + α) � (z) = � ˜y E˜x � , ∂Φ ∂z (z) (Gβ) (z) = �0 b � ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Note that MF-linearizability is stronger than the classical feedback lineariz- ability since, for the latter, Φ : TQ → R2n can be any diffeomorphism (need not be of mechanical form (3)) and yT γ(x)y +α(x) can be replaced by any function α(x, y) on TQ and β(x) by any invertible function β(x, y) on TQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' If we neglect the mechanical structure of ˙z = F(z) + G(z)u, and consider it as a general control system, we can ask if the system is F-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The well-known answer [2,3] asserts that, locally, this is the case if and only if the distributions Di = span � adj F G, 0 ≤ j ≤ i � are involutive and of constant rank for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=', 2n − 1 and D2n−1 = TQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The natural question arises whether, for F-linearizable (MS), the general feedback transformations (Φ(z), α(z), β(z)) are mechanical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' of the form (3) and (4)) or whether they can be replaced by mechanical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Example 1: Consider the mechanical system ˙x1 = y1 ˙x2 = y2 ˙y1 = −x2(y1)2 + x2 ˙y2 = u, (5) on R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' This system is locally F-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Indeed, the local diffeomorphism ˜z = Φ(z), where z = (x1, x2, y1, y2), ˜z = (˜x1, ˜x2, ˜y1, ˜y2), given by ˜x1 = x1 ˜x2 = x2 − x2(y1)2 ˜y1 = y1 ˜y2 = � (y1)2 − 1 � � 2(x2)2y1 − y2� , 5 together with the feedback u = 2(x2)3 +6(x2 −(x2)2)(y1)2 + ˜u (y1)2−1, render the original system linear and controllable ˙˜x1 = ˜y1 ˙˜x2 = ˜y2 ˙˜y1 = ˜x2 ˙˜y2 = ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore, the system is F-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Note, however, that neither the change of coordinates nor the feedback is mechanical (˜x2 depends on velocities, and the function β depends on velocities as well) so the mechanical structure is not preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Our question is whether this system can be linearized by other transformations that preserve the mechanical structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' can it be MF- linearized?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The group of mechanical transformations GMF = {(Φ, α, β, γ)} preserves trajectories, that is, maps the trajectories of (MS) into those of its MF-equivalent system ( � MS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Indeed, if z (t, z0, u(t)) is a trajectory of (MS) (passing through z0 = (x0, y0) and corresponding to a control u(t)), then ˜z (t, ˜z0, ˜u(t)) = Φ (z (t, z0, u(t))) is a trajectory of ( � MS) passing through ˜z0 = Φ(z0) = (φ(x0), ∂φ ∂x(x0)y0) and corresponding to ˜u(t), where u(t) = y(t)T γ (x(t)) y(t) + α (x(t)) + β (x(t)) ˜u(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Moreover, via φ : Q → ˜Q, it establishes a correspondence between configuration trajectories in Q and ˜Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ˜x (t, ˜z0, ˜u(t)) = φ (x(t, z0, u(t))), making the fol- lowing diagram commutative (notice, however, that π (z(t, z0, u)) = x(t, z0, u) depends on z0 = (x0, y0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' an initial configuration x0 and initial velocity y0): z(t, z0, u) ˜z(t, ˜z0, ˜u) x(t, z0, u) ˜x(t, ˜z0, ˜u) (Φ,α,β,γ) π π (φ,α,β,γ) where π : TQ → Q, π(z) = π(x, y) = x, is the canonical projection which assigns to the pair (x, y) the point x at which the velocity y is attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 3 Mechanical feedback linearization Our main result uses two basic ingredients: the covariant derivative of the con- nection ∇, see (1), and the involutivity of suitable distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We will also need the second covariant derivative of a vector field Z in the directions (X, Y ), which is a mapping ∇2 : X(Q) × X(Q) × X(Q) → X(Q) ∇2 X,Y Z = ∇X∇Y Z − ∇∇XY Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (6) For properties of the second covariant derivative see Lemma 1 in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In order to formulate the result, we associate with (MS) the following se- quence of nested distributions E0 ⊂ E1 ⊂ E2 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ⊂ Ei ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ⊂ TQ, where E0 = span {g} , Ei = span � adj eg, 0 ≤ j ≤ i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 6 Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' To analyze the behavior of the distributions Ei under mechanical feedback transformations (α, β, γ) notice, first, that Ei are invariant under γ since γ does not act on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' If the distributions Ei are involutive, then they are invariant under feedback transformations of the form (α, β, 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' for γ = 0 they remain unchanged if we replaced e and g by, respectively, e + gα and βg, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Now, we formulate our main result for MF-linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' First, we state a theorem for (MS) with n ≥ 3 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The remaining case of n = 2 degrees of freedom is treated in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For an explanation of that distinction, see the comment before Theorem 2 and Remark 3 for a comparison of both results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' By a local MF-linearization around x0 ∈ Q we mean that it holds on � x∈O TxQ, where O is a neighborhood of x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' recall that all transformations are global on tangent spaces TxQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Assume n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' A mechanical control system (MS) is, locally around x0, MF-linearizable to a controllable (LMS) if and only if (MF1) rank En−1 = n, (MF2) Ei is involutive and of constant rank, for 0 ≤ i ≤ n − 2, (MF3) ∇adieg g ∈ E0 for 0 ≤ i ≤ n − 1, (MF4) ∇2 adkeg,adj eg e ∈ E1 for 0 ≤ k, j ≤ n − 1, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Notice that (MF1)-(MF2) are the classical conditions (see [2,3,6, 7]) that assure F-linearization of the system ˙x = e(x)+g(x)u on Q via ˜x = φ(x) and u = α(x)+β(x)˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The remaining two, (MF3)-(MF4), can be interpreted as compatibility conditions that guarantee vanishing the Christoffel symbols Γi jk in the linearizing coordinates ˜x = φ(x), except for those that can be compensated by feedback u = γjk(x)yjyk + ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In the proof we will use two Lemmata 1 and 2, given in Appendix, that are of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For (LMS), we have Γi jk = 0, e = Ex and g = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It follows that adi eg = (−1)iEib and therefore, using the definitions of ∇, given by (1), and of ∇2, given by (6), we calculate ∇adiegadj eg = 0, ∇2 adkeg,adj ege = 0, (7) which implies that (MF1)-(MF4) hold for (LMS) (in particular, (MF1) holds because (LMS) is assumed controllable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' To prove necessity of (MF1)-(MF4), we will show that they are MF-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' All conditions (MF1)-(MF4) are expressed in a geometrical way, therefore they are invariant under diffeomor- phisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The conditions (MF1) and (MF2) are mechanical feedback invariant, see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It remains to show that (MF3) and (MF4) are invariant under the 7 mechanical feedback u = γjk(x)yjyk+α(x)+β(x)˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For the closed-loop system, denoted by ”∼”, the Christoffel symbols ˜Γi jk of ˜∇, ˜e, and ˜g are, respectively, given by ˜Γi jk = Γi jk − giγjk, ˜e = e + gα, ˜g = gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (8) For any X, Y ∈ X(Q), we have ˜∇XY = ∇XY − γ(X, Y )g = ∇XY mod E0, where γ(X, Y ) = γjkXjY k ∈ C∞(Q), therefore ˜∇adi ˜e˜g˜g = ∇adi ˜e˜g˜g − γ(adi ˜e˜g, ˜g)g = ∇adi ˜e˜g˜g mod E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' By ∇X˜g = ∇X (gβ) = ∇Xg + (LXβ) g, it follows that instead of calculating ∇adi ˜e˜g˜g it is enough to calculate ∇adi ˜e˜gg, since the second term (LXβ) g ∈ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For i=0, we have ∇˜gg = ∇(gβ)g = β∇gg ∈ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It is easy to show that for any 1 ≤ j ≤ n − 1, we have adj ˜e˜g = βadj eg + dj−1, (9) where dj−1 ∈ Ej−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Assume ∇adl ˜e˜gg ∈ E0, for 0 ≤ l ≤ i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Then, by formula (9), ∇adi ˜e˜gg = β∇adiegg + ∇di−1g ∈ E0, because the first term is in E0 by (MF3) and the second by the induction assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We have thus proved necessity of (MF3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' To show necessity of (MF4), using Lemma 1, calculate ˜∇2 X,Y Z = ˜∇X ˜∇Y Z − ˜∇ ˜∇XY Z = ˜∇X (∇Y Z − γ(Y, Z)g) − ˜∇(∇XY −γ(X,Y )g)Z = ∇2 X,Y Z − γ(Y, Z)∇Xg + γ(X, Y )∇gZ mod E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (10) By the above formula, we get ˜∇2 adk ˜e ˜g,adj ˜e˜g˜e =∇2 adk ˜e ˜g,adj ˜e˜g˜e − γ(adj ˜e˜g, ˜e)∇adk ˜e ˜gg + γ(adk ˜e˜g, adj ˜e˜g)∇g˜e mod E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The second term, on the right hand side, is in E0 (by (MF3) and its invariance), while the third term is a function multiplying ∇g˜e = ∇g (e + gα) = ∇ge + α∇gg + Lgα g ∈ E1, since for (LMS) we have ∇ge = −adeg = −Eb ∈ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The first term ∇2 adk ˜e ˜g,adj ˜e˜g˜e is, by (9) and Lemma 1(i), a linear combination with smooth coefficients of ∇2 adieg,adleg˜e, with 0 ≤ i ≤ k and 0 ≤ l ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Thus we calculate ∇2 adi eg,adl eg˜e = ∇2 adieg,adlege+∇2 adieg,adleg(gα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The first term vanishes since (7) holds for (LMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We calculate the second term using Lemma 1(iii), and we have ∇2 adi eg,adl eg(gα) = α∇2 adieg,adlegg + Ladi egα∇adlegg + Ladlegα∇adi egg + (∇2 adi eg,adl egα)g ∈ E0 because the first three terms vanish, due to (7), and 8 the last one is in E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Summarizing the above calculations, we conclude that ˜∇2 adk ˜e ˜g,adj ˜e˜g˜e ∈ E1 = ˜E1, which proves necessity of (MF4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We will transform the system (MS), satisfying (MF1)-(MF4), into (LMS) in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In the first step, we will normalize the vector fields e and g and show that condition (MF4) implies zeroing some of the Christoffel symbols Γi jk, which exhibit a triangular form in the normalizing coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' In the second step, we compensate the remaining Christoffel symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' By conditions (MF1)-(MF2), there exists a function h satisfying Ladj egh = 0, for 0 ≤ j ≤ n − 2, and Ladn−1 e gh ̸= 0, and thus (˜x, ˜y) = (φ(x), ∂φ ∂x(x)y) is a local mechanical diffeomorphism, where φ(x) = (Ln−1 e h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , Leh, h)T that can be completed by a feedback transformation (α, β, 0) that map, respectively, βg into ˜g = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , 0)T , e + gα into ˜e = (0, ˜x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , ˜xn−1)T , and Γi jk into ˜Γi jk, see the classical results of feedback linearization [2], [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Then, (Φ, α, β, γ) ∈ GMF , where (˜x, ˜y) = Φ(x, y) = � φ(x), ∂φ ∂x(x)y � with φ, α, β just defined and γjk = ˜Γ1 jk(˜x), brings (MS) into (we drop ”tildas” for readability) ˙x1 = y1 ˙xi = yi ˙y1 = u ˙yi = −Γi jkyjyk + xi−1, 2 ≤ i ≤ n, (11) to which Lemma 2 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We will show that the Christoffel symbols Γi jk of (11) satisfy Γi kj = 0 for 1 ≤ k ≤ n − 1, 1 ≤ j ≤ i ≤ n, Γi nj = � 0 for 1 ≤ j < i ≤ n λ(xn) for 2 ≤ j = i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (12) For system (11), we have adk−1 e g = (−1)k−1 ∂ ∂xk and, in particular, g = ∂ ∂x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Calculate ∇adk−1 e gg = (−1)k−1∇ ∂ ∂xk gi ∂ ∂xi = (−1)k−1∇ ∂ ∂xk ∂ ∂x1 = (−1)k−1Γi k1 ∂ ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It follows that Γi k1 = Γi 1k = 0, for 2 ≤ i ≤ n by (MF3), and for i = 1 by the above form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Rewrite (MF4) as ∇2 adk−1 e g,adj−1 e ge = 0 mod E1, for 1 ≤ j, k ≤ n, and apply it successively for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , n and for all 1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For j = 1, first calculate ∇ge = ∇ ∂ ∂x1 e = ∂ ∂x2 + Γi 1ses ∂ ∂xi = ∂ ∂x2 and then ∇adk−1 e g (∇ge) = (−1)k−1∇ ∂ ∂xk ∂ ∂x2 = (−1)k−1Γi k2 ∂ ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' On the other hand, ∇adk−1 e gg = (−1)k−1∇ ∂ ∂xk ∂ ∂x1 = (−1)k−1Γ1 k1 ∂ ∂x1 = 0 and hence ∇∇adk−1 e gge = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Thus, by (6), ∇2 adk−1 e g,ge = ∇adk−1 e g (∇ge) − ∇∇adk−1 e gge = = (−1)k−1Γi k2 ∂ ∂xi = 0 mod E1, 9 implying that Γi k2 = Γi 2k = 0 for any 3 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For j = 2, calculate ∇adege = −∇ ∂ ∂x2 e = − ∂ ∂x3 + Γi 2ses ∂ ∂xi = − ∂ ∂x3 − d where d = d1(x) ∂ ∂x1 + d2(x) ∂ ∂x2 ∈ E1, and then ∇adk−1 e g (∇adege) = (−1)k∇ ∂ ∂xk � ∂ ∂x3 + d � = = (−1)k � Γi k3 + Γi k1d1 + Γi k2d2� ∂ ∂xi = = (−1)kΓi k3 ∂ ∂xi mod E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' On the other hand, ∇adk−1 e gadeg = (−1)k∇ ∂ ∂xk ∂ ∂x2 = (−1)kΓi k2 ∂ ∂xi = = (−1)k � Γ1 k2 ∂ ∂x1 + Γ2 k2 ∂ ∂x2 � and ∇∇adk−1 e gadege = (−1)kΓ2 k2 ∂ ∂x3 mod E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It follows that, modulo E1, ∇2 adk−1 e g,adege = (−1)k � n � i=4 Γi k3 ∂ ∂xi + (Γ3 k3 − Γ2 k2) ∂ ∂x3 � , and, using (MF4), we conclude Γi k3 = Γi 3k = 0 for any 4 ≤ i ≤ n and Γ3 k3 = Γ2 k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Following the same line (with a more tedious calculation), one can prove the general induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Namely, assuming, for a fixed j, Γj kj = Γj−1 kj−1 Γi ks = Γi sk = 0 s + 1 ≤ i ≤ n, 1 ≤ s ≤ j, (13) one shows by calculating ∇2 adk−1 e g,adj−1 e ge, with the help of (24) of Lemma 2, that Γj+1 kj+1 = Γj kj Γi kj+1 = 0 for j + 2 ≤ i ≤ n and thus, by the induction assumption and symmetry of the Christoffel symbols, Γi ks = Γi sk = 0 s + 1 ≤ i ≤ n, 1 ≤ s ≤ j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (14) It follows that for each 1 ≤ k ≤ n the matrices consisting of Christoffel symbols (Γi kj), for 2 ≤ i, j ≤ n are upper triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' By the induction argument, (13) holds for all 2 ≤ j ≤ n and implies, for any 1 ≤ k ≤ n − 1, Γ2 k2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' = Γn−1 kn−1 = Γn kn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 10 since Γn kn = Γn nk = 0 (as n > k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' On the other hand, for k = n, (13) implies Γ2 n2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' = Γn−1 nn−1 = Γn nn = λ(x) for a function λ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore for each 1 ≤ k ≤ n the matrices (Γi kj), for 2 ≤ i, j ≤ n, are strictly upper triangular, and the last one, for k = n, is upper triangular with all diagonal elements equal to each other, which we denote by λ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The matrices read � Γi kj � = � � � � � � � � � 0 Γ2 k3 Γ2 k4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Γ2 kn−2 Γ2 kn−1 Γ2 kn 0 0 Γ3 k4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Γ3 kn−2 Γ3 kn−1 Γ3 kn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 0 Γn−2 kn−1 Γn−2 kn 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 0 0 Γn−1 kn 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 0 0 0 � � � � � � � � � , for 1 ≤ k ≤ n − 1, and � Γi nj � = � � � � � � � � � λ Γ2 n3 Γ2 n4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Γ2 nn−2 Γ2 nn−1 Γ2 nn 0 λ Γ3 n4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Γ3 nn−2 Γ3 nn−1 Γ3 nn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' λ Γn−2 nn−1 Γn−2 nn 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 0 λ Γn−1 nn 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 0 0 λ � � � � � � � � � , and are thus of the desired triangular structure (12) and it remains to prove that λ = λ(xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Note that in the above matrices we skip the first row Γ1 kj and the first column Γi k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' This is due to the fact that Γ1 kj = 0 (which can always be achieved by a suitable feedback transformation) and Γi k1 = 0 by (MF3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Notice that we have En−2 = span � ∂ ∂x1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ∂ ∂xn−1 � and thus applying (24) of Lemma 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' for j = n and any 1 ≤ k ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' we conclude (set Γn kn+1 = 0) (−1)n+k−2∇2 adk−1 e g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='adn−1 e ge = ∇2 ∂ ∂xk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ∂ ∂xn e = �∂Γn ns ∂xk es + Γn nk+1 + Γn kn+1 − Γn−1 kn + (Γd nsΓn kd − Γd knΓn ds)es � ∂ ∂xn mod En−2 = � ∂λ ∂xk en + Γn nk+1 − Γn−1 kn � ∂ ∂xn mod En−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (15) since,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' due to the triangular structure (14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Γn ns = 0 except for s = n giving Γn nn = λ and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' the equality Γd nsΓn kd−Γd knΓn ds = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Indeed, in the latter, Γn kd = 0 except d = k = n giving Γn nsΓn nn − Γn nnΓn ns = 0 and Γn ds = 0 except for d = s = n giving Γn nnΓn kn − Γn knΓn nn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 11 For (15) we will apply (MF4) in three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' First, if 1 ≤ k ≤ n − 2, then, modulo En−2, we have � ∂λ ∂xk en + Γn nk+1 − Γn−1 kn � ∂ ∂xn = � ∂λ ∂xk xn−1 � ∂ ∂xn = 0, since all Γn nk+1 = 0 and all Γn−1 kn = 0 by (14) and k ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Second, for k = n − 1, we have modulo En−2, � ∂λ ∂xn−1 en + Γn nn − Γn−1 n−1n � ∂ ∂xn = � ∂λ ∂xn−1 en + λ − λ � ∂ ∂xn = � ∂λ ∂xn−1 xn−1 � ∂ ∂xn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore ∂λ ∂xk = 0, for 1 ≤ k ≤ n − 1, implying that λ is a function of the last variable xn only, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' λ = λ(xn), which gives the system in the desired form (12) Third, for k = n, we have modulo En−2, � ∂λ ∂xn en+Γn nn+1−Γn−1 nn � ∂ ∂xn = � ∂λ ∂xn xn−1− Γn−1 nn � ∂ ∂xn = 0, implying that Γn−1 nn = Leλ, since ∂λ(xn) ∂xn xn−1 = Leλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Now, transform system (11), satisfying (12), via the local mechanical diffeo- morphism Φ : TQ → T ¯Q ¯x = φ(x) ¯y = Dφ(x)y, where φ(x) = � Ln−1 e h, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' , Leh, h �T , (16) with h(xn) = � xn 0 Λ(s2)ds2, where Λ(s2) = exp �� s2 0 λ(s1)ds1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Denote by ¯Γi jk, ¯e, ¯g the objects of the system expressed in coordinates ¯x = φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Applying feedback ¯u = −¯Γ1 jk¯yj ¯yk + Ln e h + uLgLn−1 e h, the transformed system becomes ˙¯x1 = ¯y1 ˙¯xi = ¯yi ˙¯y1 = ¯u ˙¯yi = −¯Γi jk¯yj ¯yk + ¯xi−1, 2 ≤ i ≤ n, (17) whose vector fields are ¯e = ¯xi−1 ∂ ∂¯xi , where x0 = 0, and ¯g = ∂ ∂¯x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Transformed system (17) is still of the form (11) and at the moment we ignore how Γi jk have been changed into ¯Γi jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Below we will prove that all ¯Γi jk vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' To this end, we first calculate explicitly the time-evolution of the pair (¯xn, ¯yn) ˙¯xn = d dth(xn) = Λ(xn) ˙xn = Λ(xn)yn = ¯yn ˙¯yn = d dt (Λ(xn)yn) = Λ(xn)λ(xn) ˙xnyn + Λ(xn) ˙yn = Λ(xn)λ(xn)ynyn + Λ(xn) ˙yn = Λ(xn)λ(xn)ynyn + Λ(xn) � −Γn nn(xn)ynyn + xn−1� = Λ(xn)xn−1 = ¯xn−1, 12 since ¯xn−1 = Leh = Λ(xn)xn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It follows that ¯Γn jk = 0, for all 1 ≤ k, j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For transformed system (17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' we rewrite (24) by adding ”bars” as ∇2 adk−1 ¯e ¯g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='adj−1 ¯e ¯g¯e = (−1)j+k �∂¯Γi js ∂¯xk ¯es + ¯Γi jk+1 + ¯Γi kj+1 + (¯Γd js¯Γi kd − ¯Γd kj ¯Γi ds)¯es − ¯Γi−1 kj � ∂ ∂¯xi (18) and by (MF4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' we have ∇2 adk−1 ¯e ¯g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='adj−1 ¯e ¯g¯e = (−1)j+k¯an kj(¯x) ∂ ∂¯xn = 0 mod En−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' where ¯an kj(¯x) = ∂¯Γn js ∂¯xk ¯es + ¯Γn jk+1 + ¯Γn kj+1 + (¯Γd js¯Γn kd − ¯Γd kj ¯Γn ds)¯es − ¯Γn−1 kj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' which implies (since ¯Γn kj = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' for 1 ≤ j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' k ≤ n) that ¯an kj(¯x) = ¯Γn−1 kj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Now assume ¯Γi kj = 0 for a certain 1 ≤ i ≤ n − 1 and any 1 ≤ j, k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Then (18) and (MF4) imply ¯Γi−1 kj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore we have proved that all Christoffel symbols of (17) vanish and thus the system is a linear controllable (LMS), since the vector field ¯e = ¯xi−1 ∂ ∂¯xi is linear and ¯g = ∂ ∂¯x1 is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The above theorem does not work for systems with 2 degrees of freedom, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' for n=2, as that case is too restrictive for involutivity, see Remark 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore we state the following theorem for MF-linearization of (MS) with 2 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' A mechanical system (MS) with 2 degrees of freedom is, locally around x0, MF-linearizable to a controllable linear (LMS) if and only if it satisfies in a neighborhood of x0 (MF1)’ g and adeg are independent at x0, (MF3)’ ∇g g ∈ E0 and ∇adeg g ∈ E0, (MF5)’ ∇2 g,adeg adeg − ∇2 adeg,g adeg ∈ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' If n = 2, then E0 is of rank 1, thus involutive and (MF2) is trivially satisfied, and so is (MF4) because E1 = TQ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore (MF2)’ and (MF4)’ are absent and replaced by (MF5)’ that guarantees that we can compensate the Christoffel symbols (as do (MF3)-(MF4) for n ≥ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Note that (MF1)’ is equivalent to (MF1) and (MF3)’ is (MF3) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Although Theorem 1 applies to n ≥ 3, the necessity part of its proof remains valid for any n ≥ 2 so it shows necessity of (MF1)’-(MF3)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore we need to show necessity of (MF5)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For a controllable (LMS) we have Γi jk = 0, g = b and adeg = −Eb are independent, and ∇adiegadj eg = 0, ∇2 adj eg,adkegadi eg = 0, � adj eg, adk eg � = 0, (19) 13 for 0 ≤ i, j, k ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We will use formula (10) to show that (MF5)’ is invariant under mechanical feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Denote ˜∇, ˜e, ˜g, γ as in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Then we calculate ˜∇2 ˜g,ad˜e˜gad˜e˜g =∇2 ˜g,ad˜e˜gad˜e˜g − γ(ad˜e˜g, ad˜e˜g)∇˜g˜g + γ(˜g, ad˜e˜g)∇˜gad˜e˜g mod E0, ˜∇2 ad˜e˜g,˜gad˜e˜g =∇2 ad˜e˜g,˜gad˜e˜g − γ(g, ad˜e˜g)∇ad˜e˜g˜g + γ(ad˜e˜g, ˜g)∇˜gad˜e˜g mod E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The second terms of the right hand side of both equations are in E0 due to the feedback invariance of (MF3)’, while the third terms are equal since γ(X, Y ) = γ(Y, X) is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore we conclude ˜∇2 ˜g,ad˜e˜gad˜e˜g − ˜∇2 ad˜e˜g,˜gad˜e˜g = ∇2 ˜g,ad˜e˜gad˜e˜g − ∇2 ad˜e˜g,˜gad˜e˜g mod E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Denoting ad˜e˜g = βadeg + d0g (see (9)) and by Lemma 1 (i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' we have ∇2 ˜g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='ad˜e˜gad˜e˜g = ∇2 βg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='βadeg+d0gad˜e˜g = β2∇2 g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='adegad˜e˜g + βd0∇2 g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='gad˜e˜g ∇2 ad˜e˜g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='˜gad˜e˜g = ∇2 βadeg+d0g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='βgad˜e˜g = β2∇2 adeg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='gad˜e˜g + βd0∇2 g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='gad˜e˜g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' where the last terms on the right are equal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' implying ∇2 ˜g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='ad˜e˜gad˜e˜g − ∇2 ad˜e˜g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='˜gad˜e˜g = β2 � ∇2 g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='adegad˜e˜g − β2∇2 adeg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='gad˜e˜g � and it remains to prove that ∇2 g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='adegad˜e˜g − ∇2 adeg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='gad˜e˜g ∈ E0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' which we show using Lemma 1(iii),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' where X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Y stand for either g or adeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Denote ∇Xβ = LXβ and ∇2 X,Y β = LXLY β − L∇XY β (see Lemma 1) and calculate ∇2 X,Y ad˜e˜g = ∇2 X,Y � βadeg + d0g � = β∇2 X,Y adeg + LXβ∇Y adeg + LY β∇Xadeg + � ∇2 X,Y β � adeg + d0∇2 X,Y g + LXd0∇Y g + LY d0∇Xg + � ∇2 X,Y d0� g = � ∇2 X,Y β � adeg mod E0, since all ∇2 X,Y X = 0 and ∇XY = 0 , see (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore we have ∇2 g,adegad˜e˜g − ∇2 adeg,gad˜e˜g = � ∇2 g,adegβ − ∇2 adeg,gβ � adeg mod E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Finally, we calculate ∇2 g,adegβ − ∇2 adeg,gβ = LgLadegβ − L∇gadegβ − � LadegLgβ − L∇adeggβ � = L[g,adeg]β = 0, 14 which shows necessity of (MF5)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' By (MF1)’, rank E1 = 2, and E0 = span {g} is of constant rank 1 and thus always involutive, hence the system is, locally around x0 (since g(x0) ̸= 0), MF-equivalent to (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (11)) ˙x1 = y1 ˙x2 = y2 ˙y1 = u ˙y2 = −Γ2 jkyjyk + x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We have g = ∂ ∂x1 , adeg = − ∂ ∂x2 and now we calculate ∇gg = Γ2 11 ∂ ∂x2 ∇adegg = −Γ2 12 ∂ ∂x2 , which by (MF3)’ are in E0 = span � ∂ ∂x1 � , implying Γ2 11 = Γ2 12 = Γ2 21 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It follows ∇gg = ∇adegg = ∇gadeg = 0, and ∇adegadeg = Γ2 22 ∂ ∂x2 and thus ∇2 g,adeg adeg − ∇2 adeg,g adeg = ∇g∇adegadeg − ∇∇gadegadeg − ∇adeg∇gadeg − ∇∇adeggadeg = ∇g∇adegadeg = ∇ ∂ ∂x1 Γ2 22 ∂ ∂x2 = ∂Γ2 22 ∂x1 ∂ ∂x2 implying, by (MF5)’, ∂Γ2 22 ∂x1 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Γ2 22(x2) = λ(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Now, we transform the system via the local mechanical diffeomorphism Φ : TQ → T ¯Q (compare to (16)) ¯x = φ(x) ¯y = Dφ(x)y, where φ(x) = (Leh, h)T , with h(x2) = � x2 0 Λ(s2)ds2 and Λ(s2) = exp �� s2 0 λ(s1)ds1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We calculate the evolution of the pair (¯x(t), ¯y(t)) of transformed coordinates, using d dth � x2(t) � = Λ � x2(t) � ˙x2(t) and d dtΛ � x2(t) � = λ � x2(t) � Λ � x2(t) � ˙x2(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' first we get ˙¯x2 = d dth(x2) = Λ(x2)y2 = ¯y2 ˙¯y2 = Λ(x2)λ(x2)y2y2 + Λ(x2) ˙y2 = Λ(x2)λ(x2)y2y2 + Λ(x2) � −λ(x2)y2y2 + x2� = Λ(x2)x1 = ¯x1 and then ˙¯x1 = Λ(x2)y1 + d dtΛ � x2(t) � x1y2 = ¯y1 ˙¯y1 = −¯Γ1 jk¯yj ¯yk + L2 eh + uLgLeh, where we denote by ¯Γ1 jk the Christoffel symbols in the ˙¯y1-equation of the trans- formed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Applying the feedback ¯u = −¯Γ1 jk¯yj ¯yk + L2 eh + uLgLeh, we get a controllable linear mechanical system in the canonical form ˙¯x1 = ¯y1, ˙¯y1 = ¯u, ˙¯x2 = ¯y2, ˙¯y2 = ¯x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 15 4 Examples Example 1 (cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ): For system (5), we have g = ∂ ∂x2 and adeg = − ∂ ∂x1 are in- dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We check MF-linearizability using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' A simple calculation shows that ∇gg = ∇adegg = 0 ∈ E0, but ∇2 g,adeg adeg−∇2 adeg,g adeg = ∂ ∂x1 /∈ E0, therefore the system is not MF-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Thus (5) is an example of a system that is F-linearizable but not MF- linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For such systems the choice is: either to F-linearize for the price of loosing the mechanical structure or to keep the mechanical structure but to get rid of the linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Example 2: Consider the equation of dynamics of the Inertia Wheel Pen- dulum [18] with constant parameters m0, md, J2: ˙x1 = y1, ˙x2 = y2, ˙y1 = e1 + g1u, ˙y2 = e2 + g2u, e1 = m0 md sin x1, e2 = − m0 md sin x1, g1 = − 1 md , g2 = md+J2 J2md .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We will verify whether the conditions of Theorem 2 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' First, we calculate adeg = ( m0 m2 d cos x1) ∂ ∂x1 − ( m0 m2 d cos x1) ∂ ∂x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It can be checked that g and adeg are independent for x1 ̸= ± π 2 , which corresponds to the horizontal position of the pendulum, therefore (MF1)’ is satisfied everywhere except for x1 = ± π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Next, we verify condition (MF2)’ by calculating ∇gg = ∇adegg = 0 ∈ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Finally, a direct calculation shows ∇2 g,adeg adeg = ∇2 adeg,g adeg = = (m2 0 m5 d cos2 x1) ∂ ∂x1 − (m2 0 m5 d cos2 x1) ∂ ∂x2 , thus ∇2 g,adeg adeg − ∇2 adeg,g adeg = 0 ∈ E0 satisfies (MF5)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The system is MF- linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' A linearizing function is h(x) = md+J2 J2 x1 + x2 (all others giving MF-linearization are of the form σ h(x), where σ ∈ R\\ {0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Due to the proof of Theorem 2, the linearizing diffeomorphism is (˜x, ˜y) = Φ(x, y) = (φ(x), Dφ(x)y) with φ(x) = (h, Leh)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The system in new coordinates reads ˙˜x1 = md + J2 J2 y1 + y2 = ˜y1 ˙˜y1 = md + J2 J2 �m0 md sin x1 − 1 md u � − m0 md sin x1 + md + J2 m2J2 u = m0 J2 sin x1 = Leh = ˜x2 (20) ˙˜x1 = m0 J2 cos x1y1 = ˜y2 ˙˜y2 = −m0 J2 sin x1y1y1 + m2 0 2mdJ2 sin(2x1) − m0 mdJ2 cos x1u = ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Example 3: We will study MF-linearizability of the TORA3 system (see Figure 1), which is based on the TORA system (Translational Oscillator with 16 Figure 1: The TORA3 system Rotational Actuator) studied in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' [19] (however we add gravita- tional effects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' It consists of a two dimensional spring-mass system, with masses m1, m2 and spring constants k1, k2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' A pendulum of length l3, mass m3, and moment of inertia J3 is added to the second body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The displacements of the bodies are denoted by x1 and x2, respectively, and the angle of the pen- dulum by x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The gravitational constant is a and u is a torque applied to the pendulum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The kinetic energy is T =1 2m1( ˙x1)2 + 1 2(m2 + m3)( ˙x2)2 + 1 2(J3 + m3l2 3)( ˙x3)2 + m3l3 cos x3 ˙x2 ˙x3, and the mass matrix depends on the configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The potential energy is V = 1 2k1(x1)2 + 1 2k2(x2 − x1)2 − m3l3a cos x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The equations of the dynamics read m1¨x1 + k1x1 − k2 � x2 − x1� = 0 (m2 + m3)¨x2 + m3l3 cos x3¨x3 − m3l3 sin x3( ˙x3)2 +k2 � x2 − x1� = 0 m3l3 cos x3¨x2 + (m3l2 3 + J3)¨x3 + m3l3a sin x3 = u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' which can be rewritten on TQ as ˙x1 = y1 ˙y1 = η1 ˙x2 = y2 ˙y2 = −¯Γ2 33y3y3 + η2 + τ 2u ˙x3 = y3 ˙y3 = −¯Γ3 33y3y3 + η3 + τ 3u (21) where ¯Γ2 33 = −ν0 sin x3 ν1+ν2 sin2 x3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ¯Γ3 33 = ν2 sin x3 cos x3 ν1+ν2 sin2 x3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' η1 = − k1 m1 x1 + k2 m3 � x2 − x1� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' η2 = 1 2 ν2a sin 2x3−ν3(x2−x1) ν1+ν2 sin2 x3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' η3 = ν4(x2−x1) cos x3−ν5 sin x3 ν1+ν2 sin2 x3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' τ 2 = −m3l3 cos x3 ν1+ν2 sin2 x3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' τ 3 = m2+m3 ν1+ν2 sin2 x3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' with constant parameters: ν0 = m3l3(m3l2 3 + J3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ν1 = m2m3l2 3 + J3(m2 + m3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ν2 = m2 3l2 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ν3 = k2 � m3l2 3 + J3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ν4 = m3l3k2 ν5 = m3l3a(m2 + m3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 17 -- m1 m2 ki k2 W -- W aTo simplify calculations we apply to the system a preliminary mechanical feedback1 u = 1 τ 3 �¯Γ3 33y3y3 − η3 + ¯u � which yields ˙x1 = y1 ˙x2 = y2 ˙x3 = y3 ˙y1 = −µ1x1 + µ2x2 ˙y2 = µ3 sin x3y3y3 + µ4(x1 − x2) − µ3 cos x3u ˙y3 = ¯u, (22) with µ1 = k1+k2 m1 , µ2 = k2 m1 , µ3 = m3l3 m2+m3 , µ4 = k2 m2+m3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Since conditions (MF1)-(MF4) of Theorem 1 are MF-invariant, we will check them for system (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' To summarize: Γ2 33 = −µ3 sin x3, and Γi jk = 0 otherwise, e = � −µ1x1 + µ2x2� ∂ ∂x1 + µ4 � x1 − x2� ∂ ∂x2 g = −µ3 cos x3 ∂ ∂x2 + ∂ ∂x3 = g2 ∂ ∂x2 + ∂ ∂x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We have (notice that calculations are performed on Q only) adeg = � µ2µ3 cos x3� ∂ ∂x1 − � µ3µ4 cos x3� ∂ ∂x2 , ad2 eg = µ3 cos x3 � (µ1µ2 + µ2µ4) ∂ ∂x1 − � µ2µ4 + µ2 4 � ∂ ∂x2 � , therefore rank E2 = 3 for x3 ̸= ± π 2 , and (MF1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Now [g, adeg] = − � µ2µ3 sin x3� ∂ ∂x1 + � µ3µ4 sin x3� ∂ ∂x2 ∈ E1 and (MF2) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Then, for any vector field v = vi(x) ∂ ∂xi , ∇vg = �∂g2 ∂x3 + Γ2 33 � v3 ∂ ∂x2 = 0, thus (MC3) is satisfied if we replace v by, in particular, g, adeg, ad2 eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Finally, for (MF4), we calculate ∇2 g,ge = � µ2µ3 sin x3� ∂ ∂x1 − � µ3µ4 sin x3� ∂ ∂x2 ∈ E1, ∇2 adkeg,adj ege = 0 otherwise, thus, the system is MF-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Now, choose h = µ4 µ2 x1 + x2 + µ3 sin x3 (whose differential dh annihilates g and adeg), thus we take a linearizing diffeo- morphism (˜x, ˜y) = � φ(x), ∂φ ∂x(x)y � , with φ(x) = � h, Leh, L2 eh �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The linearized 1This preliminary feedback is not necessary and it is possible to check the conditions and to linearize the system without it, since our method and conditions are feedback invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 18 system is in the form of (LMS) and reads ˙˜x1 = µ4 µ2 y1 + y2 + µ3 cos x3y3 = ˜y1 ˙˜y1 = µ4 µ2 ˙y1+ ˙y2+µ3(cos x3 ˙y3−sin x3y3y3)= µ4(µ2 − µ1) µ2 x1 = ˜x2 ˙˜x2 = µ4(µ2 − µ1) µ2 y1 = ˜y2 ˙˜y2 = µ4(µ2 − µ1) µ2 ˙y1 = µ4(µ2 − µ1) µ2 � µ2x2 − µ1x1� = ˜x3 ˙˜x3 = µ1µ4(µ1 − µ2) µ2 y1 + µ4(µ2 − µ1)y2 = ˜y3 ˙˜y3 = (µ2 − µ1)µ3µ4 sin x3y3y3 − (µ1 − µ2)(µ2 1 + µ2µ4)µ4 µ2 x1 + (µ1 − µ2)(µ1 + µ4)µ4x2 + (µ1 − µ2)µ3µ4 cos x3u = ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 5 Conclusions In this paper, we consider MF-linearization of mechanical control systems (MS) with scalar control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We formulate the problem as a particular case of feedback linearization preserving the mechanical structure of (MS) so that the trans- formed system is both linear and mechanical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' As we showed in [4] and confirmed in this paper, even in the simplest case, the class of MF-linearizable systems is substantially smaller than that of general F-linearizable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Therefore, a nat- ural question arises, namely to compare the conditions presented in this paper with those for F-linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The answer lies in the interplay between the distributions Ei = span � adj eg, 0 ≤ j ≤ i � and the ”usual” for F-linearization Di = span � adj F G, 0 ≤ j ≤ i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' We will address this problem in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' 6 Appendix The following lemma can be proved by a direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The second covariant derivative ∇2 X,Y Z satisfies the following prop- erties: (i) linearity over C∞(Q) in X and Y : ∇2 (α1X1+α2X2),Y Z = α1∇2 X1,Y Z + α2∇2 X2,Y Z ∇2 X,(α1Y1+α2Y2)Z = α1∇2 X,Y1Z + α2∇2 X,Y1Z (ii) linearity over R in Z: ∇2 X,Y (a1Z1 + a2Z2) = a1∇2 X,Y Z1 + a2∇2 X,Y Z2 19 (iii) the product rule: ∇2 X,Y (βZ) =β∇2 X,Y Z + LXβ∇Y Z + LY β∇XZ + � ∇2 X,Y β � Z, where ∇2 X,Y β = LXLY β−L∇XY β ∈ C∞(Q), Xi, Yi, Zi ∈ X(Q), αi, β ∈ C∞(Q), and ai ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' The following lemma is crucial for the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For the system ˙x1 = y1 ˙xi = yi ˙y1 = u ˙yi = −Γi jkyjyk + xi−1, 2 ≤ i ≤ n, (23) we have for any 1 ≤ k, j ≤ n, ∇2 adk−1 e g,adj−1 e ge = (−1)j+k �∂Γi js ∂xk es + Γi jk+1 + Γi kj+1 − Γi−1 kj + (Γd jsΓi kd − Γd kjΓi ds)es � ∂ ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' (24) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' For system (23) we calculate ∇2 adk−1 e g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content='adj−1 e ge = (−1)j+k∇2 ∂ ∂xk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' ∂ ∂xj e = ∇ ∂ ∂xk ∇ ∂ ∂xj e − ∇∇ ∂ ∂xk ∂ ∂xj e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' where ∇ ∂ ∂xj e = � ∂ed ∂xj + Γd jses� ∂ ∂xd ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' and ∇ ∂ ∂xk � ∇ ∂ ∂xj e � = ∇ ∂ ∂xk � ∂ed ∂xj � ∂ ∂xd + ∇ ∂ ∂xk � Γd jses� ∂ ∂xd = ∂ed ∂xj ∇ ∂ ∂xk ∂ ∂xd + L ∂ ∂xk � ∂ed ∂xj � ∂ ∂xd + � Γd jses� ∇ ∂ ∂xk ∂ ∂xd + � L ∂ ∂xk � Γd js � es + L ∂ ∂xk (es) Γd js � ∂ ∂xd = ∂ed ∂xj Γi kd ∂ ∂xi + Γd jsesΓi kd ∂ ∂xi + � ∂Γi js ∂xk es + ∂es ∂xk Γi js � ∂ ∂xi = � ∂Γi js ∂xk es + Γi jk+1 + Γi kj+1 + Γd jsΓi kdes � ∂ ∂xi ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' since ∂ed ∂xj = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' if d = j + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' and zero otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' and thus ∂ed ∂xj Γi kd = Γi kj+1 (analogously for the other derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Now, using ∇ ∂ ∂xk ∂ ∂xj = Γd kj ∂ ∂xd , we calculate ∇∇ ∂ ∂xk ∂ ∂xj e = ∇Γd kj ∂ ∂xd e = Γd kj � ∂ei ∂xd + Γi dses � ∂ ∂xi = � Γi−1 kj + Γd kjΓi dses� ∂ ∂xi , so we have 20 ∇2 ∂ ∂xk , ∂ ∂xj e = ∇ ∂ ∂xk � ∇ ∂ ∂xj e � − ∇∇ ∂ ∂xk ∂ ∂xj e = �∂Γi js ∂xk es + Γi jk+1 + Γi kj+1 − Γi−1 kj + (Γd jsΓi kd − Γd kjΓi ds)es � ∂ ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' which yields (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAyT4oBgHgl3EQfSfdv/content/2301.00087v1.pdf'} +page_content=' Brockett, ”Feedback invariants for nonlinear systems”, in Proc.' metadata={'source': 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January 13, 2023) +This paper discusses the capacity of graph neural networks to learn the functional form of ordinary +differential equations that govern dynamics on complex networks. We propose necessary elements +for such a problem, namely, inductive biases, a neural network architecture and a learning task. Sta- +tistical learning theory suggests that generalisation power of neural networks relies on independence +and identical distribution (i.i.d.) of training and testing data. Although this assumption together +with an appropriate neural architecture and a learning mechanism is sufficient for accurate out-of- +sample predictions of dynamics such as, e.g. mass-action kinetics, by studying the out-of-distribution +generalisation in the case of diffusion dynamics, we find that the neural network model: (i) has a +generalisation capacity that depends on the first moment of the initial value data distribution; (ii) +learns the non-dissipative nature of dynamics implicitly; and (iii) the model’s accuracy resolution +limit is of order O(1/√n) for a system of size n. +Introduction +Dynamics in a complex networked sys- +tem is modelled as a set of n ordinary differential equa- +tions (ODEs) that describe the rate of change of a quan- +tity xi(t) for each node i and are coupled via adjacency +matrix A ∈ Rn×n. A general form of these equations is +˙xi = L(xi(t)) + +� +j +AijQ(xi(t), xj(t)) +(1) += F(xi(t), x(t), A) +where L describes self-interactions, Q is a function that +models pairwise interactions between neighbours and � +is an aggregation function. +With appropriate choices +of functions L, Q, � this definition is a general form +for models of epidemic processes, biochemical dynamics, +birth–death processes, gene regulatory dynamics [1], as +well as dynamics that show chaotic behaviour [2]. +The initial value problem of a set of ODEs such as Eq. +1 together with an initial condition x(t0), has a solution +that satisfies +x(t) = x(t0) + +� t +t0 +FFF(x(t′), A)dt′ +(2) +and describes a set of trajectories of the dynamics, if the +system was initialised at x(t0). +Appropriately setup, a neural network ΨΨΨ(x;ωωω) has ca- +pacity to approximate any continuous function F with +compact support [3]. In practice, learning the weights is +usually done via some variant of backpropagation algo- +rithm [4]. +Notably, neural networks can also be used to approx- +imate dynamical systems [5] and find solutions of initial +and boundary value problems of differential equations [6]. +A dynamical system is that in which FFF describes the time +dependence of x in an ambient space. Notably, if FFF is +known, the description quality of the course of dynamics +∗ vvasiliau@ethz.ch +† Authors contributed equally to this work. +is independent of a coordinate in the space. For exam- +ple, Newton’s laws of motion describe the trajectory of a +bouncing ball regardless of its longitudinal and latitudi- +nal position. Recovering universal dynamical principles +from empirical data has been shown to belong to NP- +hard class [7]. +Despite, the hardness of problem, in recent years, dif- +ferent classes of neural networks were used to learn dif- +ferent parts of dynamics from empirical data, including +graph neural networks [8] and their differential [9] coun- +terparts [10]; reservoir computers [11, 12] as well as re- +gression techniques [13, 14] or to learn control dynam- +ics [15]. +Here we discuss architectural design choices and induc- +tive biases that are crucial for a neural network model +that approximates dynamics evolving on complex net- +works. We then study the model’s generalisation capac- +ity using simple models of deterministic dynamics [1]. +Lastly, we discuss our work in the context of learning +principles that govern dynamics in complex system from +perspective of generalization to unseen initial conditions. +Inductive biases for dynamics on complex networks +There are several important inductive biases and assump- +tions worth noting about the complex network dynamics +and its neural approximations. +1. +Network structure: There exists a known static +network represented as an adjacency matrix A. There- +fore it is reasonable to take a GNN [16] as the candidate +for ΨΨΨ. A single-layer graph convolution network can be +defined as +ΨΨΨgnn(x) = (σ [ΦΦΦxW + b]) Wagg. +(3) +where x ∈ Rn×d is an input, ΦΦΦ ∈ Rn×n is a graph oper- +ator (e.g. ΦΦΦ = ˜D− 1 +2 ˜A ˜D− 1 +2 [17]), W ∈ Rd×h, b ∈ Rn×1, +Wagg ∈ Rh×d are trainable parameters and σ is a non- +linear function. Different versions of GNN with respect +to different expressive power for Weisfeiler-Lehman iso- +morphism are described in [18]. +2. +Self-Interaction: +The model includes a self- +interaction part that approximates L(·). +arXiv:2301.04900v1 [cond-mat.stat-mech] 12 Jan 2023 + +2 +3. +Neighbour-Interaction: +The model includes a +neighbour interaction part that approximates Q(·, ·). +Note that a single-layer GNN, such as a convolutional +graph neural network has no mixed quadratic terms +xixj and therefore does not simply satisfy such a con- +dition. +Although theoretically it should still be pos- +sible to approximate nonlinear quadratic terms with a +single layer neural network with an arbitrary width, in +practice it can be challenging and require either a very +large number of hidden neurons, or an exotic learning +mechanism that goes beyond the standard gradient de- +scend. +Alternatively, one can improve expressivity of +the model by increasing its depth, i.e. using multi-layer +GNNs or message-passing neural networks [19] to rep- +resent ΨΨΨ(x;ωωω). +Here ωωω includes graph operator terms +ΦΦΦk, k ∈ {1, 2, ..., K} where K is the depth of the neural +network. +4. Spatiotemporal locality: The dynamical process +that follows Eq. 1 must be local, that is, the function +Q(·, ·) encodes interactions between neighbours. +How- +ever, including terms ΦΦΦk in a multi-layer graph neural +network allows for k-hop interactions via length k walks +in a network at a timescale smaller than the infinitesimal +dt thereby subdividing dt to k intervals and breaking an +assumption of temporal locality. +5. Aggregation of neighbour-interactions: The ag- +gregation can itself be non-linear. +6. Initial value condition: Initial values are preserved +during training: x0: ΨΨΨ(x0) → x0. If the neural network +straightforwardly approximates the RHS of Eq. 2, then +enconding and decoding layers must be pseudo-inverses +of each other, see App. A. +7. Conservation/dissipation laws. If the system is +closed, it does not exchange energy or mass with the en- +vironment, therefore a conservation law holds, namely +� +i +dxi(t) +dt += C +∀t. +(4) +A constraint on a neural network to satisfy conservation +laws can be imposed via a regularisation term in the loss +function, +R(D) = +1 +|D| +� +x∈D +|FFF(x)1 − ΨΨΨ(x)1| , +that penalises the model weights which produce predic- +tions which do not respect the conservation law Eq. 4. +Here D is the dataset over which the loss is calculated. +The strength of the regulariser term can be modulated +by mutiplying R(D) with a non-negative real number λ. +Architecture +Given the inductive biases for dynamics +on networks, we propose a neural network model of the +following form: +˙x = ψψψℓ(x) + ψψψ +� +(x) +(5) +ψψψ +� +(x) = vec−1� +ψψψq3� +vec +� +ΦΦΦ ⊙ +� +ψψψq1(x)⊤1 ×k ψψψq2(x)⊤2� ��� +where ψψψ(x) is a single hidden layer neural network +are given by (3). +The mappings of local interaction +are summarised in App. B. The design choices of Eq. +5 comply with the inductive biases stated earlier. +To +this end, we performed vetorisation of input to the +function ψψψ +� +[ψψψq3 (·)]. +This function can approximate +any invariant poolings of a set [20] or a multiset [18]. +Notably, we also assumed that Q(·, ·) is factorisable. +Since it can be approximated by Chebyshev polynomials, +and, according to the strictly real fundamental theorem +of algebra [21], it is possible to factorise polynomial +function to two factors. Alternatively, one can use deep +sets [20] as arguments to approximate Q(·, ·). +In order to guarantee the local existence and unique- +ness of the solution to the initial value problem, by Pi- +card–Lindel¨of theorem the neural network ΨΨΨ needs to be +Lipschitz continuous. To enforce Lipschitz continuity of +ΨΨΨ, we will be using 1-Lipschitz activation functions such +as ReLU, sigmoid, softmax, or tanh. +Learning task +We formulate two distinct statistical +learning settings that relate to an increasing strength of +generality in the approximation of a dynamical system. +1. +Regression task to approximate FFF by ΨΨΨ: An +appropriate “proto data set” here is +D = {(x(t)α, y(t)α)}, +s.t. x(t)α ∈ Rn, y(t)α ∈ Rn, x(0)α ∼ fx(0)(x), t = [0, T] ∈ R. +our labels are defined as y(t)α = FFF(x((t))α), α denotes +α-th initial condition x(0)α sampled from a predefined +distribution fx(0)(x); all others points x(t)α are obtained +following Eq. 2. Here the functional mapping that is be- +ing learnt is ˆFFF : Rn → Rn and is obtained by minimising +the loss L between the true labels y and the labels f(x) +obtained by the current model: +ˆFFF = arg +min +f:Rn→Rn +E +P(x,y) L(f(x), y). +Here E is an expectation operator, P(x, y) is the data +sampling distribution. +At the moment, samples from the “proto data set” are +not independent: those trajectories that were obtained +from the same initial condition are non-i.i.d. Such sam- +pling is compulsory for the Uniform Law of Large num- +bers, that together with capacity control ensures general- +isation from train to test set [22, 23]. To ensure statistical +independence of samples, we create finite train and test +sets of size m1, m2 by using a specific distribution P over +a “proto data set” +Dtrain ∪ Dtest ∼ P(x, y). +Specifically, +we randomly delegate (x(t)α, y(t)α) to +either Dtrain or Dtest thereby ensuring an i.i.d. condition +by dropping information on the initial conditions and +time. + +3 +Dynamics +L +Q +Ltrain +reg +Ltest +reg +≈reg +Ltrain +traj +Ltest +traj +≈traj +Heata +– +B(xj − xi) 2.03 ± 1.03 +2.14 ± 1.08 +✓ +1.39 ± 0.59 1.47 ± 0.63 +✓ +MAKb +F − Bxb +i +Rxj +0.41 ± 1.08 +0.44 ± 1.14 +✓ +1.48 ± 0.05 1.55 ± 0.04 +× +PDc +−Bxb +i +Rxa +j +4.68 ± 12.82 4.72 ± 12.89 +✓ +3.03 ± 0.03 3.04 ± 0.03 +✓ +MMd +−Bxi +R +xh +j +1+xh +j +7.68 ± 5.36 +7.83 ± 5.47 +✓ +5.93 ± 0.12 5.94 ± 0.14 +✓ +SISe +−Bxi +(1 − xi)xj +1.16 ± 3.62 +1.31 ± 4.07 +✓ +1.54 ± 0.01 1.64 ± 0.02 +× +a B = 0.05. +b B = 0.1, R = 1, f = 0.5. +c B = 2, R = 0.3, a = 1.5, b = 3. +d B = 4, R = 0.5, h = 3. +e B = 5, R = 0.5. +TABLE I. Generalisation of a neural network model Eq. 5 trained on dynamics from [1] in the regression task setting, and +the trajectory learning setting. Reported loss values are multiplied by a factor 10−2. In columns denoted “≈” we indicate for +which dynamics the train loss is approximately similar (“✓”) or different (“×”) from the test loss. +2. +Trajectory learning setting that approxi- +mates x(t): here the train set contains m1 initial condi- +tions x(0)α as inputs, while each label corresponds to tra- +jectories yα = {x(t)α}, where t = 0, ∆t, 2∆t, ....k∆t = T +that were realised from the initial condition x(0)α: +Dtrain = {(x(0)α, yα)}, +s.t. x(0)α ∈ Rn, yα ∈ Rkn, x(0)α ∼ fx(0)(x), α ∈ [1, m1], +yα = {x(0)α, x(∆t)α, ..., x(k∆t)α} +and test set Dtest is constructed analogously from m2 +initial conditions that are sampled from the same distri- +bution x(0)α ∼ fx(0)(x). The mapping learnt here is of +the following form: +ˆFFF : Rn → Rkn and is realised by +computing an initial value problem Eq. 2 using a neural +network ΨΨΨ in replacement of FFF. +Experiments and Results +We consider models with +h′ = 6, h = 8, h′′ = 5, hd = 3, trained in 1000 epochs us- +ing Adam optimiser with learning rate of 10−2 and weight +decay 10−3. All activations are ReLU. Unless otherwise +stated, the initial values in both the train set and the test +set are sampled from B[a = 5, b = 5]. For numerical in- +tegration, an explicit Runge-Kutta method of order 5(4) +is used [24]. +The training loss function is the average L1 norm. For +the regression task, the loss is +Ltrain +reg += +1 +Nreg +� +x,y∈Dtrain +� +||f(x) − y||1 + λR(x) +� +, +where Nreg = |Dtrain|(xmax − xmin). For the trajectory +learning task, the loss is defined as: +Ltrain +traj = +1 +Ntraj +� +x(0),y∈Dtrain +T/∆t +� +k=0 +(6) +� +||x(k∆t) − ˆx(k∆t)||1 + λR(x(k∆t)) +� +Here +the +normalisation +constant +is +Ntraj += +|Dtrain|nT(xmax − xmin)/∆t. +λ = 0 and the regu- +larisation terms are nil for the first part of the analysis. +The training sets include samples from 103 trajectories, +the testing sets – from 102 trajectories and the batch +size is 10. The parameters for numerical integration are +∆t = 0.01, T = 1.5. In all cases, a graph was sampled +from Erd¨os-R´enyi ensemble with p = 0.5 and � = � +j. +Tab. I shows that the trained neural network model Eq. +5 can well-approximate the true dynamics and generalise +to unseen initial values well, provided fx(0)(x) is used for +generating both, a training test and a sampling test. +Generalisation +Crucially, the universality of the neu- +ral approximation exemplified in Tab. I is only at the low- +est level that is attainable by putting strong constraints +on a test set (that are in accordance with statistical learn- +ing theory): the two sets must be statistically equivalent. +If the distribution of initial values is irrelevant for the +steady state solution, the neural model also inadvertently +universally approximates the dynamical system. +However, it seems reasonable to ask if a neural net- +work can do better. In Tab. II we propose three tiers of +universality of approximation FFF ≈ ΨΨΨ in terms of statis- +tical properties of training and testing samples. In this +context, the statistical learning theory concerns only the +lowest level of generality. +Level +Relation between f and g +Bottom fX0 ≡ gX0 or fX0,X∞ = fX0fX∞ +Mid +fX0 ̸≡ gX0, sup fX0 = sup gX0 +Top +fX0 ̸≡ gX0, sup fX0 ̸= sup gX0 +TABLE II. Generalisation levels for a neural approximation +FFF ≈ ΨΨΨ is encompassed in model’s ability to extrapolate pre- +dictions to data that was not used during training. Probabil- +ity density functions related to training data are denoted by +“f”; related to testing data are denoted by “g”. +More sophisticated, mid and top level generalisations +would enable a faithful prediction in cases where the con- +straints on statistical properties of data are relaxed, for +example, where fX0 is not the same as gX0. +Diffusion +A dynamical system whose faith and the +course of action depend on the distribution of initial val- +ues enables us to study the limits of generalisation of a + +4 +neural network. Diffusion equation on a graph is a good +example due to its simplicity and known analytical solu- +tion of the form +x(t) = +� +i +ai(0)e−Bλitvi, +ai(0) = x(t)⊤vi, +(7) +where λi, vi are ith eigenvalue and eigenvector of the +graph Laplacian and the steady state solution is given +by +lim +t→∞ xi(t) = 1 +n +� +j +xj(0) +∀i. +Perturbation of the initial value x(0) by δ ∼ fδ such +that xδ +i (0) = xi(0) + δ gives a difference in the steady +state solutions of ⟨xδ +i (0)⟩i − ⟨xi(0)⟩i = γ. +Fig. 1 shows how the loss accumulates over the inte- +gration time t for the neural network model ΨΨΨ for tra- +jectories in the train and in the test sets. In addition, +we consider a perturbation (NN,pert) where the initial +value is sampled from a different distribution, namely, +gX0(x) = B(6, 5), while the neural network was trained +using fX0(x) = B(5, 5). This figure shows that the neu- +ral network prediction is reasonable, under i.i.d. sampling +condition for an initial condition in train and test set. +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +t +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +traj(t) +NN,train +NN,test +NN,pert +Numerical +FIG. 1. +Node average loss between the analytical solution +and: +1) the numerical solution (numerical), 2) the neu- +ral network solution for a subset of initial conditions in the +training set (NN,train) as well as a subset of a testing set +(NN,test). +The original x(0) ∼ B(5, 5), whereas the per- +turbed (NN,pert) initial values x′(0) ∼ B(6, 5). +The loss +is computed for the trajectory learning task using Ntraj = +100 trajectories in each case using an equation Ltraj(t) = +1 +Ntraj +� +x(0),y∈D ||x(t)− ˆx(t)||1. The errors show one standard +deviation. +Fig. 2 follows the same analysis and shows that by +varying the parameters of the beta distribution B(a, b), +the loss in the steady state (averaged over the last 10 +steps of the simulation) is proportional to the differ- +ence in expectation value of the beta-distribution used +in training, and in testing to generate the initial values. +All in all, these results show that the neural network ap- +proximation of the differential form is exclusive to the +statistical properties of the training set. +Upsofar, conservation law (4) and the effect of the reg- +ulariser were not considered. We study it in Fig. 3 for a +small case with a graph composed of N = 2 nodes. This +figure presents two key findings: Fig. 3a) clearly shows +that ΨΨΨ is biased towards the training set; whereas in Fig. +3b) it is clear that ΨΨΨ has the property of implicit dissipa- +tive (conservation) regularization. Even in the case of no +explicit regularization of the dissipative term, the neural +network optimises towards a less dissipative regime. This +is of particular importance, since some systems in Tab. I +are non-dissipative and some are dissipative. +2 +4 +6 +8 +a +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +train +b = a +b = 5 +FIG. 2. +Generalisation of ΨΨΨ to unseen initial conditions. The +neural network was trained using initial values sampled from +x0 ∼ B(5, 5) until it achieved the loss train. Its prediction +capacity was then tested on dynamics with initial conditions +x0 ∼ B(a, b = a) (red circles) as well as x0 ∼ B(a, b = 5) (blue +triangles). The dashed orange line is a function |0.5−a|/(a+ +5)). +The loss is computed for the trajectory learning task +using Ntraj = 100 trajectories in each case using (6), omitting +the term xmax−xmin in the normalisation and considering the +last 10 timesteps. The errors show one standard deviation +across trajectories. +Next, we turn our attention to analyse the out-of- +sample loss for system of n coupled differential equations +(coupling with Erd¨os-R´enyi model) and diffusion dynam- +ics. Notably, the steady state solution is governed by the +average value ⟨x0⟩, and since we have n nodes in our sys- +tem this value has variance ∝ 1/√n. This implies that +it is easier to accurately predict dynamics with a larger +number of differential equations. In Fig. 4, we show that +indeed, test loss is inversely proportional to the system’s +size. +Discussion +In this paper, we proposed a variant of a +Neural ODE model which implements a set of inductive +biases suitable for complex dynamics on graphs and elic- +its dynamical models in complex networked systems di- +rectly from time series these systems produce. While we +showed the presence of generalisation out-of-sample for +a wide range of dynamical models, perhaps more impor- +tantly such an exercise reflects on generalisation capacity +only at the most trivial level. Multiple out-of-distribution + +5 +epoch=0 +epoch=1000 +epoch=2000 +a) +b) +𝜆 = 1 +𝜆 = 0 +Legend +FIG. 3. Learning diffusion on a fully connected n = 2 network +using the regression training paradigm and a conservation +law regulariser. The training sample consists of datapoints +obtained from trajectories generated using x0 ∼ [0.2, 0.7] + +N(0, 0.1), the testing sample: x0 ∼ [0.3, 0.8] + N(0, 0.1). a) +shows an example of a training process, namely by contrast- +ing the true (continuous lines) and learnt (dotted) trajectories +of an initial value problem as predicted after indicated train- +ing epochs, using λ = 1. +b) shows the loss and the value +of the regulariser over training period in the case where the +regulariser plays a part in training (λ = 1, same training as +in a)), and when it does not (λ = 0). The results in b) are +obtained from 10 independent runs. +tests suggest that the neural network approximation is +valid only for a specific probability distribution of initial +values, which was also used to generate the training sam- +ples. Furthermore, even if we kept the statistics intact, +we observe that it is harder to achieve accurate predic- +tions in small-size systems as opposed to large-scale ones, +due to presence of fluctuations that scale as O(1/√n) for +a system of size n. +Appendix A: Encoding and decoding layers +Preceding the differential model layer ΨΨΨ, one can en- +code the input via ΨΨΨe : x ∈ Rn×d → x ∈ Rn×de [10], +in which case, the state space is of n × de dimensions +instead of n × d. +To revert back to the original n × d +space, a decoding function ΨΨΨd is used at the end. The +embedding respects the initial values iff ΨΨΨe = +� +ΨΨΨd�−1. If +the encoding and decoding are obtained via linear lay- +ers without bias terms, they are represented by matri- +ces We ∈ Rd×de and Wd ∈ Rde×d. So after a forward +pass, the initial values are modified if WeWd ̸= I. This +only holds if the two matrices are inverses to each other. +Since these matrices are not square, one can use a Moore- +Penrose inverse, which is a generalisation of the tradi- +tional inverse. We want Wd to be a right inverse of We, +0 +50 +100 +n +0.00 +0.05 +0.10 +0.15 +FIG. 4. Test loss (computed for the last 10 time steps of the +simulation) for a regression learning task at varied network +sizes. +The training and testing datasets are sampled from +B(1, 1). Averages are evaluated using 1000 test samples; for +training, 100 trajectories were used. The figure indicates that +the larger the network, the smaller the average loss and the +variance. +defined as: Wd = W∗ +e(WeW∗ +e)−1 . Here W∗ +e denotes +a Hermitian transpose of We, however in our case it is +equivalent to a transpose, since We is defined over real +numbers. +Appendix B: Neural network mappings +The mappings of functions that constitute the neural +network model defined in Eq. 3 are defined as (here we +consider input x ∈ Rn×1×d, a three-dimensional tensor, +and tensor dimension is counted starting from 1): +1. ψψψℓ : Rn×1×d → Rn×1×d, k = 3 mode product with +W ∈ Rd×h′ i.e. Rn×1×d ×3 Rd×h′ ∈ Rn×1×h′ and +C ∈ Rh′×d. +2. ψψψq1,ψψψq2 : Rn×1×d → Rn×1×h, k = 3 mode product +with W ∈ Rd×h: Rn×1×d ×3 Rd×h ∈ Rn×1×h and +C = I. +3. x⊤1 : Rn×1×h → Rh×n×1. +4. x⊤2 : Rn×1×h → Rh×1×n. +5. +� +ψψψq1(x)⊤1 ×k ψψψq2(x)⊤2� +: +Rh×n×1 ×3 Rh×1×n +∈ +Rh×n×n. +6. 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Basu, Strictly real fundamental theorem of algebra +using polynomial interlacing, Bulletin of the Australian +Mathematical Society 104, 249 (2021). +[22] T. Hastie, R. Tibshirani, J. H. Friedman, and J. H. Fried- +man, The elements of statistical learning: data mining, +inference, and prediction, Vol. 2 (Springer, 2009). +[23] C. Cortes, V. Vapnik, and L. Saitta, Machine Leaming, +Tech. Rep. (1995). +[24] L. F. Shampine, Some practical runge-kutta formulas, +Mathematics of Computation 46, 135 (1986). + diff --git a/FtE4T4oBgHgl3EQfHAzF/content/tmp_files/load_file.txt b/FtE4T4oBgHgl3EQfHAzF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..87f4c4e806f3bf7a794ff694024ac1ed949230d1 --- /dev/null +++ b/FtE4T4oBgHgl3EQfHAzF/content/tmp_files/load_file.txt @@ -0,0 +1,467 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf,len=466 +page_content='Universality of neural dynamics on complex networks Vaiva Vasiliauskaite†∗ and Nino Antulov-Fantulin† Computational Social Science, ETH Z¨urich, 8092 Z¨urich, Switzerland (Dated: January 13, 2023) This paper discusses the capacity of graph neural networks to learn the functional form of ordinary differential equations that govern dynamics on complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' We propose necessary elements for such a problem, namely, inductive biases, a neural network architecture and a learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Sta- tistical learning theory suggests that generalisation power of neural networks relies on independence and identical distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=') of training and testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Although this assumption together with an appropriate neural architecture and a learning mechanism is sufficient for accurate out-of- sample predictions of dynamics such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' mass-action kinetics, by studying the out-of-distribution generalisation in the case of diffusion dynamics, we find that the neural network model: (i) has a generalisation capacity that depends on the first moment of the initial value data distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' (ii) learns the non-dissipative nature of dynamics implicitly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' and (iii) the model’s accuracy resolution limit is of order O(1/√n) for a system of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Introduction Dynamics in a complex networked sys- tem is modelled as a set of n ordinary differential equa- tions (ODEs) that describe the rate of change of a quan- tity xi(t) for each node i and are coupled via adjacency matrix A ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' A general form of these equations is ˙xi = L(xi(t)) + � j AijQ(xi(t), xj(t)) (1) = F(xi(t), x(t), A) where L describes self-interactions, Q is a function that models pairwise interactions between neighbours and � is an aggregation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' With appropriate choices of functions L, Q, � this definition is a general form for models of epidemic processes, biochemical dynamics, birth–death processes, gene regulatory dynamics [1], as well as dynamics that show chaotic behaviour [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The initial value problem of a set of ODEs such as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 1 together with an initial condition x(t0), has a solution that satisfies x(t) = x(t0) + � t t0 FFF(x(t′), A)dt′ (2) and describes a set of trajectories of the dynamics, if the system was initialised at x(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Appropriately setup, a neural network ΨΨΨ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='ωωω) has ca- pacity to approximate any continuous function F with compact support [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' In practice, learning the weights is usually done via some variant of backpropagation algo- rithm [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Notably, neural networks can also be used to approx- imate dynamical systems [5] and find solutions of initial and boundary value problems of differential equations [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' A dynamical system is that in which FFF describes the time dependence of x in an ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Notably, if FFF is known, the description quality of the course of dynamics ∗ vvasiliau@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='ch † Authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' is independent of a coordinate in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' For exam- ple, Newton’s laws of motion describe the trajectory of a bouncing ball regardless of its longitudinal and latitudi- nal position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Recovering universal dynamical principles from empirical data has been shown to belong to NP- hard class [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Despite, the hardness of problem, in recent years, dif- ferent classes of neural networks were used to learn dif- ferent parts of dynamics from empirical data, including graph neural networks [8] and their differential [9] coun- terparts [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' reservoir computers [11, 12] as well as re- gression techniques [13, 14] or to learn control dynam- ics [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Here we discuss architectural design choices and induc- tive biases that are crucial for a neural network model that approximates dynamics evolving on complex net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' We then study the model’s generalisation capac- ity using simple models of deterministic dynamics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Lastly, we discuss our work in the context of learning principles that govern dynamics in complex system from perspective of generalization to unseen initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Inductive biases for dynamics on complex networks There are several important inductive biases and assump- tions worth noting about the complex network dynamics and its neural approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Network structure: There exists a known static network represented as an adjacency matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' There- fore it is reasonable to take a GNN [16] as the candidate for ΨΨΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' A single-layer graph convolution network can be defined as ΨΨΨgnn(x) = (σ [ΦΦΦxW + b]) Wagg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' (3) where x ∈ Rn×d is an input, ΦΦΦ ∈ Rn×n is a graph oper- ator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' ΦΦΦ = ˜D− 1 2 ˜A ˜D− 1 2 [17]), W ∈ Rd×h, b ∈ Rn×1, Wagg ∈ Rh×d are trainable parameters and σ is a non- linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Different versions of GNN with respect to different expressive power for Weisfeiler-Lehman iso- morphism are described in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Self-Interaction: The model includes a self- interaction part that approximates L(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='04900v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='stat-mech] 12 Jan 2023 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Neighbour-Interaction: The model includes a neighbour interaction part that approximates Q(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Note that a single-layer GNN, such as a convolutional graph neural network has no mixed quadratic terms xixj and therefore does not simply satisfy such a con- dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Although theoretically it should still be pos- sible to approximate nonlinear quadratic terms with a single layer neural network with an arbitrary width, in practice it can be challenging and require either a very large number of hidden neurons, or an exotic learning mechanism that goes beyond the standard gradient de- scend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Alternatively, one can improve expressivity of the model by increasing its depth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' using multi-layer GNNs or message-passing neural networks [19] to rep- resent ΨΨΨ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='ωωω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Here ωωω includes graph operator terms ΦΦΦk, k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=', K} where K is the depth of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Spatiotemporal locality: The dynamical process that follows Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 1 must be local, that is, the function Q(·, ·) encodes interactions between neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' How- ever, including terms ΦΦΦk in a multi-layer graph neural network allows for k-hop interactions via length k walks in a network at a timescale smaller than the infinitesimal dt thereby subdividing dt to k intervals and breaking an assumption of temporal locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Aggregation of neighbour-interactions: The ag- gregation can itself be non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Initial value condition: Initial values are preserved during training: x0: ΨΨΨ(x0) → x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' If the neural network straightforwardly approximates the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 2, then enconding and decoding layers must be pseudo-inverses of each other, see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Conservation/dissipation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' If the system is closed, it does not exchange energy or mass with the en- vironment, therefore a conservation law holds, namely � i dxi(t) dt = C ∀t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' (4) A constraint on a neural network to satisfy conservation laws can be imposed via a regularisation term in the loss function, R(D) = 1 |D| � x∈D |FFF(x)1 − ΨΨΨ(x)1| , that penalises the model weights which produce predic- tions which do not respect the conservation law Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Here D is the dataset over which the loss is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The strength of the regulariser term can be modulated by mutiplying R(D) with a non-negative real number λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Architecture Given the inductive biases for dynamics on networks, we propose a neural network model of the following form: ˙x = ψψψℓ(x) + ψψψ � (x) (5) ψψψ � (x) = vec−1� ψψψq3� vec � ΦΦΦ ⊙ � ψψψq1(x)⊤1 ×k ψψψq2(x)⊤2� ��� where ψψψ(x) is a single hidden layer neural network are given by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The mappings of local interaction are summarised in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The design choices of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 5 comply with the inductive biases stated earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' To this end, we performed vetorisation of input to the function ψψψ � [ψψψq3 (·)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' This function can approximate any invariant poolings of a set [20] or a multiset [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Notably, we also assumed that Q(·, ·) is factorisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Since it can be approximated by Chebyshev polynomials, and, according to the strictly real fundamental theorem of algebra [21], it is possible to factorise polynomial function to two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Alternatively, one can use deep sets [20] as arguments to approximate Q(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' In order to guarantee the local existence and unique- ness of the solution to the initial value problem, by Pi- card–Lindel¨of theorem the neural network ΨΨΨ needs to be Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' To enforce Lipschitz continuity of ΨΨΨ, we will be using 1-Lipschitz activation functions such as ReLU, sigmoid, softmax, or tanh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Learning task We formulate two distinct statistical learning settings that relate to an increasing strength of generality in the approximation of a dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Regression task to approximate FFF by ΨΨΨ: An appropriate “proto data set” here is D = {(x(t)α, y(t)α)}, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' x(t)α ∈ Rn, y(t)α ∈ Rn, x(0)α ∼ fx(0)(x), t = [0, T] ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' our labels are defined as y(t)α = FFF(x((t))α), α denotes α-th initial condition x(0)α sampled from a predefined distribution fx(0)(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' all others points x(t)α are obtained following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Here the functional mapping that is be- ing learnt is ˆFFF : Rn → Rn and is obtained by minimising the loss L between the true labels y and the labels f(x) obtained by the current model: ˆFFF = arg min f:Rn→Rn E P(x,y) L(f(x), y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Here E is an expectation operator, P(x, y) is the data sampling distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' At the moment, samples from the “proto data set” are not independent: those trajectories that were obtained from the same initial condition are non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Such sam- pling is compulsory for the Uniform Law of Large num- bers, that together with capacity control ensures general- isation from train to test set [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' To ensure statistical independence of samples, we create finite train and test sets of size m1, m2 by using a specific distribution P over a “proto data set” Dtrain ∪ Dtest ∼ P(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Specifically, we randomly delegate (x(t)α, y(t)α) to either Dtrain or Dtest thereby ensuring an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' condition by dropping information on the initial conditions and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 3 Dynamics L Q Ltrain reg Ltest reg ≈reg Ltrain traj Ltest traj ≈traj Heata – B(xj − xi) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='03 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='14 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='08 ✓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='63 ✓ MAKb F − Bxb i Rxj 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='41 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='44 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='14 ✓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='04 × PDc −Bxb i Rxa j 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='68 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='82 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='72 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='89 ✓ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='03 ✓ MMd −Bxi R xh j 1+xh j 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='68 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='36 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='83 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='47 ✓ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='14 ✓ SISe −Bxi (1 − xi)xj 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='16 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='31 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='07 ✓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='02 × a B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' b B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='1, R = 1, f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' c B = 2, R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='3, a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='5, b = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' d B = 4, R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='5, h = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' e B = 5, R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Generalisation of a neural network model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 5 trained on dynamics from [1] in the regression task setting, and the trajectory learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Reported loss values are multiplied by a factor 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' In columns denoted “≈” we indicate for which dynamics the train loss is approximately similar (“✓”) or different (“×”) from the test loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Trajectory learning setting that approxi- mates x(t): here the train set contains m1 initial condi- tions x(0)α as inputs, while each label corresponds to tra- jectories yα = {x(t)α}, where t = 0, ∆t, 2∆t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='.k∆t = T that were realised from the initial condition x(0)α: Dtrain = {(x(0)α, yα)}, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' x(0)α ∈ Rn, yα ∈ Rkn, x(0)α ∼ fx(0)(x), α ∈ [1, m1], yα = {x(0)α, x(∆t)α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=', x(k∆t)α} and test set Dtest is constructed analogously from m2 initial conditions that are sampled from the same distri- bution x(0)α ∼ fx(0)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The mapping learnt here is of the following form: ˆFFF : Rn → Rkn and is realised by computing an initial value problem Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 2 using a neural network ΨΨΨ in replacement of FFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Experiments and Results We consider models with h′ = 6, h = 8, h′′ = 5, hd = 3, trained in 1000 epochs us- ing Adam optimiser with learning rate of 10−2 and weight decay 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' All activations are ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Unless otherwise stated, the initial values in both the train set and the test set are sampled from B[a = 5, b = 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' For numerical in- tegration, an explicit Runge-Kutta method of order 5(4) is used [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The training loss function is the average L1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' For the regression task, the loss is Ltrain reg = 1 Nreg � x,y∈Dtrain � ||f(x) − y||1 + λR(x) � , where Nreg = |Dtrain|(xmax − xmin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' For the trajectory learning task, the loss is defined as: Ltrain traj = 1 Ntraj � x(0),y∈Dtrain T/∆t � k=0 (6) � ||x(k∆t) − ˆx(k∆t)||1 + λR(x(k∆t)) � Here the normalisation constant is Ntraj = |Dtrain|nT(xmax − xmin)/∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' λ = 0 and the regu- larisation terms are nil for the first part of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The training sets include samples from 103 trajectories, the testing sets – from 102 trajectories and the batch size is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The parameters for numerical integration are ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='01, T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' In all cases, a graph was sampled from Erd¨os-R´enyi ensemble with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='5 and � = � j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' I shows that the trained neural network model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 5 can well-approximate the true dynamics and generalise to unseen initial values well, provided fx(0)(x) is used for generating both, a training test and a sampling test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Generalisation Crucially, the universality of the neu- ral approximation exemplified in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' I is only at the low- est level that is attainable by putting strong constraints on a test set (that are in accordance with statistical learn- ing theory): the two sets must be statistically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' If the distribution of initial values is irrelevant for the steady state solution, the neural model also inadvertently universally approximates the dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' However, it seems reasonable to ask if a neural net- work can do better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' II we propose three tiers of universality of approximation FFF ≈ ΨΨΨ in terms of statis- tical properties of training and testing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' In this context, the statistical learning theory concerns only the lowest level of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Level Relation between f and g Bottom fX0 ≡ gX0 or fX0,X∞ = fX0fX∞ Mid fX0 ̸≡ gX0, sup fX0 = sup gX0 Top fX0 ̸≡ gX0, sup fX0 ̸= sup gX0 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Generalisation levels for a neural approximation FFF ≈ ΨΨΨ is encompassed in model’s ability to extrapolate pre- dictions to data that was not used during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Probabil- ity density functions related to training data are denoted by “f”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' related to testing data are denoted by “g”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' More sophisticated, mid and top level generalisations would enable a faithful prediction in cases where the con- straints on statistical properties of data are relaxed, for example, where fX0 is not the same as gX0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Diffusion A dynamical system whose faith and the course of action depend on the distribution of initial val- ues enables us to study the limits of generalisation of a 4 neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Diffusion equation on a graph is a good example due to its simplicity and known analytical solu- tion of the form x(t) = � i ai(0)e−Bλitvi, ai(0) = x(t)⊤vi, (7) where λi, vi are ith eigenvalue and eigenvector of the graph Laplacian and the steady state solution is given by lim t→∞ xi(t) = 1 n � j xj(0) ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Perturbation of the initial value x(0) by δ ∼ fδ such that xδ i (0) = xi(0) + δ gives a difference in the steady state solutions of ⟨xδ i (0)⟩i − ⟨xi(0)⟩i = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 1 shows how the loss accumulates over the inte- gration time t for the neural network model ΨΨΨ for tra- jectories in the train and in the test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' In addition, we consider a perturbation (NN,pert) where the initial value is sampled from a different distribution, namely, gX0(x) = B(6, 5), while the neural network was trained using fX0(x) = B(5, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' This figure shows that the neu- ral network prediction is reasonable, under i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' sampling condition for an initial condition in train and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='50 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='07 traj(t) NN,train NN,test NN,pert Numerical FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Node average loss between the analytical solution and: 1) the numerical solution (numerical), 2) the neu- ral network solution for a subset of initial conditions in the training set (NN,train) as well as a subset of a testing set (NN,test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The original x(0) ∼ B(5, 5), whereas the per- turbed (NN,pert) initial values x′(0) ∼ B(6, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The loss is computed for the trajectory learning task using Ntraj = 100 trajectories in each case using an equation Ltraj(t) = 1 Ntraj � x(0),y∈D ||x(t)− ˆx(t)||1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The errors show one standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 2 follows the same analysis and shows that by varying the parameters of the beta distribution B(a, b), the loss in the steady state (averaged over the last 10 steps of the simulation) is proportional to the differ- ence in expectation value of the beta-distribution used in training, and in testing to generate the initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' All in all, these results show that the neural network ap- proximation of the differential form is exclusive to the statistical properties of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Upsofar, conservation law (4) and the effect of the reg- ulariser were not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' We study it in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 3 for a small case with a graph composed of N = 2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' This figure presents two key findings: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 3a) clearly shows that ΨΨΨ is biased towards the training set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' whereas in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 3b) it is clear that ΨΨΨ has the property of implicit dissipa- tive (conservation) regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Even in the case of no explicit regularization of the dissipative term, the neural network optimises towards a less dissipative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' This is of particular importance, since some systems in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' I are non-dissipative and some are dissipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 2 4 6 8 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='30 train b = a b = 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Generalisation of ΨΨΨ to unseen initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The neural network was trained using initial values sampled from x0 ∼ B(5, 5) until it achieved the loss train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Its prediction capacity was then tested on dynamics with initial conditions x0 ∼ B(a, b = a) (red circles) as well as x0 ∼ B(a, b = 5) (blue triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The dashed orange line is a function |0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='5−a|/(a+ 5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The loss is computed for the trajectory learning task using Ntraj = 100 trajectories in each case using (6), omitting the term xmax−xmin in the normalisation and considering the last 10 timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The errors show one standard deviation across trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Next, we turn our attention to analyse the out-of- sample loss for system of n coupled differential equations (coupling with Erd¨os-R´enyi model) and diffusion dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Notably, the steady state solution is governed by the average value ⟨x0⟩, and since we have n nodes in our sys- tem this value has variance ∝ 1/√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' This implies that it is easier to accurately predict dynamics with a larger number of differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 4, we show that indeed, test loss is inversely proportional to the system’s size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Discussion In this paper, we proposed a variant of a Neural ODE model which implements a set of inductive biases suitable for complex dynamics on graphs and elic- its dynamical models in complex networked systems di- rectly from time series these systems produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' While we showed the presence of generalisation out-of-sample for a wide range of dynamical models, perhaps more impor- tantly such an exercise reflects on generalisation capacity only at the most trivial level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Multiple out-of-distribution 5 epoch=0 epoch=1000 epoch=2000 a) b) 𝜆 = 1 𝜆 = 0 Legend FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Learning diffusion on a fully connected n = 2 network using the regression training paradigm and a conservation law regulariser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The training sample consists of datapoints obtained from trajectories generated using x0 ∼ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='7] + N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='1), the testing sample: x0 ∼ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='8] + N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' a) shows an example of a training process, namely by contrast- ing the true (continuous lines) and learnt (dotted) trajectories of an initial value problem as predicted after indicated train- ing epochs, using λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' b) shows the loss and the value of the regulariser over training period in the case where the regulariser plays a part in training (λ = 1, same training as in a)), and when it does not (λ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The results in b) are obtained from 10 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' tests suggest that the neural network approximation is valid only for a specific probability distribution of initial values, which was also used to generate the training sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Furthermore, even if we kept the statistics intact, we observe that it is harder to achieve accurate predic- tions in small-size systems as opposed to large-scale ones, due to presence of fluctuations that scale as O(1/√n) for a system of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Appendix A: Encoding and decoding layers Preceding the differential model layer ΨΨΨ, one can en- code the input via ΨΨΨe : x ∈ Rn×d → x ∈ Rn×de [10], in which case, the state space is of n × de dimensions instead of n × d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' To revert back to the original n × d space, a decoding function ΨΨΨd is used at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The embedding respects the initial values iff ΨΨΨe = � ΨΨΨd�−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' If the encoding and decoding are obtained via linear lay- ers without bias terms, they are represented by matri- ces We ∈ Rd×de and Wd ∈ Rde×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' So after a forward pass, the initial values are modified if WeWd ̸= I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' This only holds if the two matrices are inverses to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Since these matrices are not square, one can use a Moore- Penrose inverse, which is a generalisation of the tradi- tional inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' We want Wd to be a right inverse of We, 0 50 100 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='15 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Test loss (computed for the last 10 time steps of the simulation) for a regression learning task at varied network sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The training and testing datasets are sampled from B(1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Averages are evaluated using 1000 test samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' for training, 100 trajectories were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' The figure indicates that the larger the network, the smaller the average loss and the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' defined as: Wd = W∗ e(WeW∗ e)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Here W∗ e denotes a Hermitian transpose of We, however in our case it is equivalent to a transpose, since We is defined over real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Appendix B: Neural network mappings The mappings of functions that constitute the neural network model defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 3 are defined as (here we consider input x ∈ Rn×1×d, a three-dimensional tensor, and tensor dimension is counted starting from 1): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' ψψψℓ : Rn×1×d → Rn×1×d, k = 3 mode product with W ∈ Rd×h′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Rn×1×d ×3 Rd×h′ ∈ Rn×1×h′ and C ∈ Rh′×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' ψψψq1,ψψψq2 : Rn×1×d → Rn×1×h, k = 3 mode product with W ∈ Rd×h: Rn×1×d ×3 Rd×h ∈ Rn×1×h and C = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' x⊤1 : Rn×1×h → Rh×n×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' x⊤2 : Rn×1×h → Rh×1×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' � ψψψq1(x)⊤1 ×k ψψψq2(x)⊤2� : Rh×n×1 ×3 Rh×1×n ∈ Rh×n×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' ΦΦΦ ⊙ � ψψψq1(x)⊤1 ×k ψψψq2(x)⊤2� : Rn×n ⊙ Rh×n×1 ×3 Rh×1×n ∈ Rh×n×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' Here an operator ⊙ denotes a standard “broadcasted” element-wise multiplica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' vec(·): Rh×n×n → Rn2h×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' ψψψq3 : Rn2h×1 → Rn2h×1, W ∈ R1×h′′ and C ∈ Rh′′×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' vec−1(·): Rn2h×1 → Rn×nh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' 6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' ψψψ � = ψ(�(·)), where we use �(·) as invariant pooling layer Rn×nh → Rn×1 and then apply de- coding layer ψ that maps Rn×1 → Rn×d, with W ∈ R1×hd and C ∈ Rhd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'} +page_content=' [1] B.' metadata={'source': 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Antonov(1)∗, A. Leonidov(1,2), and A. Semenov(1,3) +(1) P.N. Lebedev Physical Institute, Moscow, Russia +(2) Moscow Institute of Physics and Technology, Dolgoprudny, Russia +(3) Higher School of Economics, Moscow, Russia +Abstract +Transitions between metastable equilibria in the low-temperature phase +of dynamical Ising game with activity spillover are studied in the infinite +time limit. +It is shown that exponential enhancement due to activity +spillover previously found for finite-time transitions in [1] is absent in the +infinite time limit. Analytical description for infinite time trajectory is +developed and compared with results of exact numerical analysis. +∗antonov@lpi.ru +1 +arXiv:2301.01285v1 [cond-mat.stat-mech] 3 Jan 2023 + +1. Introduction +Studies of noisy binary choice games are of special interest because of the +existence of close parallels to statistical physics of spin systems, in particular to +static and dynamic properties of phase transitions in them [2, 3, 4]. These par- +allels are particularly intriguing because of the fundamentally different origins +of equilibria in game theory and statistical physics: in game theory equilibration +is a result of balancing individual interests while in statistical physics equilibra- +tion is a search of a global minimum of free energy. For the noisy binary choice +problem on complete graphs it is long known, see [2] and references therein, +that for a special choice of noise game-theoretic equilibria are characterised by +the same mean-field Curie-Weiss equation as that describing phase transitions +in magnetics, see e.g. [4]. The properties of static and dynamic equilibria in +noisy binary choice games were studied in [5, 6, 7] for arbitrary noise, and com- +plete and random graph topologies. It was established in particular that static +game-theoretic equilibria in noisy binary choice games on graphs correspond to +the so-called quantal response/expectation equilibria [8]. +The dynamics of games can, however, be fundamentally different from con- +ventional spin dynamics due to a variety of possible mechanisms. One of these +is a possibility of activity spillover (self-excitation) that was intensively studied +for so-called Hawkes processes [9] with applications to finance [10, 11], earth- +quakes [12] and other subjects, see the recent review in [13]. A master equation +formalism for such processes was developed in [14, 15]. The effects of an ac- +tivity spillover different from the Hawkes self-excitation mechanism for a noisy +binary choice game (Ising game) on complete graphs was studied in [1]. The +main focus of [1] was in studying transitions between metastable equilibria in +the low-temperature phase taking place at finite time. It was observed that +activity spillover leads to an exponential acceleration of such transitions. The +present paper complements the analysis of [1] by studying transitions between +metastable equilibria in the limit of infinite time. The importance of studying +this limit is, first, in establishing a link with a rich literature on Kramers rate +[16] and, second, in that in this limit the exponential enhancement is absent +and an analysis of pre-exponential contribution is necessary. In analysing this +problem we develop an analytical description of the infinite-limit trajectory and +suggest an analytical formula for the transition rate that is compared with the +results of exact numerical simulations. +2. Model +We consider a dynamical noisy binary choice game of N agents on a complete +graph topology. Each agent i has two possible strategies si = ±1 so the system +is fully described by the vector st = (s1, . . . , sn)t at given time t. The temporal +evolution of the strategies configuration st → st+δt within a small time interval +δt is assumed to be driven by a strategy flip si → −si of some agent i with the +flip probability +Prob[si → −si|(t; t + δt)] = λi(t)δt γi(si → −si|s−i,t) +(1) +where λi(t) is an activity rate of the agent i, i.e. λi(t)δt is a time-dependent +probability for an agent i to be active and have a possibility to change a strategy +2 + +within a time interval (t, t + δt) while γ(si → −si|s−i,t) is a probability, for an +active agent i, of a strategy flip dependent of the current configuration s−i,t of +strategies in the neighbourhood of this node. In what follows we shall assume +a noisy best response (Ising-Glauber) flip rate1. For a complete graph topology +at large N, it is the same for all agents +γ(m(t)) = 1 +2 [1 − si tanh (βJm(t))] +→ +γ±(m(t)) = 1 +2 [1 ± tanh(βJm(t))] +(2) +where β = 1/T is an inverse temperature, J is an Ising coupling constant, +γ± = γ(∓s → ±s) and m(t) = +1 +N +�N +i=1 si. For the compete graph topology +activity rates {λi} are also the same for all agents, λi(t) = λ(t) for any i. +The time-dependence of the activity rate λ(t) is due to the spillover effect +driven by the past events of strategy flips assumed to be described by the Hawkes +process [9] with an exponential kernel: +λ(t) = λ0 + µ +N +� +τk 1 it has two symmetrical (meta)stable equlibria at +meq = ±m0(β) as well as the unstable one at m = 0 serving as a separatrix +separating the two stable ones. +The λ - equilibria are more complicated and depend on both temperature +βJ and self-excitation memory kernel parameter b. +In the high-temperature phase βJ < 1 and b > 1, we have the equilibrium +configuration of the form +m = 0, λ = bλ0 +b − 1 +while for b < 1 we have a blow-up solution with λ → ∞ for m = 0. +In the low-temperature phase βJ > 1, three following modes are possible: +• Mode 1 “calm agents”: if b > 1, then we have two (meta)stable equilibrium +configurations at +m = ±m0(β), λ = +bλ0 +b − 1 + m2 +0(β) = ˜λ(m0) +as well as the unstable saddle one at +m = 0, λ = bλ0 +b − 1 +• Mode 2 “excited agents”: if 1 − m2 +0(β) < b < 1, then we still have equilib- +rium configurations at +m = ±m0(β), λ = ˜λ(m0), +but the saddle configuration is now absent: +m = 0, λ → ∞ +• Mode 3 “physcho agents”: if b < 1 − m2 +0(β), then +λ → ∞ +for all extrema of the m-axis. +4 + +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +βJ +b +Mode 2 +Mode 3 +Single equilibrium +Mode 1 +Figure 1. Phase diagram of all possible modes in the (b, βJ) plane for λ0 = 1. +The red area (single equilibrium) here denotes the presence of only equilibrium +along the m-axis. Blue, yellow and green areas correspond to Modes 1, 2 and +3, respectively (see the description in the main text). +5 + +The phase diagram showing the above modes is given in Fig. 1. +At the timescale of τλ ∼ 1/b, in Modes 1 and 2 the system relaxes to the ap- +propriate temperature-dependent equilibrium while the Mode 3 does not corre- +spond to any equilibrium. The dependence of such a relaxation on temperature +βJ and Hawkes parameter b in the Modes 1 and 2 was studied in [1]. +At low temperatures βJ > 1, the equilibrium configurations for Modes 1 and +2 are in fact metastable due to noise-induced transitions of the type m0(β) ↔ +−m0(β) taking place at large timescale τ ≫ τλ. +The saddle we introduced +for Mode 1 then has the following physical meaning: it is the point where the +transition trajectory from one equilibrium to another at the infinite time limit +crosses the separatrix m = 0. [18] +To consider these transitions, here and in what follows we fix βJ = 1.5 to +establish the mode with two metastable equilibria (Mode 1 or 2, see Fig. 1). +For our convenience, in what follows we shall consider the transition −m0(β) → +m0(β). +3. Transition between metastable equilibria +3.1. Long-time behaviour of probability density function +The subject of our study is a comparison of the transition probability be- +tween the states (m(ta), λ(ta)) and (m(tb), λ(tb)) within the time interval [ta, tb] +for Hawkes and Poisson Ising games. In what follows we shall use a condensed +notation xa,b = (m(ta,b), λ(ta,b)) and fix [ta, tb] = [0, τ] so that the transition +probability between two metastable states is +P(xb, t|xa, 0) ≡ P(xb, t) +��� +x(0)=xa +(7) +where m(0) = −m0(β), λ(0) = ˜λ(m0) and m(τ) = m0(β), λ(τ) = ˜λ(m0). The +transition probability (7) obeys [1] the Fokker-Planck equation (4). +In the previous paper we have compared the probabilities of transition be- +tween metastable equilibria in Hawkes and Poisson Ising games within a finite +time interval [0, τ] and demonstrated an exponential acceleration of this tran- +sition in the Hawkes case. The main goal of the present paper is to calculate +this transition probability in the limit τ → ∞. To discuss this limit let us use, +following [1], the analogy with classical mechanics. A formal justification for it +can be found, e.g., in [19]. +As the diffusion coefficient in (4) is proportional to 1/N, in the limit of N → +∞, for solving the Fokker-Planck equation we can use the WKB approximation. +Introducing an analogue of action S(x, t) through P(x, t) ∝ e−NS(x,t), we get +the following Hamilton-Jacobi equation for S: +∂tS(x, t) = fi(x(t))∂S(x, t) +∂xi +− gij(x(t))∂S(x, t) +∂xi +∂S(x, t) +∂xj +. +(8) +On can also introduce an analogue of the Hamiltonian +H(p, x; t) = −fi(x(t))pi(t) + gij(x(t))pi(t)pj(t), +pi = ∂S +∂xi +(9) +6 + +The time evolution of the system is then given by the corresponding Hamilton +equations +˙xi(t) + fi(x(t)) += +2gij(x(t))pj(t) +˙pi(t) − pj∂ifj(x(t)) += +−pj∂igjk(x(t))pk(t). +(10) +The system of Hamilton equations (10) has the first integral H(p, x) = E. As +will be shown later, the value of E implicitly sets conditions on the transition +time τ from one metastable equilibrium to another in the classical problem. +The leading contribution to the transition probability has the form +P(xi, xf; τ) ∝ e−NS +(11) +where the exponential factor S can be calculated by implementing the Mopertui +principle [20] +S = S0 − Eτ = +� +i +� ∞ +0 +pi(t) ˙xi(t)dt − Eτ = +� +i +� +trajectory +pidxi − Eτ. +(12) +The transition trajectory itself is determined by equations (10), the first +integral H(p, x) = E and, obviously, should minimise the trajectory-depended +term S0. Transition time is set by E via relation τ = ∂S0/∂E. [20] +In [1] we considered transition probability from one metastable equilibrium +to another in finite time (E ̸= 0) and found out that the probability exponen- +tially increases due to activity spillover. In the present study we augment the +results of [1] by considering introducing transition rates in the infinite time limit +corresponding to E = 0. +The system of differential equation (10) for E = 0 is solvable in quadratures. +The corresponding solution for the transition trajectory can naturally be broken +into two pieces. +The first piece corresponding to transition from the initial equilibrium to +separatrix −m0(β) → 0. The corresponding formulae read +˙m(t) += +λ(t)[m − tanh(βJm)], +(13) +˙λ(t) += +λ(t)[1 − m tanh(βJm)] − b(λ(t) − λ0) +− +λ(t)[m − tanh(βJm)]2 +1 − m tanh(βJm) +, +(14) +pm += +m − tanh(βJm) +1 − m tanh(βJm), +(15) +pλ += +0. +(16) +The second piece corresponding to transition from the separatrix to another +equilibrium 0 → m0(β) . The corresponding formulae read +˙m(t) += +−λ(t)[m − tanh(βJm)], +(17) +˙λ(t) += +λ(t)[1 − m tanh(βJm)] − b(λ(t) − λ0), +(18) +pm += +0, +(19) +pλ += +0. +(20) +7 + +We note that despite the symmetry with respect to m-axis, the transition tra- +jectory is asymmetric as the external field is non-gradient. +In accordance with the classification of modes introduced in Section 2, for +different values of the parameter b the Hawkes transition trajectory does either +pass through the saddle point where it has the discontinuity (Mode 1) or di- +verges at the separatrix m = 0 (Mode 2). The trajectories for various values of +parameter b are shown in Fig. 2. +1 +2 +3 +4 +5 +6 +7 +−1 +0 +m0 +−m0 +1 +λ +m +b = 2.0 +b = 1.5 +b = 1.2 +b = 0.5 +Figure 2. +Transition trajectories +−m0(β) → m0(β) at the infinite time +limit E = 0 given by Eqs. 13-16 (left half) and Eqs. 17-20 (right half) for +b = 0.5, 1.2, 1.5, 2.0 at βJ = 1.5, λ0 = 1. +The trajectories for Mode 1 +(b = 1.2, 1.5, 2.0) are defined and have a discontinuity at the saddle, and the +trajectory for Mode 2 (b = 0.5) diverges at m = 0. +From Eqs. (12),(13-20) it follows that in the infinite time limit for which +E = 0, the exponential factor S for Poisson and Hawkes Ising games is the +same is equal for Poisson and Hawkes games for all b: +S = +0 +� +−m0(β) +m − tanh(βJm) +1 − m tanh(βJm)dm +(21) +Therefore, for understanding a possible difference between the Hawkes and +Poisson Ising games in the infinite time limit, an analysis of pre-exponential +factor of the transition rate is required. +8 + +3.2. Pre-exponential factor of the transition rate +The calculation of the pre-exponential factor for the one-dimensional Pois- +son game closely follows the original calculation by Kramers [16] and can be +done analytically, see e.g. [21, 22]. A more general result for larger number +of dimensions, including the case of non-potential fields, was obtained in [23]. +However, this result is not applicable in our the two-dimensional Hawkes game, +since the transition trajectory in the non-gradient field has a discontinuity, see +a related discussion in [24]. +When the trajectory is defined (Mode 1), we can use analogies with one- +dimensional motion. In the Kramers’ problem for the potential with smooth +barrier the pre-exponential factor of escape rate depends on second derivatives +of the potential both for stationary attractor and a saddle. +However, if the +potential barrier is edge-shaped, the result depends only on the second derivative +of the potential at stationary attractor.[25]. That leads us to an assumption that +in the Hawkes game acceleration with respect to Poisson one is caused only by +a corresponding change in the activity of agents in the equilibrium state, with +the rest of motion having non-significant effect on the transition time. That +means average transition times in the Hawkes and Poisson games with intensity +˜λ(m0) are equal. Therefore a ratio of transition times in the original Hawkes +and Poisson games can be written in the following form: +ttr,P +ttr,H +≃ +b +b − 1 + m2 +0(β). +(22) +To check the above-formulated assumption we have performed computer +simulations of Hawkes and Poisson games as well as those of Langevin equations +that correspond to Eq. 4. We have also checked that the transition time ratio +does not depend on number of agents for N ≥ 20, i.e. when the number of +agents is sufficiently large. A comparison of the results of these simulations +with Eq. 22 is shown in Fig. 3. +From Fig. 3 we see that the activity in the Hawkes game as compared to the +Poisson one is indeed enhanced. A more detailed conclusion is that in the regime +corresponding to Mode 1 the formula in Eq. (22) works well for the Mode 1 for +both continuum and discrete cases, but in the regime corresponding to Mode +2 it is, due to the presence of divergence the continuum generalisation of the +game, not in agreement with the exact discrete formulation. Despite this, the +shape of the transition trajectory still provides us a qualitatively correct insight +into the behaviour of agents, see Fig. 4. +Let us note that the decision process does significantly intensify around the +separatrix, i.e. when are uncertain of which of the two (quasi)stable equilibria +to choose. Once the decision is made, the agents calm down. +4. Conclusions +We have studied the self-excited Ising game on a complete graph. Inspite +of its simplicity, it has rich dynamics exhibiting various types of behaviour. +Competition of “calming down” and “activation” in the Hawkes self-excitation +mechanism at different levels of noise results in three possible modes (phases). +9 + +1 +1.2 +1.4 +1.6 +1.8 +2 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +ttr,P/ttr,H +b +game +langevin +analytics +Mode 1 +Mode 2 +Figure 3. +Ratio of transition times in Hawkes and Poisson cases. +Triangles +show simulation results for games (discrete model), and squares show results +for Langevin equations (continuum model). The line refers to the theoretical +prediction given by Eq. (22). Dashed line b = 1 separates Mode 1 (blue area) +from Mode 2 (yellow area). +10 + +2 +4 +6 +8 +10 +12 +14 +−1 +0 +1 +λ +m +Analytics +Agent model +Figure 4. Example of transition in Hawkes game (red line) for b = 0.5, λ0 = +1, N = 20. As in the corresponding transition trajectory (blue line), the inten- +sity of decision making process increases near m = 0. +We expect that this competition might play an important role in other situ- +ations, e.g. for non-exponential Hawkes kernels [26] or for more complicated +graph topology. +Another focus in this work was to investigate the probability of transition +between metastable equilibria in the infinite time limit. This is a very challeng- +ing task for a multi-dimensional case when the external field is non-gradient and +has a discontinuity. Also, since in the relevant one-dimensional case (i.e. when +the potential field only has the discontinuity) it is known that the dynamics for +such fields is rather different that for smooth potential fields [27], it would be +natural to assume a similar situation in the multi-dimensional case. However, +based on the intuitive understanding of the considered model, we have presented +an approach that allows us to reduce the problem to calculating the transition +time in the corresponding one-dimensional model. The analytically calculated +transition trajectory also gave us a qualitative insight into the behaviour of +agents in the corresponding discrete system. +As for further developments of the suggested approach, an interesting idea +would be working out its generalisation for two- and multi-dimensional systems. +Compared to other another existing approaches for treating the case of non- +gradient external field (see e.g. [28, 29]), this newly introduced method could +present a workable alternative due to its simplicity. +11 + +References +[1] A. Antonov, A. Leonidov, and A. Semenov. Self-excited ising game. Physica +A: Statistical Mechanics and its Applications, 561:125305, 2021. +[2] Lawrence Blume and Steven Durlauf. 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Kramers’ activation rate for a sharp edged potential barrier: +The double oscillator. Physica A: Statistical Mechanics and its Applica- +tions, 136(1):124–146, 1986. +[28] Nicholas Paskal and Maria Cameron. An efficient jet marcher for computing +the quasipotential for 2d sdes, 2021. +[29] Peter Ashwin, +Jennifer Creaser, +and Krasimira Tsaneva-Atanasova. +Quasipotentials for coupled escape problems and the gate-height bifurca- +tion, 2022. +13 + diff --git a/KNAzT4oBgHgl3EQfVfxu/content/tmp_files/load_file.txt b/KNAzT4oBgHgl3EQfVfxu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a51bbb709197ebedae21cdeaccd88016aed29aff --- /dev/null +++ b/KNAzT4oBgHgl3EQfVfxu/content/tmp_files/load_file.txt @@ -0,0 +1,356 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf,len=355 +page_content='Transition between metastable equilibra: applications to binary-choice games A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' Antonov(1)∗, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' Leonidov(1,2), and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' Semenov(1,3) (1) P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' Lebedev Physical Institute, Moscow, Russia (2) Moscow Institute of Physics and Technology, Dolgoprudny, Russia (3) Higher School of Economics, Moscow, Russia Abstract Transitions between metastable equilibria in the low-temperature phase of dynamical Ising game with activity spillover are studied in the infinite time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' It is shown that exponential enhancement due to activity spillover previously found for finite-time transitions in [1] is absent in the infinite time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' Analytical description for infinite time trajectory is developed and compared with results of exact numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' ∗antonov@lpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content='ru 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content='01285v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content='stat-mech] 3 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' Introduction Studies of noisy binary choice games are of special interest because of the existence of close parallels to statistical physics of spin systems, in particular to static and dynamic properties of phase transitions in them [2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' These par- allels are particularly intriguing because of the fundamentally different origins of equilibria in game theory and statistical physics: in game theory equilibration is a result of balancing individual interests while in statistical physics equilibra- tion is a search of a global minimum of free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' For the noisy binary choice problem on complete graphs it is long known, see [2] and references therein, that for a special choice of noise game-theoretic equilibria are characterised by the same mean-field Curie-Weiss equation as that describing phase transitions in magnetics, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' The properties of static and dynamic equilibria in noisy binary choice games were studied in [5, 6, 7] for arbitrary noise, and com- plete and random graph topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' It was established in particular that static game-theoretic equilibria in noisy binary choice games on graphs correspond to the so-called quantal response/expectation equilibria [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' The dynamics of games can, however, be fundamentally different from con- ventional spin dynamics due to a variety of possible mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' One of these is a possibility of activity spillover (self-excitation) that was intensively studied for so-called Hawkes processes [9] with applications to finance [10, 11], earth- quakes [12] and other subjects, see the recent review in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' A master equation formalism for such processes was developed in [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' The effects of an ac- tivity spillover different from the Hawkes self-excitation mechanism for a noisy binary choice game (Ising game) on complete graphs was studied in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' The main focus of [1] was in studying transitions between metastable equilibria in the low-temperature phase taking place at finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' It was observed that activity spillover leads to an exponential acceleration of such transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' The present paper complements the analysis of [1] by studying transitions between metastable equilibria in the limit of infinite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' The importance of studying this limit is, first, in establishing a link with a rich literature on Kramers rate [16] and, second, in that in this limit the exponential enhancement is absent and an analysis of pre-exponential contribution is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' In analysing this problem we develop an analytical description of the infinite-limit trajectory and suggest an analytical formula for the transition rate that is compared with the results of exact numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' Model We consider a dynamical noisy binary choice game of N agents on a complete graph topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' Each agent i has two possible strategies si = ±1 so the system is fully described by the vector st = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' , sn)t at given time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' The temporal evolution of the strategies configuration st → st+δt within a small time interval δt is assumed to be driven by a strategy flip si → −si of some agent i with the flip probability Prob[si → −si|(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' t + δt)] = λi(t)δt γi(si → −si|s−i,t) (1) where λi(t) is an activity rate of the agent i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' λi(t)δt is a time-dependent probability for an agent i to be active and have a possibility to change a strategy 2 within a time interval (t, t + δt) while γ(si → −si|s−i,t) is a probability, for an active agent i, of a strategy flip dependent of the current configuration s−i,t of strategies in the neighbourhood of this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' In what follows we shall assume a noisy best response (Ising-Glauber) flip rate1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' For a complete graph topology at large N, it is the same for all agents γ(m(t)) = 1 2 [1 − si tanh (βJm(t))] → γ±(m(t)) = 1 2 [1 ± tanh(βJm(t))] (2) where β = 1/T is an inverse temperature, J is an Ising coupling constant, γ± = γ(∓s → ±s) and m(t) = 1 N �N i=1 si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' For the compete graph topology activity rates {λi} are also the same for all agents, λi(t) = λ(t) for any i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'} +page_content=' The time-dependence of the activity rate λ(t) is due to the spillover effect driven by the past events of strategy flips assumed to be described by the Hawkes process [9] with an exponential kernel: λ(t) = λ0 + µ N � τk 300 km s−1). The outflow morphology suggests the gas propagation along a collimated structure before it fragments into two filaments, +giving it a "tuning-fork" resemblance. The green contours in the middle panel shows the [O iii] outflow contours within the collimated structure to highlight +that the outflowing structure itself shows the presence of clumps. Thanks to the 0.1′′ spatial resolution achieved with the NFM observations, such structures are +barely visible in archival WFM data shown in the right panel. In all the panels, the blue "X" marks the AGN location and the black bar on the bottom left shows +the 20 pc scale. +Figure 3. The top left panel shows the stellar velocity map obtained from the stellar continuum fitting. The stellar velocity map shows a smooth rotation-like +profile of the host galaxy. The [O iii] centroid velocity profile (top centre panel) approximately mimics the stellar velocity field, suggesting that the bulk of the +ionised gas cone co-rotates with the host galaxy. The top right panel shows the residual map after subtracting the stellar velocity map from the [O iii] velocity +map. We observe residuals at the locations of the "tuning-fork" structure (red contour), suggesting that it is a part of the non-rotation component. The positive +residuals in the filament directed towards the West and the negative residuals in the filament towards North shows that the outflow itself is co-rotating with the +ionised gas and the host galaxy. The bottom panels show the non-parametric velocities, 𝑣10 and 𝑤80, described in Sect. 3. Both these velocities confirm that +the high velocity regions are along the collimated structure that fragments into two filaments ∼1.5′′ from the AGN location. Furthermore, the presence of this +structure in the 𝑣10 map shows that the dominant component of the outflow is blue-shifted. The black "X" in all maps indicate the AGN location. +et al. 2021; Kakkad et al. 2022). The systemic flux dominates the +bulk of the ionised gas flux in the host galaxy by nearly two orders of +magnitude, compared to the [O iii] outflow flux. The [O iii] outflow +map (middle panel, Fig. 2), on the other hand, shows a collimated +structure that originates close to the AGN location and extends to- +wards the NW of the nucleus (same direction and approximately the +same PA as the ionisation cone). The collimated structure itself is +not uniform and shows multiple clumps. Such clumps have also been +previously reported in extended radial ionised gas filaments of the +Circinus galaxy (e.g., Veilleux & Bland-Hawthorn 1997). We note +that the location of the first clump is not coincident with the AGN +location, but ≈0.4′′ NW of the nucleus. In Section 5, we further dis- +cuss the origin of these clumps and whether they could be produced +by the shocks within the outflowing wind. +Beyond ∼1.5′′ from the AGN location (∼30 pc) in the NW direc- +tion, the collimated structure then fragments into two filaments, one +towards the West and another towards North, which gives the overall +outflow morphology a "tuning-fork" resemblance. The impact of the +high resolution NFM observations is clear from these observations +as such pc-scale filaments and fragmenting structures are not visible +in the archival low resolution (∼0.5′′) MUSE WFM data, as shown +in the right panel of Fig. 2. +Figure 3 shows velocity maps of the stellar component and the +ionised gas of the Circinus galaxy, derived from the NFM observa- +tions. The top left panel in Fig. 3 shows the stellar velocity distribution +MNRAS 000, 1–9 (2021) + +1e-17. +10 +1e-19 +9 +3 +3 +8 +8 +2 +2 +arcsec] +6 +1 +5 +0 +0 +4 +4 +-1 +m +2 +2 +2 +2 +20 pc +-3 +20 pc +-3 +1 +[ol] systemic flux +[olll] outflowflux +C +-2 +0 +2 +-2 +0 +2 +Ax [arcsec] +△x [arcsec][Olll] outflow flux WFM +1e-18 +6 +3 +5 +2 - +4 +y [arcsec] +1 +erg/s/cm2 +0 - +X +-11 +2 +-2 - +1 +-3 +0 +-2 +0 +2 +△x [arcsec]100 +100 +30 +75 +75 +2 +2 +2 +20 +50 +50 +Ay [arcsec] +V[ol] [km s-1] +10 +25 +[t-s +25 +V[OI] - V* +0 +0 +[km +0 +0 +0 +0 +-25 +-25 +-10 +-50 +-50 +-2 +-2 +-2 +-75 +-75 +-20 +100 +-100 +-2 +30 +-2 +0 +2 +0 +2 +-2 +0 +2 +Ax[arcsec] +Ax[arcsec] +Ax[arcsec][OI] V10 +[OI|] W80 +-50 +380 +2 +2 +[arcsec] +-95 +320 +-1 +-1 +'s +0 +0 +S +km +km +-135 +260 +-2 +-2 +-180 +200 +-2 +0 +2 +-2 +0 +2 +Ax[arcsec] +Ax[arcsec]NFM observations of Circinus +5 +Figure 4. The map shows the mass outflow rate distribution for the ionised +gas derived from the [O iii]𝜆5007 line in the MUSE-NFM observations of +the Circinus galaxy. The mass outflow rates are higher in regions with higher +outflow velocity. +in the host galaxy obtained from the stellar continuum modelling. The +velocity map shows a smooth gradient indicating a rotation-like pro- +file, with the axis of rotation aligned approximately along the axis of +the ionisation cone. The [O iii] centroid map, shown in the top centre +panel of Fig. 3, mimics the stellar velocity map i.e., the ionised gas +co-rotates with the host galaxy. The [O iii] centroid velocity profile is +also consistent with previous MUSE-WFM results from the literature +(e.g., Fonseca-Faria et al. 2021). On subtracting the stellar velocity +component from the [O iii] centroid velocity, we see clear residuals +at the locations of the "tuning-fork" structure, as shown in the top +right panel of Fig. 3. This proves that the outflow flux shown in the +middle panel in Fig. 2 is indeed part of the non-systemic component +of the host galaxy. Furthermore, the positive and negative residuals +in the West and North arms respectively in the residual map in the +top right panel of Fig. 3 indicates that the fork structure itself is +co-rotating with the host galaxy and the ionisation cone. The bottom +panels in Fig. 3 show the [O iii] 𝑣10 and the 𝑤80 maps (left and right +panels respectively). Both the 𝑣10 and 𝑤80 maps confirm the results +seen in the outflow maps i.e., the high velocity regions are collimated +up to ∼1.5′′ from the AGN location before they fragment into two +filaments. The presence of the tuning-fork structure in the 𝑣10 map +suggests that most of the observed outflow flux is dominated by the +blue-shifted emission. We note that the stellar velocity map in the +top left panel of Fig. 3 also shows a "V-shaped" structure at approx- +imately the same location where the high velocity regions fragment +into the two arms, suggesting that the material within the cone is both +outflowing and co-rotating with the host galaxy. +We also derived the ionised gas mass outflow rate using the +[O iii] line adopting methods from the literature (e.g., Rupke et al. +2005; Genzel et al. 2011; Veilleux et al. 2020; Kakkad et al. 2022). +We report two kinds of outflow rate values: Instantaneous outflow +rates ( �𝑀inst) is the sum of mass outflow rates calculated for each +pixel, and time-averaged mass outflow rate (𝑀Tavg) calculated by +taking averaged quantities over the whole outflowing region. These +quantities can be computed using the following equations: +�𝑀inst = +∑︁ +pix +𝑀out · 𝑣out +Δ𝑅 +(1) +�𝑀Tavg = 𝑀out· < 𝑣out > +𝑅 +(2) +In Equation 1, the mass of the outflowing ionised gas, 𝑀out and the +velocity of the ionised gas, 𝑣out is computed for each pixel and Δ𝑅 +is the size of the pixel. In Equation 2, on the other hand, 𝑀out is the +total outflowing gas mass computed from the outflowing [O iii] flux +and 𝑣out is the average velocity over the outflowing region (∼300 km +s−1). The parameter, 𝑅, in Eq. 2 is the distance of the outflow from +the AGN location. As we are using spatially-resolved observations, +we do not need to assume an outflow geometry or outflow density +for the time-averaged quantity. The outflow density in both cases +is obtained from the flux ratio of the outflowing components of +[S ii]𝜆𝜆6716, 6731. +The mass outflow rate map, representing the instantaneous mass +outflow rates (Eq. 1), is shown in Fig. 4, where the mass outflow rate +was calculated for each pixel. The advantage of using this method is +that variation in the outflow parameters such as outflow density and +velocity can be incorporated without the need for any assumptions +on outflow models. We find the median outflow density across the +field-of-view, calculated using the flux ratio of [S ii]𝜆𝜆6716, 6731 +to be ∼200 cm−3 (e.g., Sanders et al. 2016; Kaasinen et al. 2017; +Kakkad et al. 2018). The total summed instantaneous outflow rate +is 0.01 M⊙ yr−1 (an average of 3×10−7 M⊙ yr−1 per pixel where +outflow is detected), which is two orders of magnitude less than the +total instantaneous outflow rate value reported with MUSE-WFM +observations (Kakkad et al. 2022). The time-averaged outflow rate +computed using Eq. 2 is 10−4 M⊙ yr−1. The obscured star formation +rate (SFR) in the Circinus galaxy is reported to be between 3–8 M⊙ +yr−1. The orders of magnitude difference between the SFR and the +ionised outflow rate within a radius of ∼100 pc of the AGN location, +therefore, suggests that the observed ionised outflow is not expected +to shut down star formation in the host galaxy. However, this may not +be true for kiloparsec-scale molecular outflows where the outflow +rate in the molecular gas phase has been reported to be ∼0.35–12.3 +M⊙ yr−1 (see Zschaechner et al. 2016). The high molecular outflow +rate in kiloparsec-scales can, therefore, regulate star formation. A +multi-phase approach to high resolution gas kinematics, by tracing +warm and cold molecular gas components, is required to robustly +confirm whether these AGN outflows affect star formation within +∼100 pc of the AGN. +Lastly, using spatially resolved Baldwin, Phillips & Terlevich di- +agrams (e.g., Baldwin et al. 1981; Veilleux & Osterbrock 1987), we +infer that the dominant source of ionisation across the NFM field-of- +view is the AGN and the ionisation by star formation is negligible or +absent (Figure 5). The ionisation structure is consistent with previ- +ous WFM results in the literature (e.g., Mingozzi et al. 2019; Kakkad +et al. 2022). The ionisation by the AGN is observed for both systemic +as well as outflowing components. Therefore, the current observa- +tions also do not support a scenario where these outflows trigger star +formation activity in the vicinity of the AGN. +4.2 The dust-outflow connection in Circinus +Previous mid-infrared observations of the Circinus galaxy estab- +lished that a major fraction of dust emission is coming from the +polar region, tentatively associated with dusty winds driven by radi- +ation pressure (e.g., Stalevski et al. 2017; Venanzi et al. 2020). Even +though far away from the central engine, dust and gas are expected to +be coupled and co-spatial, and until recently the models of infrared +emission ignored this polar dust component. The spatially-resolved +optical spectra from the MUSE-NFM mode can be used to derive +extinction maps from Balmer decrement (H𝛼/H𝛽) to confirm the +presence of dust along the polar direction. Therefore, we derived the +host galaxy extinction, 𝐴V, across the NFM field-of-view using the +MNRAS 000, 1–9 (2021) + +OutflowRatemap +1e-6 +3 +4 +2 +[arcsec] +1 +3 +0 +2 +-1 +-2 +1 +-3 +0 +-2 +0 +2 +△x[arcsec]6 +D. Kakkad et al. +Figure 5. The left panel shows the ionisation structure (AGN in red and composite in orange) in the field-of-view probed by the MUSE-NFM data. The right +panel shows the location of each pixel in the classical [N ii] BPT diagram. The solid and dashed black curves are obtained from Kauffmann et al. (2003) and +Kewley et al. (2001) and divide the plots between regions ionised by AGN, star formation and composite processes. The systemic flux of the emission lines was +used while plotting this diagram. However, the results are similar if the outflowing components is used. The figure highlights that the gas is ionised primarily by +the AGN. +Figure 6. Dust extinction (𝐴V) map of the Circinus galaxy using the NFM +observations. The background image shows the extinction map obtained from +the systemic components ofH𝛼 and H𝛽, thecyan contoursshow theextinction +from the outflowing components and the magenta contours show the location +of high velocity ionised gas outflow. The dust extinction is dominant along +the polar direction, consistent with previous mid-infrared observations in the +literature. +Balmer Decrement parameter. We assumed a Calzetti et al. (2000) +dust attenuation law with 𝑅V = 4.05 and a fixed temperature of 10,000 +K, which is the typical electron temperature in the NLR. We note +that the Circinus galaxy suffers from Galactic extinction of 𝐴V ∼2 +(see For et al. 2012). However, the [O iii] outflow morphology and +the associated velocities and mass outflow rates will not change on +correcting the Galactic extinction. The extinction map is shown in +Fig. 6. The background map in Fig. 6 shows the extinction map ob- +tained from the systemic components of the flux ratio, H𝛼/H𝛽. The +cyan contours show the extinction (𝐴V > 1) obtained from the out- +flowing component of H𝛼 and H𝛽, and the magenta contours show +the location of the [O iii] ionised gas outflow from the middle panel +in Figure 2. +While the map in Fig. 6 shows potential dust distribution both along +the disk (consistent with the results reported in Mingozzi et al. (2019) +and Fonseca-Faria et al. (2021)) as well as the polar direction, the +overall distribution is dominant along the polar direction. This result +is consistent with the previous results with mid-infrared emission, +which was also dominant along the polar direction (e.g., Jaffe et al. +2004; Tristram et al. 2007, 2014; Asmus et al. 2016). This suggests +that the dust along the polar direction might be a part of lower velocity +gas (compared to the high velocity collimated outflow observed here) +surrounding the ionised gas outflow. The extinction from the systemic +components peaks at the location of the AGN and gradually falls off +to 𝐴V = 0 at a distance of ≈3′′ i.e., ≈60 pc. +The extinction obtained from the outflowing H𝛼 and H𝛽 compo- +nents shows non-uniform clumps scarcely distributed along the polar +direction. We attribute these clumps to be a part of the outflowing +gas and dust. It is worth noting that one of these clumps is almost +at the tip of the collimated component of the outflow, approximately +where the outflow filament fragments into two arms. The observa- +tion, therefore, might support a picture where the ionised gas outflow +chooses the path of least resistance and therefore fragments into the +two filaments, avoiding the region radially outward towards where +the dust clump is present. +5 DISCUSSION +The results presented in Section 4 highlights the complex structures +within an outflow that are revealed from high resolution observations +in the vicinity of the AGN. The presence of an outflow in the form of +an ionisation cone in the Circinus galaxy has been known for nearly +three decades, and thanks to the current state-of-the-art instrumen- +tation that we are now able to resolve parsec scale emission using +optical spectra. The origin of the initial clumpy collimated structure +observed in the NFM data could be due to a small-scale radio jet. +The Circinus galaxy is known to host a radio jet, although its PA is +aligned closer to the edge of the ionisation cone (e.g., Elmouttie et al. +1998). However, these are lower spatial resolution radio imaging ob- +servations and the regions close to the AGN torus is not resolved. +Due to the precession and interaction with the surrounding medium, +jets in AGN are known to bend and change directons at larger scales +(e.g., as in the case of NGC 1068 Gallimore et al. 2004). Therefore, +future work will target the regions closer to the AGN to search for +the potential impact of small-scale jets that may be aligned along the +MNRAS 000, 1–9 (2021) + +1.5 +3 +AGN +1.0 +2 +0.5 +1 +Log [OI]/Hβ +[arcsec] +0 +0.0 +-0.5 +-2 +-1.0 +Star +-3 +formation +Composite +-1.5 +-3 +-2 +0 +1 +2 +3 +0.8 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +Ax [arcsec] +Log [NII]/Hα6 +3 +5 +2 +4 +[arcsec] +7 +0 +m +以 +-1 +2 +-2 +1 +[ooutflow +-3 +0 +2 +Ax [arcsec]NFM observations of Circinus +7 +Figure 7. The background image shows the [O iii]𝜆5007 flux map from the +MUSE-WFM data, which shows an overall asymmetric conical morphology +with two distinct filaments on hundreds of parsec scales. The red contours +show the [O iii] outflow morphology obtained from the NFM observations +in the middle panel of Fig. 2. Comparing the NFM contours with that of the +WFM [O iii] flux map, it appears that the two filaments observed in the WFM +data have their origins in parsec-scale outflow traced with the NFM. +Figure 8. A cartoon model of the ionised outflow observed with the MUSE- +NFM data in the Circinus galaxy. The outflow might be launched as a col- +limated structure by the radio jet, which then fragments into two filaments +probably due to the presence of a dense clump at the tip of the collimated +structure. The dust is distributed along the ionisation cone, apparent from the +extinction maps and also consistent with archival mid-infrared observations. +axis of the ionisation cone. The relative orientations of the radio jet +and the ionisation cone in the Circinus galaxy are depicted in Fig. 8. +The outflow itself is not composed of uniformly distributed ionised +gas, but shows the presence of clumps, confirming previous observa- +tions from lower resolution data of other targets that the outflowing +media are non-uniform in nature (e.g., Kakkad et al. 2018, 2022). +The presence of clumps or knots along outflow/ionised gas filaments +in the Circinus galaxy has also been previously reported in Veilleux +& Bland-Hawthorn (1997) and these structures have been attributed +to bow-shocked features that resemble Herbig Haro objects that in- +teract strongly with the surrounding ISM. The observed morphology +in the NFM data could be a smaller-scale version of the observations +in larger scales. +The collimated component of the outflow most likely fragments +into two components because of the presence of an obstruction in +the path of the outflow. In the NFM data, there is an indication of +the presence of a dust clump via extinction maps obtained from the +outflowing component of H𝛼 and H𝛽 emission. Infrared observations +targeting primarily the dust emission could confirm this scenario. +Filament structures are also observed in images that trace gas across +larger scales (e.g., Marconi et al. 1994; Veilleux & Bland-Hawthorn +1997) and it is probable that the NFM data presented in this paper +reveals the origin of the kiloparsec-scale filaments. Fig. 7 shows +the ionised gas morphology traced by the archival MUSE-WFM +observations (background) and the emission from the outflowing +[O iii] component from the MUSE-NFM data (red contours). The +spatial distribution of the filaments observed in WFM is strongly +suggestive that their origin is to be found in the fragmented arms of +the tuning fork observed in the NFM observations. +Fig. 8 shows an overall cartoon model of the ionised outflow in the +Circinus galaxy, that shows the ionised outflow that fragments into +two filaments, probably due to the presence of a dense clump at the +tip of the collimated structure. The dust itself is distributed around the +ionisation cone and it envelops the lower velocity ionised gas within +the conical structure. We speculate that the outflow itself might be +launched due to the radio jet, which cannot be robustly confirmed +based on the available archival radio observations, as mentioned +earlier. +As the NFM observations have only recently targeted the nearby +galaxies, it is unclear if such outflow structures and fragmentations +within outflows are common. If this is indeed observed for majority +of the galaxies, conventional outflow models that use ionisation cone +morphology may need to be revised to account for the results from +these high spatial resolution data. +6 SUMMARY & CONCLUSIONS +We presented the MUSE NFM observations of the Circinus galaxy at +a spatial resolution of 0.1′′ (∼2 pc) that resolves the regions close to +the AGN torus. We derived the properties of the ionised gas outflow +using the [O iii]𝜆5007 emission line and the dust distribution using +Balmer Decrement. We follow a non-parametric approach to analyse +the emission lines, so the derived properties are independent of the +fitting functions used. The main results of this work is summarized +below: +• The flux distribution of the systemic component of [O iii] emis- +sion, defined by the velocity components within ±300 km s−1, shows +a conical morphology, which is also observed in larger spatial scales +up to hundreds of parsecs in the literature. Archival radio observa- +tions show that the radio jet is aligned approximately with the edge +of the ionisation cone. The flux distribution of the outflowing com- +ponent of the [O iii] emission, on the other hand, shows a collimated +structure up to ∼30 pc before fragmenting into two arms that overall +mimics a “tuning-fork” shape. The outflowing structure itself is not +smooth and shows clumps at several locations. +• A comparison between the stellar kinematic map and the +[O iii] centroid map suggests that the ionised gas (both the systemic +as well as the outflowing component) co-rotates with the host galaxy. +Both the non-parametric velocity distributions, 𝑣10 and 𝑤80 show +similar structures as the outflow flux i.e., the tuning-fork shapes. Most +of this outflow is blue-shifted, consistent with the outflow models of +the Circinus galaxy reported in the literature. +• We find a total instantaneous mass outflow rate of ∼0.01 M⊙ +yr−1 (3×10−7 on average per pixel of size 0.5 pc) and a time-average +mass outflow rate of 10−4 M⊙ yr−1. This is much lower than the star +formation within the galaxy and therefore, the ionised gas outflow +is not expected to regular star formation within a radius of ∼100 pc +from the AGN location. +MNRAS 000, 1–9 (2021) + +Dust +clumps +Radio jet +lonisationcone +AGN20 +15 +[arcsec] +10 +5 +Ay I +0 +-5 +-10 +-10 +-5 +0 +5 +10 +15 +20 +△x [arcsec]8 +D. Kakkad et al. +• The extinction maps using the systemic components of the H𝛽 +and H𝛼 line show that the dust distribution is concentrated along the +ionisation cone i.e., the polar direction, consistent with the archival +mid-infrared observations. The extinction map obtained from the +outflowing components of H𝛽 and H𝛼 shows a scarce distribution, +with a clump approximately at the tip of the collimated part of the +outflow. This dust clump might explain the fragmentation in the +outflowing ionised gas. +• We combine previous reported results gathered from the litera- +ture and the observed outflows from the NFM data presented in this +paper to present a model of the ionised gas outflow in the Circinus +galaxy. We suggest that the observed outflow in the Circinus galaxy +is composed of high velocity collimated gas that is enveloped by +lower velocity dusty ionised gas. The presence of a dust clump at the +tip of the collimated part of the outflow might be responsible for the +fragmentation in the outflowing gas. The collimated outflow might +be launched by a radio jet. Although the jet, observed in low spatial +resolution radio observations, does not show a 1:1 alignment with +the outflowing cone PA, they are known to bend and change at larger +scales. +While the MUSE-NFM only provides the kinematic information +in the ionised gas phase, outflows have been known in exist also in +the molecular gas phase in the Circinus galaxy. A multi-wavelength +approach to trace gas in the other phases such as warm and cold +molecular, at the same spatial resolution of ∼2 pc, will be key to +obtaining a holistic view into the outflow-AGN connection in the +Circinus galaxy. Furthermore, high spatial resolution radio obser- +vations will verify if the observed collimated outflow is a result of +jet-ISM interaction on small scales. +ACKNOWLEDGEMENTS +We would like to thank the anonymous referee for insightful com- +ments and suggestions to improve this manuscript. The authors also +thank T. Fischer for useful discussions. Based on observations from +the ESO programme ID 0103.B-0396. M.S. and S.K. acknowledge +support by the Science Fund of the Republic of Serbia, PROMIS +6060916, BOWIE and by the Ministry of Education, Science and +Technological Development of the Republic of Serbia through con- +tract No. 451-03-9/2022-14/200002. D.A. acknowledges funding +through the European Union’s Horizon 2020 and Innovation pro- +gramme under the Marie Sklodowska-Curie grant agreement no. +793499 (DUSTDEVILS). +DATA AVAILABILITY +The data presented in this paper is available with ESO archive under +the programme ID: 0103.B-0396. +REFERENCES +Antonucci R., 1993, ARA&A, 31, 473 +Arévalo P., et al., 2014, ApJ, 791, 81 +Asmus D., 2019, MNRAS, 489, 2177 +Asmus D., Hönig S. F., Gandhi P., 2016, ApJ, 822, 109 +Bacon R., et al., 2010, in Ground-based and Airborne Instrumentation for +Astronomy III. p. 773508, doi:10.1117/12.856027 +Baldwin J. A., Phillips M. M., Terlevich R., 1981, PASP, 93, 5 +Bock J. 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K., et al., 2016, ApJ, 832, 142 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–9 (2021) + diff --git a/L9AyT4oBgHgl3EQf6voI/content/tmp_files/load_file.txt b/L9AyT4oBgHgl3EQf6voI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1269fbb977b894c37496a9ac42c9f55849698072 --- /dev/null +++ b/L9AyT4oBgHgl3EQf6voI/content/tmp_files/load_file.txt @@ -0,0 +1,1047 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf,len=1046 +page_content='MNRAS 000, 1–9 (2021) Preprint 4 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='0 Dissecting the active galactic nucleus in Circinus IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' MUSE 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We present the ionised gas outflow morphology in the Circinus galaxy using the Narrow Field Mode (NFM) of the MUSE instrument on board the Very Large Telescope (VLT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The NFM observations provide a spatial resolution of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='1′′, corresponding to a physical scale of ∼2 pc, one of the highest spatial resolution achievable using ground-based AO-assisted observations in the optical wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The MUSE observations reveal a collimated clumpy outflow profile originating near the AGN location and extending up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5′′ (∼30 pc) in the NW direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The collimated structure then fragments into two filaments, giving the entire outflowing gas a “tuning-fork” morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' These structures remain undetected in the lower spatial resolution MUSE Wide Field Mode data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We explain the origin of this tuning-fork structure to the interaction of the outflow with a dense clump in the interstellar medium (ISM) as the outflow propagates outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The origin of the collimated structure itself could be from jet-ISM interactions on small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' These observations also provide evidence to the origin of the ionised gas filaments previously observed in the Circinus galaxy out to kiloparsec scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We find instantaneous and time-averaged mass outflow rates of 10−2 M⊙ yr−1 and 10−4 M⊙ yr−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Based on the star formation rate in the Circinus galaxy reported in the literature, the observed ionised outflows are not expected to regulate star formation within the ∼100 pc scales probed by the NFM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Key words: galaxies:active – galaxies:individual – galaxies:ISM – galaxies: kinematics and dynamics – galaxies: nuclei – galaxies: Seyfert 1 INTRODUCTION The so-called Unified Model (UM) of the active galactic nuclei (AGN) consists of a central black hole surrounded by an equato- rial torus-like structure, which is responsible for the angle-dependent obscuration of the accretion disk and in some cases, may include col- limated jets along the polar directions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Antonucci 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Urry & Padovani 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Netzer 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The equatorial torus has been be- lieved to dominate the infrared emission from the AGN (see Ramos Almeida & Ricci 2017, and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' There have been intensive efforts in the literature, both from an observational as well as modelling perspective, to study the nature of this torus, such as its geometry (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Hönig 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' García-Burillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2021), what are the typical dust covering factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Elitzur 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Stalevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Toba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2021) and whether the material is clumpy or smooth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Dullemond & van Bemmel 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Marin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' García-González et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' High resolution mid-infrared observations over the past decade have now challenged these simplified torus model that have dusty ★ E-mail: dkakkad@stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='edu clumps only in the equatorial region (Nenkova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Several studies in the literature now show strong infrared emission along the polar direction on the scales of a few parsecs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Hönig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2012, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' López-Gonzaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Hönig & Kishimoto 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Leftley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2018) to hundreds of parsecs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Braatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Bock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Asmus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Asmus 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' As a result, the dust emission around the AGN is believed to consist of two components: an equatorial thin disk and a polar extended feature that could originate from the winds from the central engine (see Hönig 2019, and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The polar wind is believed to have a multi-phase composition ranging from dust to ionised and molecular gas components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' In fact, recent high spatial resolution observations of nearby AGN have tar- geted the molecular gas distribution around the torus using ALMA, revealing high velocity outflows in the molecular gas phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Gallimore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Combes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' García-Burillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Lopez-Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' In order to get a holistic view of the multi-phase dusty gas flows around the torus, it is imperative to obtain the morphology and kinematics of the ionised gas on the same scales as the molecular gas and infrared emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Furthermore, obtaining outflow morphology at such small spatial scales can also © 2021 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='00825v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='GA] 2 Jan 2023 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' give clues into the outflow launching mechanism and connect them to the observed kiloparsec scales structure or outflows, whenever available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Thanks to the Narrow Field Mode (NFM) capabilities of the Multi Unit Spectroscopic Explorer (MUSE Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2010) at the Very Large Telescope (VLT), such high spatial resolution observations can now be performed with ground-based Integral Field Spectroscopic instruments operating at optical wave- lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The optical wavelengths provide access to bright emission lines such as the [O iii]𝜆5007 that trace ionised gas in the Narrow Line Region (NLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Furthermore, emission lines such as H𝛼, H𝛽, [N ii]𝜆𝜆6549, 6585 and [S ii]𝜆𝜆6716, 6731 help derive dust extinc- tion maps and diagnostic diagrams that trace the source of ionisation across the field-of-view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The Circinus galaxy is the closest Seyfert 2 galaxy (∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='2 Mpc away, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='001 Freeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 1977) and hosts an infrared-bright AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Jarrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The polar axis of the AGN in the Circinus galaxy is seen almost edge-on and is therefore an ideal target to study the relative gas and dust structure in a typical obscured AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Recent mid-infrared (MIR) observations using the upgraded VISIR instrument at the Very Large Telescope (VLT) suggests the presence of dust in the form of a hollow cone at the edges of the ionised outflow (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Stalevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The polar elongation of the infrared emission has also been reported on parsec scales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Tristram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Stalevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2019) using Mid-Infrared Interferometric Instrument (MIDI) and the Multi AperTure mid-infrared Spectro- Scopic Experiment (MATISSE, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Isbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2022) at the VLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The presence of dust in the form of a hollow cone in the polar region is also confirmed by multi-band optical polarimetry with VLT/FORS2 (Stalevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', submitted) The galaxy also hosts powerful outflows in the ionised gas phase, visible in the form of a one-sided ionisation cone extended up to ∼1 kiloparsec (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Marconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Veilleux & Bland-Hawthorn 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Mingozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Fonseca-Faria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The Circinus galaxy hosts an obscured AGN with nuclear star formation that dominates the dust emission on scales of hundreds of parsec (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Matt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Arévalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' In this fourth paper in the series, we map the morphology and kinematics of ionised gas in the Circinus galaxy at ∼2 pc resolution using MUSE-NFM observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We present a model of the ionisa- tion cone and the resulting outflowing gas structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The observed ionised outflow morphology obtained from the NFM observations is compared with the larger scale outflows observed with the Wide Field Mode (WFM) of MUSE, to understand outflow propagation across the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We locate the regions with high dust extinc- tion and compare this with the archival mid-infrared images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Lastly, the MUSE data is compared with other archival radio observations to infer the presence of jet-ISM interaction in the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Throughout this paper, e adopt the following ΛCDM cosmological parameters: 𝐻0 = 70 km s−1, ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='3 and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' All the maps use the following convention: North is up and East is to left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2 MUSE-NFM OBSERVATIONS & DATA REDUCTION The observations were performed using the Laser Tomographic Adaptive Optics (LTAO) assisted Narrow Field Mode of the MUSE instrument, on board Unit Telescope 4 of the VLT1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The observa- tions were carried out on the nights of 29 and 30 April 2019 with a DIMM seeing in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='89–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='01′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We observed the galaxy with 1 ESO programme ID: 0103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='B-0396(A) an optimised sequence: O-S-O-O-S-O (O = Object, S = Sky), where the Sky was obtained at an offset position ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5 arcmin away outside the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We performed small dithering between the individual science exposures and rotated the field by 90 degrees on each subse- quent exposure to eliminate the impact of bad pixels and to average out the patterns of slicers and channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The nucleus of the Circinus galaxy has an H-band magnitude of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='4 and therefore, served as the Adaptive Optics (AO) reference for the Wavefront Sensor (WFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The total on-source exposure time was ∼4000s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The raw data was reduced using the standard MUSE pipeline (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Weilbacher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2014, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The pipeline performs bias correction, flat fielding, wavelength and astrometry calibration, sky subtraction and flux calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The final data cube consisted of a field-of-view of ∼7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5×7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5 arcsec2 centred on the nucleus (AGN) with a spatial sampling of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='025′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' As the observations were performed in the Nominal mode, this provided a uniform wavelength coverage between 4800–9300 Å with a gap between 5780–6050 Å due to the presence of a notch filter that suppresses the Sodium laser light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The spectral PSF (also known as the Line Spread Function, LSF) is in the range 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='9 Å, with the best resolution obtained at the redder end of the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' This corresponds to a velocity resolution of ∼150 km s−1 at the location of [O iii]𝜆5007 line, one of the emission lines that will be analysed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The LTAO-assisted observations resulted in a spatial PSF of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='1′′, determined using one of the point sources in the field-of-view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' This spatial resolution is one of the highest that can be achieved using ground-based IFS observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' At the redshift of the Circinus, this corresponds to a physical scale of ∼2 pc, which means that the observations can potentially resolve the region near the AGN torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' With a field-of-view of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5′′×7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5′′, the NFM observations trace spatial scales up to ∼100 pc from the AGN location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 3 ANALYSIS In order to derive the flux and velocity maps from the optical emis- sion lines, we first subtract the stellar continuum emission across the MUSE field-of-view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We perform the stellar continuum emission us- ing the LZIFU tool (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Kreckel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2018), which adopts the penalized pixel fitting routine (PPXF Cappellari & Em- sellem 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Cappellari 2017) to fit the stellar continuum using input stellar spectrum templates (from González Delgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' (2005)) or modelled simple stellar populations (SSPs) that are convolved with parametrised velocity distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' In doing so, regions in the spectra with strong skylines and emission lines intrinsic to the host galaxy were masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The key emission lines that were masked were H𝛽, [O iii]𝜆𝜆4959, 5007, [N ii]𝜆𝜆6549, 6585, H𝛼 and [S ii]𝜆𝜆6716, 6731 ([S ii] doublet hereafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We also mask the notch filter region in the spectrum that is contaminated by the sodium doublet emission from the lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Being a Seyfert 2 galaxy, the Circinus galaxy does not display broad emission lines from the Broad Line Region (BLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The resulting stellar continuum-subtracted data cubes were then used to analyse the morphology, kinematics and the ionisation mech- anism of the gas using strong emission lines in the optical spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' For instance, we used the [O iii]𝜆5007 line ([O iii] hereafter) to trace the ionised gas in the Narrow Line Region (NLR), the Balmer lines H𝛼 and H𝛽 lines are used to trace potential star formation and dust extinc- tion in the host galaxy (using Balmer decrement), the [N ii]𝜆𝜆6549, 6585 lines are used in investigating the ionisation source in each pixel (AGN or star formation) and the [S ii] doublet are used to trace the ionised gas electron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We model these emission lines with multiple Gaussian functions using the scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='curve-fit package in MNRAS 000, 1–9 (2021) NFM observations of Circinus 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The top left panel shows an RGB colour image of the Circinus galaxy, derived from the NFM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The field-of-view of the RGB image is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5′′×7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The red hue indicated the H𝛼 emission, while the ionised gas cone is apparent in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The white square at the edge of the ionisation cone shows the pixel from where the example spectra shown in this figure was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The middle and right panels in the top row shows the example of a stellar continuum fit and the middle and right panels in the bottom row shows the emission line fitting results of H𝛽, [O iii]𝜆𝜆4959,5007, [N ii]𝜆𝜆6549,6585, H𝛼 and [S ii]𝜆𝜆6716,6731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' In all the spectra, the grey curve shows the extracted data from the pixel location shown in the median image, the magenta curve shows the stellar continuum model, the green and blue curves show the individual Gaussian components used to model the emission lines and the red curve shows the overall model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The bottom left panel illustrates the definition of outflow and systemic flux used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Further details are given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' python (see Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Initially a single Gaussian is fitted to the emission line profile, and additional Gaussian functions were added if the 𝜒2 is minimised, and until the line fluxes are stable within ∼10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Circinus does not display BLR emission in its H𝛽 or H𝛼 pro- files and the maximum number of Gaussians required to model an emission line was two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We do not assign any physical significance to the individual Gaussian components as we follow a non-parametric approach to derive the properties associated with the systemic and outflow components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The non-parametric approach was chosen over a parametric model as it does not depend on the choice of the models used for the emission lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Gaussian, Lorentzian or a power- law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' In addition, we tied the line centroids of [O iii] line with that of H𝛽 and the [N ii] and [S ii] lines with that of H𝛼 based on the expected location of the respective atomic species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The emission line ratios [O iii]𝜆5007:[O iii]𝜆4959 and [N ii]𝜆6585:[N ii]𝜆6549 were set ap- proximately equal to 3:1 based on expectations from theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Osterbrock & Ferland 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Dimitrijević et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Lastly, the FWHM of the individual Gaussian components of the [O iii] line was coupled with H𝛽 and the H𝛼 FWHM with that of [N ii].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' From the line fitting procedure, we are interested in the following parameters: The 10th percentile velocity: 𝑣10 (blue-shifted [O iii] ve- locity that contains 10% of the overall [O iii] flux), the width of the emission line: 𝑤80 (width containing 80% of the flux of the emission line, see Harrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Wylezalek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2020) and the flux of the outflowing and systemic components of each of the emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' To determine the fluxes, we first define the zero velocity location in the emission line spectra, which is the location of the peak of the emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The flux within ±300 km s−1 on either side of this zero velocity location is considered to be the systemic component of the emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The choice of 300 km s−1 is based on the line width (FWHM) cut of ∼600 km s−1 which is often employed in the literature to distinguish between outflow- ing and non-outflowing gas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Using similar arguments, we use the flux outside of the ±300 km s−1 channels to define the flux associated with outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Using lower velocity cuts such as 250 or 200 km s−1 yield similar results, but due to the pos- sibility of contamination from the non-outflowing gas at the lower velocities, we make a conservative cut of 300 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Figure 1 shows an example of the analysis methods and the fitting results presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 4 RESULTS & DISCUSSION In this section, we show the results of the analysis methods described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The main aim of this section is to derive the ionised gas outflow morphology and kinematics close to the AGN torus and compare this with archival multi-wavelength data, including the low spatial resolution MUSE-WFM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='1 The parsec-scale ionised outflow in the Circinus galaxy Figure 2 shows the flux maps of the systemic (left panel) and outflow- ing component (middle panel) of the [O iii] emission line from the NFM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The systemic [O iii] flux map traces the ionisation cone originating from the AGN location and extends towards the NW di- rection from the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The presence of the ionisation cone in the Circinus galaxy has previously also been reported in the literature, in- cluding MUSE Wide Field Mode (WFM) observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Marconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Mingozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Fonseca-Faria MNRAS 000, 1–9 (2021) 3500 Data 3500 Stellar continuum 3000 3000 erg/s/cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='/A) 2500 2500 2000 2000 [e-17 1500 1500 Flux 1000 1000 500 500 0 0 4800 4850 4900 4950 5000 5050 6550 6600 6650 6700 6750 Rest-frame wavelength[A] Rest-frame wavelength[A] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5 Outflow Data Systemic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='0 Component 1 Component2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='0 erg/s/cm²/A) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5 Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5 [e-17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='0 300 0 300 4800 4850 4900 4950 5000 5050 65506600 665067006750 v [km/s] Rest-frameWavelength[A] Rest-frameWavelength[A]4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The left panel shows the systemic [O iii] flux (|𝑣 | < 300 km s−1) that traces the ionisation cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The middle panel shows the outflowing component of the [O iii] flux (|𝑣 | > 300 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The outflow morphology suggests the gas propagation along a collimated structure before it fragments into two filaments, giving it a "tuning-fork" resemblance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The green contours in the middle panel shows the [O iii] outflow contours within the collimated structure to highlight that the outflowing structure itself shows the presence of clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Thanks to the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='1′′ spatial resolution achieved with the NFM observations, such structures are barely visible in archival WFM data shown in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' In all the panels, the blue "X" marks the AGN location and the black bar on the bottom left shows the 20 pc scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The top left panel shows the stellar velocity map obtained from the stellar continuum fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The stellar velocity map shows a smooth rotation-like profile of the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The [O iii] centroid velocity profile (top centre panel) approximately mimics the stellar velocity field, suggesting that the bulk of the ionised gas cone co-rotates with the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The top right panel shows the residual map after subtracting the stellar velocity map from the [O iii] velocity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We observe residuals at the locations of the "tuning-fork" structure (red contour), suggesting that it is a part of the non-rotation component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The positive residuals in the filament directed towards the West and the negative residuals in the filament towards North shows that the outflow itself is co-rotating with the ionised gas and the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The bottom panels show the non-parametric velocities, 𝑣10 and 𝑤80, described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Both these velocities confirm that the high velocity regions are along the collimated structure that fragments into two filaments ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5′′ from the AGN location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Furthermore, the presence of this structure in the 𝑣10 map shows that the dominant component of the outflow is blue-shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The black "X" in all maps indicate the AGN location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The systemic flux dominates the bulk of the ionised gas flux in the host galaxy by nearly two orders of magnitude, compared to the [O iii] outflow flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The [O iii] outflow map (middle panel, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2), on the other hand, shows a collimated structure that originates close to the AGN location and extends to- wards the NW of the nucleus (same direction and approximately the same PA as the ionisation cone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The collimated structure itself is not uniform and shows multiple clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Such clumps have also been previously reported in extended radial ionised gas filaments of the Circinus galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Veilleux & Bland-Hawthorn 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We note that the location of the first clump is not coincident with the AGN location, but ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='4′′ NW of the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' In Section 5, we further dis- cuss the origin of these clumps and whether they could be produced by the shocks within the outflowing wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Beyond ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5′′ from the AGN location (∼30 pc) in the NW direc- tion, the collimated structure then fragments into two filaments, one towards the West and another towards North, which gives the overall outflow morphology a "tuning-fork" resemblance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The impact of the high resolution NFM observations is clear from these observations as such pc-scale filaments and fragmenting structures are not visible in the archival low resolution (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5′′) MUSE WFM data, as shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Figure 3 shows velocity maps of the stellar component and the ionised gas of the Circinus galaxy, derived from the NFM observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The top left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 3 shows the stellar velocity distribution MNRAS 000, 1–9 (2021) 1e-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='1e-19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='8 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The map shows the mass outflow rate distribution for the ionised gas derived from the [O iii]𝜆5007 line in the MUSE-NFM observations of the Circinus galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The mass outflow rates are higher in regions with higher outflow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' in the host galaxy obtained from the stellar continuum modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The velocity map shows a smooth gradient indicating a rotation-like pro- file, with the axis of rotation aligned approximately along the axis of the ionisation cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The [O iii] centroid map, shown in the top centre panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 3, mimics the stellar velocity map i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', the ionised gas co-rotates with the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The [O iii] centroid velocity profile is also consistent with previous MUSE-WFM results from the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Fonseca-Faria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' On subtracting the stellar velocity component from the [O iii] centroid velocity, we see clear residuals at the locations of the "tuning-fork" structure, as shown in the top right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' This proves that the outflow flux shown in the middle panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2 is indeed part of the non-systemic component of the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Furthermore, the positive and negative residuals in the West and North arms respectively in the residual map in the top right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 3 indicates that the fork structure itself is co-rotating with the host galaxy and the ionisation cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The bottom panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 3 show the [O iii] 𝑣10 and the 𝑤80 maps (left and right panels respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Both the 𝑣10 and 𝑤80 maps confirm the results seen in the outflow maps i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', the high velocity regions are collimated up to ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5′′ from the AGN location before they fragment into two filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The presence of the tuning-fork structure in the 𝑣10 map suggests that most of the observed outflow flux is dominated by the blue-shifted emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We note that the stellar velocity map in the top left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 3 also shows a "V-shaped" structure at approx- imately the same location where the high velocity regions fragment into the two arms, suggesting that the material within the cone is both outflowing and co-rotating with the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We also derived the ionised gas mass outflow rate using the [O iii] line adopting methods from the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Rupke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Veilleux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We report two kinds of outflow rate values: Instantaneous outflow rates ( �𝑀inst) is the sum of mass outflow rates calculated for each pixel, and time-averaged mass outflow rate (𝑀Tavg) calculated by taking averaged quantities over the whole outflowing region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' These quantities can be computed using the following equations: �𝑀inst = ∑︁ pix 𝑀out · 𝑣out Δ𝑅 (1) �𝑀Tavg = 𝑀out· < 𝑣out > 𝑅 (2) In Equation 1, the mass of the outflowing ionised gas, 𝑀out and the velocity of the ionised gas, 𝑣out is computed for each pixel and Δ𝑅 is the size of the pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' In Equation 2, on the other hand, 𝑀out is the total outflowing gas mass computed from the outflowing [O iii] flux and 𝑣out is the average velocity over the outflowing region (∼300 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The parameter, 𝑅, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2 is the distance of the outflow from the AGN location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' As we are using spatially-resolved observations, we do not need to assume an outflow geometry or outflow density for the time-averaged quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The outflow density in both cases is obtained from the flux ratio of the outflowing components of [S ii]𝜆𝜆6716, 6731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The mass outflow rate map, representing the instantaneous mass outflow rates (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 1), is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 4, where the mass outflow rate was calculated for each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The advantage of using this method is that variation in the outflow parameters such as outflow density and velocity can be incorporated without the need for any assumptions on outflow models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We find the median outflow density across the field-of-view, calculated using the flux ratio of [S ii]𝜆𝜆6716, 6731 to be ∼200 cm−3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Kaasinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The total summed instantaneous outflow rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='01 M⊙ yr−1 (an average of 3×10−7 M⊙ yr−1 per pixel where outflow is detected), which is two orders of magnitude less than the total instantaneous outflow rate value reported with MUSE-WFM observations (Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The time-averaged outflow rate computed using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2 is 10−4 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The obscured star formation rate (SFR) in the Circinus galaxy is reported to be between 3–8 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The orders of magnitude difference between the SFR and the ionised outflow rate within a radius of ∼100 pc of the AGN location, therefore, suggests that the observed ionised outflow is not expected to shut down star formation in the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' However, this may not be true for kiloparsec-scale molecular outflows where the outflow rate in the molecular gas phase has been reported to be ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='35–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='3 M⊙ yr−1 (see Zschaechner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The high molecular outflow rate in kiloparsec-scales can, therefore, regulate star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' A multi-phase approach to high resolution gas kinematics, by tracing warm and cold molecular gas components, is required to robustly confirm whether these AGN outflows affect star formation within ∼100 pc of the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Lastly, using spatially resolved Baldwin, Phillips & Terlevich di- agrams (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Baldwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Veilleux & Osterbrock 1987), we infer that the dominant source of ionisation across the NFM field-of- view is the AGN and the ionisation by star formation is negligible or absent (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The ionisation structure is consistent with previ- ous WFM results in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Mingozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The ionisation by the AGN is observed for both systemic as well as outflowing components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Therefore, the current observa- tions also do not support a scenario where these outflows trigger star formation activity in the vicinity of the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='2 The dust-outflow connection in Circinus Previous mid-infrared observations of the Circinus galaxy estab- lished that a major fraction of dust emission is coming from the polar region, tentatively associated with dusty winds driven by radi- ation pressure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Stalevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Venanzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Even though far away from the central engine, dust and gas are expected to be coupled and co-spatial, and until recently the models of infrared emission ignored this polar dust component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The spatially-resolved optical spectra from the MUSE-NFM mode can be used to derive extinction maps from Balmer decrement (H𝛼/H𝛽) to confirm the presence of dust along the polar direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Therefore, we derived the host galaxy extinction, 𝐴V, across the NFM field-of-view using the MNRAS 000, 1–9 (2021) OutflowRatemap 1e-6 3 4 2 [arcsec] 1 3 0 2 1 2 1 3 0 2 0 2 △x[arcsec]6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The left panel shows the ionisation structure (AGN in red and composite in orange) in the field-of-view probed by the MUSE-NFM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The right panel shows the location of each pixel in the classical [N ii] BPT diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The solid and dashed black curves are obtained from Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' (2003) and Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' (2001) and divide the plots between regions ionised by AGN, star formation and composite processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The systemic flux of the emission lines was used while plotting this diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' However, the results are similar if the outflowing components is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The figure highlights that the gas is ionised primarily by the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Dust extinction (𝐴V) map of the Circinus galaxy using the NFM observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The background image shows the extinction map obtained from the systemic components ofH𝛼 and H𝛽, thecyan contoursshow theextinction from the outflowing components and the magenta contours show the location of high velocity ionised gas outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The dust extinction is dominant along the polar direction, consistent with previous mid-infrared observations in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Balmer Decrement parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We assumed a Calzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' (2000) dust attenuation law with 𝑅V = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='05 and a fixed temperature of 10,000 K, which is the typical electron temperature in the NLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We note that the Circinus galaxy suffers from Galactic extinction of 𝐴V ∼2 (see For et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' However, the [O iii] outflow morphology and the associated velocities and mass outflow rates will not change on correcting the Galactic extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The extinction map is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The background map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 6 shows the extinction map ob- tained from the systemic components of the flux ratio, H𝛼/H𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The cyan contours show the extinction (𝐴V > 1) obtained from the out- flowing component of H𝛼 and H𝛽, and the magenta contours show the location of the [O iii] ionised gas outflow from the middle panel in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' While the map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 6 shows potential dust distribution both along the disk (consistent with the results reported in Mingozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' (2019) and Fonseca-Faria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' (2021)) as well as the polar direction, the overall distribution is dominant along the polar direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' This result is consistent with the previous results with mid-infrared emission, which was also dominant along the polar direction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Jaffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Tristram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2007, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Asmus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' This suggests that the dust along the polar direction might be a part of lower velocity gas (compared to the high velocity collimated outflow observed here) surrounding the ionised gas outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The extinction from the systemic components peaks at the location of the AGN and gradually falls off to 𝐴V = 0 at a distance of ≈3′′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', ≈60 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The extinction obtained from the outflowing H𝛼 and H𝛽 compo- nents shows non-uniform clumps scarcely distributed along the polar direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We attribute these clumps to be a part of the outflowing gas and dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' It is worth noting that one of these clumps is almost at the tip of the collimated component of the outflow, approximately where the outflow filament fragments into two arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The observa- tion, therefore, might support a picture where the ionised gas outflow chooses the path of least resistance and therefore fragments into the two filaments, avoiding the region radially outward towards where the dust clump is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 5 DISCUSSION The results presented in Section 4 highlights the complex structures within an outflow that are revealed from high resolution observations in the vicinity of the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The presence of an outflow in the form of an ionisation cone in the Circinus galaxy has been known for nearly three decades, and thanks to the current state-of-the-art instrumen- tation that we are now able to resolve parsec scale emission using optical spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The origin of the initial clumpy collimated structure observed in the NFM data could be due to a small-scale radio jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The Circinus galaxy is known to host a radio jet, although its PA is aligned closer to the edge of the ionisation cone (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Elmouttie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' However, these are lower spatial resolution radio imaging ob- servations and the regions close to the AGN torus is not resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Due to the precession and interaction with the surrounding medium, jets in AGN are known to bend and change directons at larger scales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', as in the case of NGC 1068 Gallimore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Therefore, future work will target the regions closer to the AGN to search for the potential impact of small-scale jets that may be aligned along the MNRAS 000, 1–9 (2021) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5 3 AGN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5 1 Log [OI]/Hβ [arcsec] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='0 Star 3 formation Composite 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5 3 2 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='4 Ax [arcsec] Log [NII]/Hα6 3 5 2 4 [arcsec] 7 0 m 以 1 2 2 1 [ooutflow 3 0 2 Ax [arcsec]NFM observations of Circinus 7 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The background image shows the [O iii]𝜆5007 flux map from the MUSE-WFM data, which shows an overall asymmetric conical morphology with two distinct filaments on hundreds of parsec scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The red contours show the [O iii] outflow morphology obtained from the NFM observations in the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Comparing the NFM contours with that of the WFM [O iii] flux map, it appears that the two filaments observed in the WFM data have their origins in parsec-scale outflow traced with the NFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' A cartoon model of the ionised outflow observed with the MUSE- NFM data in the Circinus galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The outflow might be launched as a col- limated structure by the radio jet, which then fragments into two filaments probably due to the presence of a dense clump at the tip of the collimated structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The dust is distributed along the ionisation cone, apparent from the extinction maps and also consistent with archival mid-infrared observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' axis of the ionisation cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The relative orientations of the radio jet and the ionisation cone in the Circinus galaxy are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The outflow itself is not composed of uniformly distributed ionised gas, but shows the presence of clumps, confirming previous observa- tions from lower resolution data of other targets that the outflowing media are non-uniform in nature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 2018, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The presence of clumps or knots along outflow/ionised gas filaments in the Circinus galaxy has also been previously reported in Veilleux & Bland-Hawthorn (1997) and these structures have been attributed to bow-shocked features that resemble Herbig Haro objects that in- teract strongly with the surrounding ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The observed morphology in the NFM data could be a smaller-scale version of the observations in larger scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The collimated component of the outflow most likely fragments into two components because of the presence of an obstruction in the path of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' In the NFM data, there is an indication of the presence of a dust clump via extinction maps obtained from the outflowing component of H𝛼 and H𝛽 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Infrared observations targeting primarily the dust emission could confirm this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Filament structures are also observed in images that trace gas across larger scales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', Marconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Veilleux & Bland-Hawthorn 1997) and it is probable that the NFM data presented in this paper reveals the origin of the kiloparsec-scale filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 7 shows the ionised gas morphology traced by the archival MUSE-WFM observations (background) and the emission from the outflowing [O iii] component from the MUSE-NFM data (red contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The spatial distribution of the filaments observed in WFM is strongly suggestive that their origin is to be found in the fragmented arms of the tuning fork observed in the NFM observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 8 shows an overall cartoon model of the ionised outflow in the Circinus galaxy, that shows the ionised outflow that fragments into two filaments, probably due to the presence of a dense clump at the tip of the collimated structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The dust itself is distributed around the ionisation cone and it envelops the lower velocity ionised gas within the conical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We speculate that the outflow itself might be launched due to the radio jet, which cannot be robustly confirmed based on the available archival radio observations, as mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' As the NFM observations have only recently targeted the nearby galaxies, it is unclear if such outflow structures and fragmentations within outflows are common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' If this is indeed observed for majority of the galaxies, conventional outflow models that use ionisation cone morphology may need to be revised to account for the results from these high spatial resolution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 6 SUMMARY & CONCLUSIONS We presented the MUSE NFM observations of the Circinus galaxy at a spatial resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='1′′ (∼2 pc) that resolves the regions close to the AGN torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We derived the properties of the ionised gas outflow using the [O iii]𝜆5007 emission line and the dust distribution using Balmer Decrement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We follow a non-parametric approach to analyse the emission lines, so the derived properties are independent of the fitting functions used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The main results of this work is summarized below: The flux distribution of the systemic component of [O iii] emis- sion, defined by the velocity components within ±300 km s−1, shows a conical morphology, which is also observed in larger spatial scales up to hundreds of parsecs in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Archival radio observa- tions show that the radio jet is aligned approximately with the edge of the ionisation cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The flux distribution of the outflowing com- ponent of the [O iii] emission, on the other hand, shows a collimated structure up to ∼30 pc before fragmenting into two arms that overall mimics a “tuning-fork” shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The outflowing structure itself is not smooth and shows clumps at several locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' A comparison between the stellar kinematic map and the [O iii] centroid map suggests that the ionised gas (both the systemic as well as the outflowing component) co-rotates with the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Both the non-parametric velocity distributions, 𝑣10 and 𝑤80 show similar structures as the outflow flux i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', the tuning-fork shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Most of this outflow is blue-shifted, consistent with the outflow models of the Circinus galaxy reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We find a total instantaneous mass outflow rate of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='01 M⊙ yr−1 (3×10−7 on average per pixel of size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='5 pc) and a time-average mass outflow rate of 10−4 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' This is much lower than the star formation within the galaxy and therefore, the ionised gas outflow is not expected to regular star formation within a radius of ∼100 pc from the AGN location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' MNRAS 000, 1–9 (2021) Dust clumps Radio jet lonisationcone AGN20 15 [arcsec] 10 5 Ay I 0 5 10 10 5 0 5 10 15 20 △x [arcsec]8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Kakkad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The extinction maps using the systemic components of the H𝛽 and H𝛼 line show that the dust distribution is concentrated along the ionisation cone i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=', the polar direction, consistent with the archival mid-infrared observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The extinction map obtained from the outflowing components of H𝛽 and H𝛼 shows a scarce distribution, with a clump approximately at the tip of the collimated part of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' This dust clump might explain the fragmentation in the outflowing ionised gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We combine previous reported results gathered from the litera- ture and the observed outflows from the NFM data presented in this paper to present a model of the ionised gas outflow in the Circinus galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' We suggest that the observed outflow in the Circinus galaxy is composed of high velocity collimated gas that is enveloped by lower velocity dusty ionised gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The presence of a dust clump at the tip of the collimated part of the outflow might be responsible for the fragmentation in the outflowing gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The collimated outflow might be launched by a radio jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Although the jet, observed in low spatial resolution radio observations, does not show a 1:1 alignment with the outflowing cone PA, they are known to bend and change at larger scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' While the MUSE-NFM only provides the kinematic information in the ionised gas phase, outflows have been known in exist also in the molecular gas phase in the Circinus galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' A multi-wavelength approach to trace gas in the other phases such as warm and cold molecular, at the same spatial resolution of ∼2 pc, will be key to obtaining a holistic view into the outflow-AGN connection in the Circinus galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Furthermore, high spatial resolution radio obser- vations will verify if the observed collimated outflow is a result of jet-ISM interaction on small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We would like to thank the anonymous referee for insightful com- ments and suggestions to improve this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' The authors also thank T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Fischer for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' Based on observations from the ESO programme ID 0103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='B-0396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' acknowledge support by the Science Fund of the Republic of Serbia, PROMIS 6060916, BOWIE and by the Ministry of Education, Science and Technological Development of the Republic of Serbia through con- tract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 451-03-9/2022-14/200002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' acknowledges funding through the European Union’s Horizon 2020 and Innovation pro- gramme under the Marie Sklodowska-Curie grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' 793499 (DUSTDEVILS).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} +page_content=' MNRAS 000, 1–9 (2021)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQf6voI/content/2301.00825v1.pdf'} diff --git a/R9E4T4oBgHgl3EQf_g7i/content/tmp_files/2301.05372v1.pdf.txt b/R9E4T4oBgHgl3EQf_g7i/content/tmp_files/2301.05372v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e2a912b3c2d83b0c77d88802d05454e0703e1cb --- /dev/null +++ b/R9E4T4oBgHgl3EQf_g7i/content/tmp_files/2301.05372v1.pdf.txt @@ -0,0 +1,964 @@ +Text to Point Cloud Localization with Relation-Enhanced Transformer +Guangzhi Wang1, Hehe Fan2, Mohan Kankanhalli2 +1Institute of Data Science, National University of Singapore +2School of Computing, National University of Singapore +guangzhi.wang@u.nus.edu, hehe.fan@nus.edu.sg, mohan@comp.nus.edu.sg +Abstract +Automatically localizing a position based on a few natural +language instructions is essential for future robots to commu- +nicate and collaborate with humans. To approach this goal, +we focus on the text-to-point-cloud cross-modal localization +problem. Given a textual query, it aims to identify the de- +scribed location from city-scale point clouds. The task in- +volves two challenges. 1) In city-scale point clouds, similar +ambient instances may exist in several locations. Searching +each location in a huge point cloud with only instances as +guidance may lead to less discriminative signals and incor- +rect results. 2) In textual descriptions, the hints are provided +separately. In this case, the relations among those hints are +not explicitly described, leading to the difficulties of learn- +ing relations. To overcome these two challenges, we propose +a unified Relation-Enhanced Transformer (RET) to improve +representation discriminability for both point cloud and nat- +ural language queries. The core of the proposed RET is a +novel Relation-enhanced Self-Attention (RSA) mechanism, +which explicitly encodes instance (hint)-wise relations for the +two modalities. Moreover, we propose a fine-grained cross- +modal matching method to further refine the location predic- +tions in a subsequent instance-hint matching stage. Experi- +mental results on the KITTI360Pose dataset demonstrate that +our approach surpasses the previous state-of-the-art method +by large margins. +Introduction +Understanding natural language instructions in the 3D real +world is a fundamental skill for future artificial intelligence +assistants to collaborate with humans. In this paper, we fo- +cus on the outdoor environment and study the task of natural +language-based localization from city-scale point clouds. As +shown in Figure 1, given a linguistic description of a posi- +tion, which contains several hints, the goal of the task is to +find out the target location from a large-scale point cloud. +This task can effectively help mobile robots, such as self- +driving cars and autonomous drones, cooperate with humans +to coordinate actions and plan their trajectories. By under- +standing the destination from natural language instructions, +it reduces the human effort required for manual operation. +However, this task is intrinsically challenging. Precise lo- +calization requires both correct language interpretation and +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +Heading to a place: +[hint1] east of a dark-green terrain. +[hint2] south of a gray road. +[hint3] west of a dark-green traffic sign. +[hint4] south of a green terrain. +Textual Query +Localization +Figure 1: Illustration of the text to point cloud localization +task. Given a textual query, which usually contains several +independent hints, the goal is to localize the point of interest +in a huge city-scale point cloud. +effective large-scale point cloud understanding. Considering +the difficulties, an existing method (Kolmet et al. 2022) first +divides a city-wide point cloud into several cells, and then +solves this task in a Coarse-to-Fine manner. +The goal of the ‘coarse’ stage is to find out the target +cell that contains the queried location according to the given +natural language descriptions. In this stage, the instances +included in point cloud cells and those mentioned in lan- +guage descriptions are mainly used for text-to-point-cloud +retrieval based on their types, without considering their rela- +tions. In the ‘fine’ stage, each object in the textual query is +matched with an in-cell point cloud instance, whereby a tar- +get location will be predicted from each hint. This pioneer- +ing method sets up a significant starting point for tackling +the challenging task. However, it fails to consider the intrin- +sic relations in both stages, resulting in sub-optimal perfor- +mance. +For the coarse stage, because similar ambient instances +may exist in several cells, performing retrieval based on only +the cell-contained and query-related instance types without +considering their relations may lead to low discriminabil- +arXiv:2301.05372v1 [cs.CV] 13 Jan 2023 + +ity for both cell and query representations, which inevitably +leads to ambiguity. Based on those low-discriminability rep- +resentations, it is difficult to find out the correct cell. In the +fine stage, we observe that insufficient cross-modal collabo- +ration leads to difficulties in location refinement. Given the +retrieved cell, precise location prediction requires joint un- +derstanding of both point clouds and textual queries. How- +ever, in the previous method (Kolmet et al. 2022), the cross- +modal collaboration is only performed from textual queries +to point clouds in a single step, which results in optimization +difficulty for multi-task learning. +In this work, we aim to solve the aforementioned short- +comings in both stages. For the coarse stage, we pro- +pose to encode pairwise instance relations to improve rep- +resentation discriminability for both modalities, which is +achieved through a novel Relation-Enhanced Transformer +(RET) architecture. In particular, the in-cell point cloud in- +stance relations are modeled as their geometric displace- +ments, while computed as the fusion of hint representations +in the linguistic domain. These relations from two modali- +ties are respectively incorporated into their representation in +a unified manner, which is achieved through the proposed +Relation-enhanced Self-Attention (RSA) mechanism. For +the fine stage, we perform Cascaded Matching and Refine- +ment (CMR) to enhance cross-modal collaboration. In par- +ticular, different from (Kolmet et al. 2022) which achieves +this objective in a single step, we perform description- +instance matching and position refinement in two sequential +steps. Such formulation allows us to minimize the optimiza- +tion difficulty of multi-objective learning and noisy interme- +diate results, thereby improving cross-modal collaboration. +We validated the effectiveness of our method on the +KITTI360Pose benchmark (Kolmet et al. 2022). Extensive +experiments demonstrate that the proposed method can sur- +pass the previous approach by a large margin, leading to new +state-of-the-art results. Our contributions are three-fold: +• We propose a novel Relation-Enhanced Transformer +(RET) to improve representation discriminability for +both point clouds and textual queries. The core com- +ponent of RET is the Relation-enhanced Self-Attention +(RSA) mechanism, which encodes instance (hint) rela- +tions for the two modalities in a unified manner. +• We propose to perform cross-modal instance matching +and position refinement in two sequential steps. This for- +mulation allows us to minimize the optimization diffi- +culty of multi-task learning and the influence of noisy +intermediate results, thereby improving cross-modal col- +laboration for fine-grained location prediction. +• We perform extensive experiments on the KITTI360Pose +dataset (Kolmet et al. 2022). The results show that our +approach can surpass previous method by a large margin, +resulting in new state-of-the-art performance. Additional +ablation studies further demonstrate the effectiveness of +each component in the proposed method. +Related Work +Transformer and Attention Mechanism. Transformer and +self-attention mechanism (Vaswani et al. 2017; Fan, Yang, +and Kankanhalli 2021) has become increasingly popular in +recent years. Although first proposed for natural language +processing, with architectural adaptation, Transformer has +been widely applied to many vision tasks including visual +recognition (Dosovitskiy et al. 2020; Liu et al. 2021), object +detection (Carion et al. 2020; Zhu et al. 2020) and seman- +tic segmentation (Cheng, Schwing, and Kirillov 2021). Be- +sides, the transformer-based architectures are also utilized to +model cross-modal (e.g., vision and language) relations (Tan +and Bansal 2019; Lu et al. 2019; Li et al. 2019; Zhang et al. +2021; Li et al. 2022). In these architectures, the attention +mechanism is widely employed to implicitly learn relations +among the input tokens. Nevertheless, without explicit rela- +tion encoding, the vanilla Transformer can only encode rela- +tions implicitly with the help of positional encoding (Doso- +vitskiy et al. 2020). To facilitate better relation modeling, +some works modulate the attention computation process +by explicitly incorporating element relations. For example, +(Wu et al. 2021) modified the attention mechanism via uni- +fied relative position bias to improve visual recognition. For +object detection, spatial relations between bounding boxes +are introduced to modulate the attention weights (Liu et al. +2022; Gao et al. 2021). For dynamic point cloud analy- +sis, displacement between points (Fan, Yang, and Kankan- +halli 2022) is utilized for point-specific attention computa- +tion. In this work, we propose to model relations for both +point clouds and language queries by explicitly incorporat- +ing intra-modality relations in a unified manner. +Visual Localization. The task that is most related to ours is +vision-based localization (Arandjelovic et al. 2016; Brach- +mann et al. 2017; Hausler et al. 2021), which is to estimate a +pose based on an image or image sequence. Existing meth- +ods mostly solve this task in two stages (Sarlin et al. 2019; +Sattler, Leibe, and Kobbelt 2016; Zhou et al. 2020). The first +stage finds a subset of all images using image retrieval-based +techniques (Arandjelovic et al. 2016; Hausler et al. 2021; +Torii et al. 2015), while the second stage establishes pixel- +wise correspondence between the query image and the re- +trieved one to predict the precise pose. In this work, we also +study the task of localization in a coarse-to-fine manner, but +differ from visual localization in that: 1) we try to infer the +location from city-wide point clouds instead of images. 2) +we try to estimate the pose from textual query rather than +images. Compared to visual localization, our task requires +multi-modal understanding and is more challenging to solve. +3D Language Grounding. As we humans live in a 3D +world and communicate through natural language, recent +work has begun to investigate the tasks on the cross-modal +understanding of 3D vision and natural language. Among +these tasks, the one that is most related to ours is 3D lan- +guage grounding, which aims at localizing an object in +point clouds from a given natural language query. For ex- +ample, ScanRefer (Chen, Chang, and Nießner 2020) stud- +ies 3D language grounding from real-life in-door scenes. +ReferIt3D (Achlioptas et al. 2020) studies a related task un- +der a simpler setting, which assumes the object instances +are segmented in advance. InstanceRefer (Yuan et al. 2021) +improves previous methods by adopting a 3D panoptic seg- +mentation backbone, utilizing multi-level visual context. Re- + +east of a dark-green terrain. +south of a gray road. +south of a green terrain. +west of a dark-green traffic sign. +Split +... +Cells +Textual Query +Hint Encoder +north of a dark-green smallpole . +east of a green pole. +... +Instance Encoder +Hints +Instances +Relation-Enhanced +Self-Attention +Add & LayerNorm +Feed Foward Network +Add & LayerNorm +Relation-Enhanced +Self-Attention +Add & LayerNorm +Feed Foward Network +Add & LayerNorm +Instance-wise +Relation +x +x +... +(a) Hint-Instance Matching +... +... +... +... +Feature +Pooling +(b) Offset Prediction +Offsets +Matching +Coarse Stage +Fine Stage +Cross-modal +Fusion +Multi-Layer +Perceptron +Figure 2: Framework of the proposed method. The city-scale point cloud is first divided into individual cells. Then, in the +coarse stage, the cells and the textual query are respectively encoded with the proposed Relation-Enhanced Transformer (RET), +which are later used for query-cell matching. In the fine stage, each hint is matched with an in-cell instance. Then, cross-modal +fusion dynamically aggregates hints and instance representations for offset prediction. The target location is predicted based on +matching results and offset predictions. +cently, graph structure (Feng et al. 2021) is also utilized to +improve the representation learning qualities. +Methodology +Preliminaries +Given a textual query, our goal is to identify the position it +describes from a city-scale point cloud. To handle the large- +scale point cloud, we divide each scene into a set of cubic +cells of fixed size by a preset stride. Each cell C contains a +set of p point cloud instances, which are encoded by Point- +Net++ (Qi et al. 2017) into vector representations {pi}p +i=1. +Following (Kolmet et al. 2022), the textual query T is repre- +sented as a set of hints {hj}h +j=1, each encoding the direction +relation between the target location and an instance. +Inspired by the existing work (Kolmet et al. 2022), given +the cell splits, we solve this task in a coarse-to-fine manner +with two stages. The coarse stage is formulated as textual +query based cell retrieval. The goal of this stage is to train +a model that encodes C and T into a joint embedding space +whereby matched query-cell pairs are close while those un- +matched are pulled apart (Kiros, Salakhutdinov, and Zemel +2014). In the fine stage, given a retrieved cell, we aim to +refine the position prediction by utilizing fine-grained cross- +modal information. In particular, we first match each hint +in the query with an in-cell instance by formulating it as an +optimal transport problem (Liu et al. 2020). After that, with +the matching results, we predict the target location through +a cross-modal fusion of point cloud instance and hint repre- +sentations. Based on the fused representation, we predict the +target location for each matched instance. Finally, we obtain +the target location prediction based on a weighted combi- +nation of the matching and location prediction results. The +framework of our method is shown in Figure 2. In the fol- +lowing of this section, we will explain the proposed method +for coarse stage and fine stage. After that, our training and +inference procedure will be detailed. +Coarse Stage: Relation-Enhanced Transformer +After the cell split, the goal of the coarse stage is to suc- +cessfully retrieve the cell C given a textual query T . To ap- +proach this objective, we need to encode C and T into a joint +embedding space. An intuitive solution is to encode both +C and T based on the instances they contained as is done +in (Kolmet et al. 2022). However, with such representations, +the low discriminability for cells and textual queries results +in poor retrieval performance. We argue that this can be at- +tributed to the following two reasons. On the one hand, the +outdoor scenes are often of low diversity, whereby a group +of mentioned instances can appear at multiple different lo- +cations. Thus, simply describing a cell with its contained in- +stances can result in less discriminative representations. On +the other hand, the textual queries often contain limited clues +compared to the point clouds, making this cross-modality re- +trieval especially challenging. To this end, we propose to ex- +plicitly encode instance-relations to provide more discrimi- +native representations for both modalities. +The Transformer (Vaswani et al. 2017) has been widely +utilized for relation-based representation learning in vari- +ous tasks (Hu et al. 2018; Liu et al. 2021; Fan, Yang, and +Kankanhalli 2022). The key component of the Transformer +is the Self-Attention (SA) operation: +Attn(Q, K, V ) = Softmax(QKT / +√ +d)V , +(1) + +Pooling +Matmul +Add +Figure 3: Illustration of the proposed Relation-enhanced +Self-Attention (RSA) mechanism. Pairwise relations are ex- +plicitly encoded into the value computation process. +where d is the representation dimension and Q, K, V +∈ +RN×d are the query, key and value matrices by transform- +ing in-cell instances (or hints for textual queries) with corre- +sponding linear transformations: +Q = W QX, K = W KX, V = W V X, +(2) +with W ∗ ∈ Rd×d are learnable matrices and X = P ∈ +Rp×d or H ∈ Rh×d represents stacked instances1. +Despite its generality, the vanilla SA lacks explicit rela- +tions in both modalities, thus is less informative to represent +the cell and query. To this end, we propose a novel Relation- +Enhanced Transformer (RET) to model explicit instance re- +lations in both point clouds and textual descriptions. Our +RET is a stack of multiple Transformer encoder layers, ex- +cept that, in place of SA, we propose a Relation-enhanced +Self-Attention (RSA) to explicitly incorporate relation in- +formation into value computation. The computation process +is shown as follows and illustrated in Figure 3. +RSA(Q, K, V , R) = Softmax(QKT / +√ +d)(V +Pool(R, 1)), +(3) +where R ∈ RN×N×d captures pairwise relations with +Rij ∈ Rd representing the relation between the i-th and j- +th instance (hint). Pool(R, 1) indicates pooling tensor R +along dimension 1. In this way, our model can explicitly +encode instance relations through this computation process, +leading to more informative representations. +The definition of relation varies flexibly with task objec- +tive and input modality. For point cloud data, we take the +geometric displacement of two instances as their relations, +as direction is often mentioned in textual queries and thus +informative for retrieval:2 +RV +ij = W V (ci − cj), +(4) +where ci ∈ R3 represents the center coordinate of the i-th +instance and W v ∈ Rd×3 transforms the displacement into +1Note that the attention operation is often performed in different +subspaces with multiple heads, which is omitted for simplicity. +2We have also tried other features such as number of points +and bounding boxes of instances but didn’t observe performance +improvement. +embedding space. For the linguistic description, we compute +the hint relation as the concatenation of their embeddings: +RL +ij = W L[hi; hj], +(5) +where W L ∈ Rd×2d transforms the linguistic feature into +representation space. With the computation of RSA, the +instance-wise relations for different modalities can be uni- +formly incorporated into query or cell representations +Finally, the cell (description) representations Cm (Tm) are +obtained via a pooling operation over all instances (hints) +output from the RET for cross-modal retrieval. +Fine Stage: Cascaded Matching and Refinement +Following the coarse stage, we aim to refine the location pre- +diction within the retrieved cell in the fine stage. Inspired +by (Kolmet et al. 2022), we perform instance matching and +location refinement to utilize the fine-grained visual and lin- +guistic information, which involves the following two objec- +tives: (1) For each hint, we find the in-cell instance it refers +to via a matching process. (2) For each matched pair (i, j), +a regressor predicts an offset ˆti ∈ R2 for each matched hint +hj, which represents the offset from the instance center ci +to the target location.3 +Previous method (Kolmet et al. 2022) achieves the two +objectives within a single step. However, given the objec- +tive of both hint-instance matching and offset prediction, +the multi-task learning process introduces optimization dif- +ficulty. Furthermore, in the early training steps, the matcher +is only partially trained, which produces noisy matching re- +sults. The regressor learns and makes predictions based on +this noisy results, leading to unstable learning process and +sub-optimal performance. +To this end, we propose a Cascaded Matching and Refine- +ment (CMR) strategy for the fine stage, where hint-instance +matching and offset regression are sequentially performed. +Specifically, following (Kolmet et al. 2022), we first train +the SuperGlue (Sarlin et al. 2020) matcher for hint-instance +matching, which is formulated as an optimal-transport prob- +lem. Given the trained matcher, we obtain a set of hint- +instance matching results {pi, hj, wi}h +j=1, where wi repre- +sents the confidence of the match. Then, to reduce the noise +for regression, we predict the target location according to +matched instances only. +Precise location prediction requires proper understand- +ing on both point cloud (what and where the referred in- +stance is, e.g., dark-green terrain) and language de- +scription (what is the relation between the matched instance +and the target location, e.g., east of). For this, we pro- +pose to facilitate cross-modal collaboration via the Cross- +Attention (CA) mechanism, which is commonly used for +cross-modality information fusion. +CA(H, P ) = Attn(W QH, W KP , W V P ), +(6) +where H, P represent hints and instances, respectively, and +W ∗ are learnable transformation matrices. Shortcut connec- +tion and layer normalization (Ba, Kiros, and Hinton 2016) +3For position prediction, we ignore the height information and +considers 2D coordinates only. + +Table 1: Performance comparison on the KITTI360Pose. +Method +Localization Recall (ϵ < 5/10/15m) ↑ +Validation Set +Test Set +k = 1 +k = 5 +k = 10 +k = 1 +k = 5 +k = 10 +Text2Pos (Kolmet et al. 2022) +0.14/0.25/0.31 +0.36/0.55/0.61 +0.48/0.68/0.74 +0.13/0.21/0.25 +0.33/0.48/0.52 +0.43/0.61/0.65 +RET (Ours) +0.19/0.30/0.37 +0.44/0.62/0.67 +0.52/0.72/0.78 +0.16/0.25/0.29 +0.35/0.51/0.56 +0.46/0.65/0.71 +follows the cross-attention operation. With these operations, +the hint representation hi is accordingly updated to ˜hi by +dynamically fusing visual information. As such, the infor- +mation in the two modalities are joint utilized with the help +of cross-modal collaboration. +Then, we predict the offset (the direction vector from in- +stance center to target location) from the updated hint: +ˆti = MLP(˜hj). +(7) +To utilize the matching results, the final prediction is ob- +tained via a weighted combination of each hint’s prediction: +ˆg = +� +i +wi +� +m wm +(ci + ˆti), +(8) +where wi ∈ [0, 1] is the confidence score of the match +(pi, hj, wi) and is set to 0 for non-matched instances. To +filter out noisy matches, we consider only matches with con- +fidence score greater than 0.2. +Training and Inference +Training. For the coarse stage, we train the proposed RET +for cross-modal retrieval with pairwise ranking loss (Kiros, +Salakhutdinov, and Zemel 2014): +Lcoarse = +Nb +� +m=1 +Nb +� +n̸=m +[α − ⟨Cm, Tm⟩ + ⟨Cm, Tn⟩]+ ++ +Nb +� +m=1 +Nb +� +n̸=m +[α − ⟨Tm, Cm⟩ + ⟨Tm, Cn⟩]+, +(9) +where Nb is the batch size, α is a hyper-parameter to con- +trol the separation strength and ⟨·, ·⟩ represents inner product +between vectors. This loss function encourages the represen- +tation of matched description-cell pair to be by a margin α +closer than those unmatched. For the fine stage, we employ +the loss in (Sarlin et al. 2020) to train the matcher, while L2 +loss is applied to train the offset regressor. +Inference. We first encode all cells and queries into a joint +embedding space with the proposed Relation-Enhanced +Transformer. Then, for each query representation, we re- +trieve top-k cells with highest similarity. For each retrieved +cell, we use the SuperGlue matcher trained in the fine stage +to match each hint with an in-cell instance, which is fol- +lowed by offset prediction based on the fused representa- +tions. Finally, the position prediction is given by Eq. 8. +Experiments +Dataset and Implementation Details +Dataset Details. We evaluate our method on the recently +proposed KITTI360Pose dataset (Kolmet et al. 2022), which +is built upon the KITTI360 dataset (Liao, Xie, and Geiger +2021) with sampled locations and generated hints. It con- +tains point clouds of a total of 9 scenes, covering 14,934 +positions with a total area of 15.51km2. We follow (Kol- +met et al. 2022) to use five scenes for training, one for val- +idation, and the remaining three for testing. We sample the +cells of size 30m with a stride of 10m. For more details on +the dataset preprocessing, please refer to our supplementary +material. +Implementation Details For the coarse stage, we trained +the model with AdamW optimizer (Loshchilov and Hutter +2018) with a learning rate of 2e-4. The models are trained +for a total of 18 epochs while the learning rate is decayed +by 10 at the 9-th epoch. The α is set to 0.35. For the fine +stage, we first train the matcher with a learning rate of 5e- +4 for a total of 16 epochs. Afterwards, we fix the matcher +and train the regressor based on the matching results for 10 +epochs with a learning rate of 1e-4. The regressor is for- +mulated as a 3 layer Multi-Layer Perceptron. Both of the +two steps adopt an Adam (Kingma and Ba 2014) optimizer. +The RET has 2 encoder layers for both point cloud part and +linguistic part, each utilizing the Relation-enhanced Atten- +tion (RSA) mechanism with 4 heads and hidden dimension +2048. For the two stages, we encode each instance in the cell +with PointNet++ (Qi et al. 2017) provided by Text2Pos (Kol- +met et al. 2022) for a fair comparison. The hint representa- +tions are obtained by concatenating learned word embed- +dings. More details are provided in our appendix.4 +Comparison with the State-of-the-art +We compared our method with Text2Pos (Kolmet et al. +2022) on the KITTI360Pose dataset. Following (Kolmet +et al. 2022), we report top-k (k = 1/5/10) recall rate of dif- +ferent error ranges ϵ < 5/10/15m for comprehensive com- +parison. The results are shown in Table 1. Text2Pos gives +a recall of 0.14 when k = 1 and ϵ < 5m. In contrast, +our method can significantly improve the recall rate to 0.19, +which amounts to 35.7% relative improvement upon the +baseline. Furthermore, when we relax the localization error +constraints or increase k, consistent improvements upon the +baseline can also be observed. For example, with ϵ < 5m, +our method achieves top-5 recall rate of 0.44, which is 8% +higher than previous state-of-the-art. Similar improvements +can also be seen on the test set, showing our method is su- +perior to the baseline method. +Ablation Studies +In this section, we perform ablation studies for both stages +to investigate the effectiveness of each proposed component +4Code available at: https://github.com/daoyuan98/text2pos-ret + +Table 2: Ablation study of the Relation-Enhanced Trans- +former (RET) on KITTI360Pose validation set. ”wo X rela- +tion” indicates replacing the proposed RSA with the vanilla +Self-Attention in corresponding modality. +Method +k = 1 ↑ +k = 3 ↑ +k = 5 ↑ +w/o both relations +0.11 +0.24 +0.32 +w/o linguistic relation +0.14 +0.28 +0.37 +w/o visual relation +0.16 +0.30 +0.40 +Full (Ours) +0.18 +0.34 +0.44 +Table 3: The effects of #layers of RET and #heads of RSA. +#Layers +#Heads +k = 1 ↑ +k = 3 ↑ +k = 5 ↑ +1 +4 +0.16 +0.31 +0.40 +1 +8 +0.16 +0.30 +0.40 +2 +2 +0.17 +0.32 +0.42 +2 +4 +0.18 +0.34 +0.44 +2 +8 +0.16 +0.31 +0.40 +3 +4 +0.16 +0.32 +0.39 +3 +8 +0.15 +0.29 +0.37 +in our method. The ablation studies for coarse stage and fine +stage are provided separately for clear investigation. +Coarse Stage. We study the importance of explicit relation +incorporation in the coarse stage. Since the coarse stage is +formulated as a retrieval task, we use top-1/3/5 recall rate as +evaluation metric, whereby the cell that contains the ground +truth location is defined as positive. +Relation Incorporation. We first study the necessity of ex- +plicit relation modeling for both point cloud and textual +queries. The results are shown in Table 2. It can be observed +that relation modeling contributes significantly to successful +retrieval. In particular, without any relation incorporation, +the top-5 recall rate is 0.32. With the explicit fusion of lin- +guistic relation, we observe an increase of 0.05 recall rate +under same condition. Besides, with the incorporation of vi- +sual (point cloud instance) relations only, the top-5 recall +rate can be improved by 0.08, indicating explicit relations +in the point clouds play a more important role. Finally, with +both relations, we achieve an improvement of 0.12 at top-5 +recall rate upon that without any relation, showing that both +visual and linguistic relations are necessary and complemen- +tary to improve the cell retrieval performance. +RET Hyper-parameters. We also studied the importance of +the hyper-parameters involved in RET, namely the number +of layers of RET and the number of heads of RSA. The re- +sults are shown in Table 3. It can be observed that, thanks to +the strong relation modeling capacity of the proposed RET, +we can obtain the best performance with 2 layers and 4 heads +in the RSA. Decreasing and increasing the number of layers +both lead to worse performance, which may be attributed to +underfitting and overfitting, respectively. +Fine Stage. The objective of the fine stage is to correctly +match linguistic hints and point cloud instances and regress +the target location. Thus, we study the performance of the +matcher and regressor, respectively. +Table 4: Comparison of training strategy and matcher per- +formance on the KITTI360Pose dataset. +Strategy +Train +Validation +Precision ↑ +Recall ↑ +Precision ↑ +Recall ↑ +joint +98.12 +98.16 +86.67 +87.59 +cascade(ours) +98.89 +99.04 +92.18 +93.01 +Table 5: Ablation study on the regression error of the fine- +stage on the KITTI360Pose dataset. +Method +Train Error ↓ +Validation Error ↓ +w/o cascade training +10.24 (+1.72) +10.01 (+0.86) +w/o cross-attention +9.57 (+1.05) +9.56 (+0.41) +w/o confidence weighting +9.02 (+0.50) +9.23 (+0.08) +Ours +8.52 +9.15 +Matcher. Following (Sarlin et al. 2020), we take precision +and recall as the the evaluation metric of the matcher. With +an identical matcher architecture, we investigate the impact +of training strategy on the matcher performance. The results +are shown in Table 4. It can be seen that compared with joint +training (Kolmet et al. 2022), our cascaded training achieves +not only high precision and recall in the training set, but +also stronger generalization on the validation set. The re- +sults demonstrate that the cascade training strategy is able to +mitigate the multi-task optimization difficulty. +Regressor. The regressor predicts the target location based +on the the matching results. We study the effects of cas- +caded training, cross-attention based cross-modal fusion and +confidence weighting for final location prediction. We use +regression error as evaluation metric and compare different +versions on both KITTI360Pose training and validation set. +The results are shown in Table. 5. Without cascaded training +strategy, the regressor achieves an error of 10.24 and 10.01 +on the training and validation set, respectively, which is 1.72 +and 0.86 higher than that with cascaded training. This re- +sult suggests that our cascaded training strategy also allevi- +ates the optimization difficulty of the regressor, which was +caused by the noisy intermediate results. Furthermore, with- +out cross-attention mechanism, the regression error also in- +creases by a considerable margin, showing that cross-modal +collaboration is important for precise location prediction. Fi- +nally, with confidence-based weighting, we can further re- +duce the regression error on both the training and validation +set, suggesting this information from the trained matcher can +be further utilized to improve performance. +Visualizations +Embedding Space Visualization. We visualize the learned +embedding space via T-SNE (Van der Maaten and Hin- +ton 2008) in Figure 5. It can be observed that the base- +line method Text2Pos (Kolmet et al. 2022) results in a less +discriminative space, where positive cells are relatively far +away from the query and sometimes separated across the +embedding space. In contrast, our method draw positive cell +and query representations closer in the embedding space, re- +sulting in a more informative embedding space for retrieval. + +Ground Truth +Top-1 +Top-2 +Top-3 +Ground Truth +Top-1 +Top-2 +Top-3 +(a) +(b) +(c) +(e) +(d) +(f) +557.85 +10.00 +20.00 +0.00 +10.00 +50.99 +10.00 +0.0 +819.08 +10.00 +0.00 +64.03 +14.14 +211.90 +221.36 +455.41 +1150.00 +218.40 +Building +Pole +Traffic Light +Traffic Sign +Parking +Sidewalk +Vegetation +Terrain +Road +Wall +Garage +Figure 4: Qualitative retrieval results on KITTI360Pose validation set. The red dot in the ground truth cell indicates the target +location. In each retrieved cell, the number in the lower right indicates the center distance between this cell and the ground +truth. Green box indicates positive cell which contains the target location, while red box indicates negative cells. +Text2Pos +Ours +Textual Query +Negative Cell +Positive Cell +Figure 5: T-SNE visualization of embedding space for the +coarse stage. A cell is considered as positive if it contains +the location described by the query. Compared with baseline +method (Kolmet et al. 2022), our method can produce better +representation where positive cells are closer to the target. +Qualitative Cell Retrieval Results. We show some exam- +ple text to point cloud retrieval results in Figure. 4. For a +given query, we visualize the top-3 retrieved cells. A re- +trieved cell is defined as positive if it contains the target lo- +cation. It can be observed that, our method can retrieve the +ground truth cell or those close in most cases. Sometimes, +negative cells can also be retrieved, e.g., top-1 in (a) and +top-3 in (e). It can be seen that these retrieved negative cells +exhibit high semantic similarity with the ground truth cell, +even though far away from it. We also show a failure case (f), +where the retrieved cells are all negative. It can be seen that +even though far away from the target location, all these neg- +ative cells have instances similar to the ground truth. These +observations suggest that outdoor scenes are indeed of low +diversity, indicating that successful retrieval requires highly +discriminative representations to disambiguate the cells. +Conclusion +In this work, we proposed a novel method for precise +text-based localization from large-scale point clouds. Our +method employs a coarse-to-fine principle and pipelines this +process into two stages. For the coarse stage which is formu- +lated as a textual query based cell retrieval task, we aim to +improve representation discriminability for both point cloud +and query representations. This is achieved through explicit +modeling of instance relations and implemented via a newly +proposed Relation-Enhanced Transformer (RET). The core +of RET is a novel Relation-enhanced Self-Attention (RSA) +mechanism, whereby the instance relations for the two +modalities are explicitly incorporated into the value com- +putation process in a unified manner. For the fine stage, +our method performs description-instance matching and +position refinement in a cascaded way, whereby cross- +modal information collaboration is enhanced through the +cross-attention mechanism. Extensive experiments on the +KITTI360Pose dataset validated the effectiveness of the pro- +posed method, which achieves new state-of-the-art perfor- +mance. Additional ablation studies further corroborate the +effectiveness of each component in the proposed method. +Acknowledgement +This research is supported by the National Research Foun- +dation, Singapore under its Strategic Capability Research +Centres Funding Initiative. Any opinions, findings and con- +clusions or recommendations expressed in this material are +those of the author(s) and do not reflect the views of National +Research Foundation, Singapore. + +tReferences +Achlioptas, P.; Abdelreheem, A.; Xia, F.; Elhoseiny, M.; and +Guibas, L. 2020. Referit3d: Neural listeners for fine-grained +3d object identification in real-world scenes. +In ECCV. +Springer. +Arandjelovic, R.; Gronat, P.; Torii, A.; Pajdla, T.; and Sivic, +J. 2016. NetVLAD: CNN architecture for weakly supervised +place recognition. In CVPR. +Ba, J. L.; Kiros, J. R.; and Hinton, G. E. 2016. 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In ICLR. + diff --git a/R9E4T4oBgHgl3EQf_g7i/content/tmp_files/load_file.txt b/R9E4T4oBgHgl3EQf_g7i/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7cd7ca13a780e3b71764fd01aa782cb8430c6abf --- /dev/null +++ b/R9E4T4oBgHgl3EQf_g7i/content/tmp_files/load_file.txt @@ -0,0 +1,952 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf,len=951 +page_content='Text to Point Cloud Localization with Relation-Enhanced Transformer Guangzhi Wang1, Hehe Fan2, Mohan Kankanhalli2 1Institute of Data Science, National University of Singapore 2School of Computing, National University of Singapore guangzhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='wang@u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='edu, hehe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='fan@nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='sg, mohan@comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='sg Abstract Automatically localizing a position based on a few natural language instructions is essential for future robots to commu- nicate and collaborate with humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' To approach this goal, we focus on the text-to-point-cloud cross-modal localization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Given a textual query, it aims to identify the de- scribed location from city-scale point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The task in- volves two challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 1) In city-scale point clouds, similar ambient instances may exist in several locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Searching each location in a huge point cloud with only instances as guidance may lead to less discriminative signals and incor- rect results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2) In textual descriptions, the hints are provided separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In this case, the relations among those hints are not explicitly described, leading to the difficulties of learn- ing relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' To overcome these two challenges, we propose a unified Relation-Enhanced Transformer (RET) to improve representation discriminability for both point cloud and nat- ural language queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The core of the proposed RET is a novel Relation-enhanced Self-Attention (RSA) mechanism, which explicitly encodes instance (hint)-wise relations for the two modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Moreover, we propose a fine-grained cross- modal matching method to further refine the location predic- tions in a subsequent instance-hint matching stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Experi- mental results on the KITTI360Pose dataset demonstrate that our approach surpasses the previous state-of-the-art method by large margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Introduction Understanding natural language instructions in the 3D real world is a fundamental skill for future artificial intelligence assistants to collaborate with humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In this paper, we fo- cus on the outdoor environment and study the task of natural language-based localization from city-scale point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' As shown in Figure 1, given a linguistic description of a posi- tion, which contains several hints, the goal of the task is to find out the target location from a large-scale point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' This task can effectively help mobile robots, such as self- driving cars and autonomous drones, cooperate with humans to coordinate actions and plan their trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' By under- standing the destination from natural language instructions, it reduces the human effort required for manual operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' However, this task is intrinsically challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Precise lo- calization requires both correct language interpretation and Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Heading to a place: [hint1] east of a dark-green terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' [hint2] south of a gray road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' [hint3] west of a dark-green traffic sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' [hint4] south of a green terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Textual Query Localization Figure 1: Illustration of the text to point cloud localization task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Given a textual query, which usually contains several independent hints, the goal is to localize the point of interest in a huge city-scale point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' effective large-scale point cloud understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Considering the difficulties, an existing method (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022) first divides a city-wide point cloud into several cells, and then solves this task in a Coarse-to-Fine manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The goal of the ‘coarse’ stage is to find out the target cell that contains the queried location according to the given natural language descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In this stage, the instances included in point cloud cells and those mentioned in lan- guage descriptions are mainly used for text-to-point-cloud retrieval based on their types, without considering their rela- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In the ‘fine’ stage, each object in the textual query is matched with an in-cell point cloud instance, whereby a tar- get location will be predicted from each hint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' This pioneer- ing method sets up a significant starting point for tackling the challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' However, it fails to consider the intrin- sic relations in both stages, resulting in sub-optimal perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For the coarse stage, because similar ambient instances may exist in several cells, performing retrieval based on only the cell-contained and query-related instance types without considering their relations may lead to low discriminabil- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='05372v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='CV] 13 Jan 2023 ity for both cell and query representations, which inevitably leads to ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Based on those low-discriminability rep- resentations, it is difficult to find out the correct cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In the fine stage, we observe that insufficient cross-modal collabo- ration leads to difficulties in location refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Given the retrieved cell, precise location prediction requires joint un- derstanding of both point clouds and textual queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' How- ever, in the previous method (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022), the cross- modal collaboration is only performed from textual queries to point clouds in a single step, which results in optimization difficulty for multi-task learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In this work, we aim to solve the aforementioned short- comings in both stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For the coarse stage, we pro- pose to encode pairwise instance relations to improve rep- resentation discriminability for both modalities, which is achieved through a novel Relation-Enhanced Transformer (RET) architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In particular, the in-cell point cloud in- stance relations are modeled as their geometric displace- ments, while computed as the fusion of hint representations in the linguistic domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' These relations from two modali- ties are respectively incorporated into their representation in a unified manner, which is achieved through the proposed Relation-enhanced Self-Attention (RSA) mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For the fine stage, we perform Cascaded Matching and Refine- ment (CMR) to enhance cross-modal collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In par- ticular, different from (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022) which achieves this objective in a single step, we perform description- instance matching and position refinement in two sequential steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Such formulation allows us to minimize the optimiza- tion difficulty of multi-objective learning and noisy interme- diate results, thereby improving cross-modal collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We validated the effectiveness of our method on the KITTI360Pose benchmark (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Extensive experiments demonstrate that the proposed method can sur- pass the previous approach by a large margin, leading to new state-of-the-art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Our contributions are three-fold: We propose a novel Relation-Enhanced Transformer (RET) to improve representation discriminability for both point clouds and textual queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The core com- ponent of RET is the Relation-enhanced Self-Attention (RSA) mechanism, which encodes instance (hint) rela- tions for the two modalities in a unified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We propose to perform cross-modal instance matching and position refinement in two sequential steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' This for- mulation allows us to minimize the optimization diffi- culty of multi-task learning and the influence of noisy intermediate results, thereby improving cross-modal col- laboration for fine-grained location prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We perform extensive experiments on the KITTI360Pose dataset (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The results show that our approach can surpass previous method by a large margin, resulting in new state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Additional ablation studies further demonstrate the effectiveness of each component in the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Related Work Transformer and Attention Mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Transformer and self-attention mechanism (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Fan, Yang, and Kankanhalli 2021) has become increasingly popular in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Although first proposed for natural language processing, with architectural adaptation, Transformer has been widely applied to many vision tasks including visual recognition (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2021), object detection (Carion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2020) and seman- tic segmentation (Cheng, Schwing, and Kirillov 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Be- sides, the transformer-based architectures are also utilized to model cross-modal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=', vision and language) relations (Tan and Bansal 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In these architectures, the attention mechanism is widely employed to implicitly learn relations among the input tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Nevertheless, without explicit rela- tion encoding, the vanilla Transformer can only encode rela- tions implicitly with the help of positional encoding (Doso- vitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' To facilitate better relation modeling, some works modulate the attention computation process by explicitly incorporating element relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For example, (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2021) modified the attention mechanism via uni- fied relative position bias to improve visual recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For object detection, spatial relations between bounding boxes are introduced to modulate the attention weights (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For dynamic point cloud analy- sis, displacement between points (Fan, Yang, and Kankan- halli 2022) is utilized for point-specific attention computa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In this work, we propose to model relations for both point clouds and language queries by explicitly incorporat- ing intra-modality relations in a unified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Visual Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The task that is most related to ours is vision-based localization (Arandjelovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Brach- mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Hausler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2021), which is to estimate a pose based on an image or image sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Existing meth- ods mostly solve this task in two stages (Sarlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Sattler, Leibe, and Kobbelt 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The first stage finds a subset of all images using image retrieval-based techniques (Arandjelovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Hausler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Torii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2015), while the second stage establishes pixel- wise correspondence between the query image and the re- trieved one to predict the precise pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In this work, we also study the task of localization in a coarse-to-fine manner, but differ from visual localization in that: 1) we try to infer the location from city-wide point clouds instead of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2) we try to estimate the pose from textual query rather than images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Compared to visual localization, our task requires multi-modal understanding and is more challenging to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 3D Language Grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' As we humans live in a 3D world and communicate through natural language, recent work has begun to investigate the tasks on the cross-modal understanding of 3D vision and natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Among these tasks, the one that is most related to ours is 3D lan- guage grounding, which aims at localizing an object in point clouds from a given natural language query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For ex- ample, ScanRefer (Chen, Chang, and Nießner 2020) stud- ies 3D language grounding from real-life in-door scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' ReferIt3D (Achlioptas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2020) studies a related task un- der a simpler setting, which assumes the object instances are segmented in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' InstanceRefer (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2021) improves previous methods by adopting a 3D panoptic seg- mentation backbone, utilizing multi-level visual context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Re- east of a dark-green terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' south of a gray road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' south of a green terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' west of a dark-green traffic sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Split .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Cells Textual Query Hint Encoder north of a dark-green smallpole .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' east of a green pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Instance Encoder Hints Instances Relation-Enhanced Self-Attention Add & LayerNorm Feed Foward Network Add & LayerNorm Relation-Enhanced Self-Attention Add & LayerNorm Feed Foward Network Add & LayerNorm Instance-wise Relation x x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' (a) Hint-Instance Matching .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Feature Pooling (b) Offset Prediction Offsets Matching Coarse Stage Fine Stage Cross-modal Fusion Multi-Layer Perceptron Figure 2: Framework of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The city-scale point cloud is first divided into individual cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Then, in the coarse stage, the cells and the textual query are respectively encoded with the proposed Relation-Enhanced Transformer (RET), which are later used for query-cell matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In the fine stage, each hint is matched with an in-cell instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Then, cross-modal fusion dynamically aggregates hints and instance representations for offset prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The target location is predicted based on matching results and offset predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' cently, graph structure (Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2021) is also utilized to improve the representation learning qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Methodology Preliminaries Given a textual query, our goal is to identify the position it describes from a city-scale point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' To handle the large- scale point cloud, we divide each scene into a set of cubic cells of fixed size by a preset stride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Each cell C contains a set of p point cloud instances, which are encoded by Point- Net++ (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2017) into vector representations {pi}p i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Following (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022), the textual query T is repre- sented as a set of hints {hj}h j=1, each encoding the direction relation between the target location and an instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Inspired by the existing work (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022), given the cell splits, we solve this task in a coarse-to-fine manner with two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The coarse stage is formulated as textual query based cell retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The goal of this stage is to train a model that encodes C and T into a joint embedding space whereby matched query-cell pairs are close while those un- matched are pulled apart (Kiros, Salakhutdinov, and Zemel 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In the fine stage, given a retrieved cell, we aim to refine the position prediction by utilizing fine-grained cross- modal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In particular, we first match each hint in the query with an in-cell instance by formulating it as an optimal transport problem (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' After that, with the matching results, we predict the target location through a cross-modal fusion of point cloud instance and hint repre- sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Based on the fused representation, we predict the target location for each matched instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Finally, we obtain the target location prediction based on a weighted combi- nation of the matching and location prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The framework of our method is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In the fol- lowing of this section, we will explain the proposed method for coarse stage and fine stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' After that, our training and inference procedure will be detailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Coarse Stage: Relation-Enhanced Transformer After the cell split, the goal of the coarse stage is to suc- cessfully retrieve the cell C given a textual query T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' To ap- proach this objective, we need to encode C and T into a joint embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' An intuitive solution is to encode both C and T based on the instances they contained as is done in (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' However, with such representations, the low discriminability for cells and textual queries results in poor retrieval performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We argue that this can be at- tributed to the following two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' On the one hand, the outdoor scenes are often of low diversity, whereby a group of mentioned instances can appear at multiple different lo- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Thus, simply describing a cell with its contained in- stances can result in less discriminative representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' On the other hand, the textual queries often contain limited clues compared to the point clouds, making this cross-modality re- trieval especially challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' To this end, we propose to ex- plicitly encode instance-relations to provide more discrimi- native representations for both modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The Transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2017) has been widely utilized for relation-based representation learning in vari- ous tasks (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Fan, Yang, and Kankanhalli 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The key component of the Transformer is the Self-Attention (SA) operation: Attn(Q, K, V ) = Softmax(QKT / √ d)V , (1) Pooling Matmul Add Figure 3: Illustration of the proposed Relation-enhanced Self-Attention (RSA) mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Pairwise relations are ex- plicitly encoded into the value computation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' where d is the representation dimension and Q, K, V ∈ RN×d are the query, key and value matrices by transform- ing in-cell instances (or hints for textual queries) with corre- sponding linear transformations: Q = W QX, K = W KX, V = W V X, (2) with W ∗ ∈ Rd×d are learnable matrices and X = P ∈ Rp×d or H ∈ Rh×d represents stacked instances1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Despite its generality, the vanilla SA lacks explicit rela- tions in both modalities, thus is less informative to represent the cell and query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' To this end, we propose a novel Relation- Enhanced Transformer (RET) to model explicit instance re- lations in both point clouds and textual descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Our RET is a stack of multiple Transformer encoder layers, ex- cept that, in place of SA, we propose a Relation-enhanced Self-Attention (RSA) to explicitly incorporate relation in- formation into value computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The computation process is shown as follows and illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' RSA(Q, K, V , R) = Softmax(QKT / √ d)(V +Pool(R, 1)), (3) where R ∈ RN×N×d captures pairwise relations with Rij ∈ Rd representing the relation between the i-th and j- th instance (hint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Pool(R, 1) indicates pooling tensor R along dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In this way, our model can explicitly encode instance relations through this computation process, leading to more informative representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The definition of relation varies flexibly with task objec- tive and input modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For point cloud data, we take the geometric displacement of two instances as their relations, as direction is often mentioned in textual queries and thus informative for retrieval:2 RV ij = W V (ci − cj), (4) where ci ∈ R3 represents the center coordinate of the i-th instance and W v ∈ Rd×3 transforms the displacement into 1Note that the attention operation is often performed in different subspaces with multiple heads, which is omitted for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2We have also tried other features such as number of points and bounding boxes of instances but didn’t observe performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For the linguistic description, we compute the hint relation as the concatenation of their embeddings: RL ij = W L[hi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' hj], (5) where W L ∈ Rd×2d transforms the linguistic feature into representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' With the computation of RSA, the instance-wise relations for different modalities can be uni- formly incorporated into query or cell representations Finally, the cell (description) representations Cm (Tm) are obtained via a pooling operation over all instances (hints) output from the RET for cross-modal retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Fine Stage: Cascaded Matching and Refinement Following the coarse stage, we aim to refine the location pre- diction within the retrieved cell in the fine stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Inspired by (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022), we perform instance matching and location refinement to utilize the fine-grained visual and lin- guistic information, which involves the following two objec- tives: (1) For each hint, we find the in-cell instance it refers to via a matching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' (2) For each matched pair (i, j), a regressor predicts an offset ˆti ∈ R2 for each matched hint hj, which represents the offset from the instance center ci to the target location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='3 Previous method (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022) achieves the two objectives within a single step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' However, given the objec- tive of both hint-instance matching and offset prediction, the multi-task learning process introduces optimization dif- ficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Furthermore, in the early training steps, the matcher is only partially trained, which produces noisy matching re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The regressor learns and makes predictions based on this noisy results, leading to unstable learning process and sub-optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' To this end, we propose a Cascaded Matching and Refine- ment (CMR) strategy for the fine stage, where hint-instance matching and offset regression are sequentially performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Specifically, following (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022), we first train the SuperGlue (Sarlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2020) matcher for hint-instance matching, which is formulated as an optimal-transport prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Given the trained matcher, we obtain a set of hint- instance matching results {pi, hj, wi}h j=1, where wi repre- sents the confidence of the match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Then, to reduce the noise for regression, we predict the target location according to matched instances only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Precise location prediction requires proper understand- ing on both point cloud (what and where the referred in- stance is, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=', dark-green terrain) and language de- scription (what is the relation between the matched instance and the target location, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=', east of).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For this, we pro- pose to facilitate cross-modal collaboration via the Cross- Attention (CA) mechanism, which is commonly used for cross-modality information fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' CA(H, P ) = Attn(W QH, W KP , W V P ), (6) where H, P represent hints and instances, respectively, and W ∗ are learnable transformation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Shortcut connec- tion and layer normalization (Ba, Kiros, and Hinton 2016) 3For position prediction, we ignore the height information and considers 2D coordinates only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Table 1: Performance comparison on the KITTI360Pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Method Localization Recall (ϵ < 5/10/15m) ↑ Validation Set Test Set k = 1 k = 5 k = 10 k = 1 k = 5 k = 10 Text2Pos (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='14/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='25/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='36/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='55/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='48/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='43/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='61/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='65 RET (Ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='19/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='30/0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='65/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='71 follows the cross-attention operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' With these operations, the hint representation hi is accordingly updated to ˜hi by dynamically fusing visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' As such, the infor- mation in the two modalities are joint utilized with the help of cross-modal collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Then, we predict the offset (the direction vector from in- stance center to target location) from the updated hint: ˆti = MLP(˜hj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' (7) To utilize the matching results, the final prediction is ob- tained via a weighted combination of each hint’s prediction: ˆg = � i wi � m wm (ci + ˆti), (8) where wi ∈ [0, 1] is the confidence score of the match (pi, hj, wi) and is set to 0 for non-matched instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' To filter out noisy matches, we consider only matches with con- fidence score greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Training and Inference Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For the coarse stage, we train the proposed RET for cross-modal retrieval with pairwise ranking loss (Kiros, Salakhutdinov, and Zemel 2014): Lcoarse = Nb � m=1 Nb � n̸=m [α − ⟨Cm, Tm⟩ + ⟨Cm, Tn⟩]+ + Nb � m=1 Nb � n̸=m [α − ⟨Tm, Cm⟩ + ⟨Tm, Cn⟩]+, (9) where Nb is the batch size, α is a hyper-parameter to con- trol the separation strength and ⟨·, ·⟩ represents inner product between vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' This loss function encourages the represen- tation of matched description-cell pair to be by a margin α closer than those unmatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For the fine stage, we employ the loss in (Sarlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2020) to train the matcher, while L2 loss is applied to train the offset regressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We first encode all cells and queries into a joint embedding space with the proposed Relation-Enhanced Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Then, for each query representation, we re- trieve top-k cells with highest similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For each retrieved cell, we use the SuperGlue matcher trained in the fine stage to match each hint with an in-cell instance, which is fol- lowed by offset prediction based on the fused representa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Finally, the position prediction is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Experiments Dataset and Implementation Details Dataset Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We evaluate our method on the recently proposed KITTI360Pose dataset (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022), which is built upon the KITTI360 dataset (Liao, Xie, and Geiger 2021) with sampled locations and generated hints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' It con- tains point clouds of a total of 9 scenes, covering 14,934 positions with a total area of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='51km2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We follow (Kol- met et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022) to use five scenes for training, one for val- idation, and the remaining three for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We sample the cells of size 30m with a stride of 10m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For more details on the dataset preprocessing, please refer to our supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Implementation Details For the coarse stage, we trained the model with AdamW optimizer (Loshchilov and Hutter 2018) with a learning rate of 2e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The models are trained for a total of 18 epochs while the learning rate is decayed by 10 at the 9-th epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The α is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For the fine stage, we first train the matcher with a learning rate of 5e- 4 for a total of 16 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Afterwards, we fix the matcher and train the regressor based on the matching results for 10 epochs with a learning rate of 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The regressor is for- mulated as a 3 layer Multi-Layer Perceptron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Both of the two steps adopt an Adam (Kingma and Ba 2014) optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The RET has 2 encoder layers for both point cloud part and linguistic part, each utilizing the Relation-enhanced Atten- tion (RSA) mechanism with 4 heads and hidden dimension 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For the two stages, we encode each instance in the cell with PointNet++ (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2017) provided by Text2Pos (Kol- met et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022) for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The hint representa- tions are obtained by concatenating learned word embed- dings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' More details are provided in our appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='4 Comparison with the State-of-the-art We compared our method with Text2Pos (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022) on the KITTI360Pose dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Following (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022), we report top-k (k = 1/5/10) recall rate of dif- ferent error ranges ϵ < 5/10/15m for comprehensive com- parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Text2Pos gives a recall of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='14 when k = 1 and ϵ < 5m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In contrast, our method can significantly improve the recall rate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='19, which amounts to 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='7% relative improvement upon the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Furthermore, when we relax the localization error constraints or increase k, consistent improvements upon the baseline can also be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For example, with ϵ < 5m, our method achieves top-5 recall rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='44, which is 8% higher than previous state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Similar improvements can also be seen on the test set, showing our method is su- perior to the baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Ablation Studies In this section, we perform ablation studies for both stages to investigate the effectiveness of each proposed component 4Code available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='com/daoyuan98/text2pos-ret Table 2: Ablation study of the Relation-Enhanced Trans- former (RET) on KITTI360Pose validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' ”wo X rela- tion” indicates replacing the proposed RSA with the vanilla Self-Attention in corresponding modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Method k = 1 ↑ k = 3 ↑ k = 5 ↑ w/o both relations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='32 w/o linguistic relation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='37 w/o visual relation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='40 Full (Ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='44 Table 3: The effects of #layers of RET and #heads of RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' #Layers #Heads k = 1 ↑ k = 3 ↑ k = 5 ↑ 1 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='40 1 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='40 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='42 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='44 2 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='40 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='39 3 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='37 in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The ablation studies for coarse stage and fine stage are provided separately for clear investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Coarse Stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We study the importance of explicit relation incorporation in the coarse stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Since the coarse stage is formulated as a retrieval task, we use top-1/3/5 recall rate as evaluation metric, whereby the cell that contains the ground truth location is defined as positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Relation Incorporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We first study the necessity of ex- plicit relation modeling for both point cloud and textual queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' It can be observed that relation modeling contributes significantly to successful retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In particular, without any relation incorporation, the top-5 recall rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' With the explicit fusion of lin- guistic relation, we observe an increase of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='05 recall rate under same condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Besides, with the incorporation of vi- sual (point cloud instance) relations only, the top-5 recall rate can be improved by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='08, indicating explicit relations in the point clouds play a more important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Finally, with both relations, we achieve an improvement of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='12 at top-5 recall rate upon that without any relation, showing that both visual and linguistic relations are necessary and complemen- tary to improve the cell retrieval performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' RET Hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We also studied the importance of the hyper-parameters involved in RET, namely the number of layers of RET and the number of heads of RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The re- sults are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' It can be observed that, thanks to the strong relation modeling capacity of the proposed RET, we can obtain the best performance with 2 layers and 4 heads in the RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Decreasing and increasing the number of layers both lead to worse performance, which may be attributed to underfitting and overfitting, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Fine Stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The objective of the fine stage is to correctly match linguistic hints and point cloud instances and regress the target location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Thus, we study the performance of the matcher and regressor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Table 4: Comparison of training strategy and matcher per- formance on the KITTI360Pose dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Strategy Train Validation Precision ↑ Recall ↑ Precision ↑ Recall ↑ joint 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='12 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='16 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='67 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='59 cascade(ours) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='89 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='04 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='18 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='01 Table 5: Ablation study on the regression error of the fine- stage on the KITTI360Pose dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Method Train Error ↓ Validation Error ↓ w/o cascade training 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='24 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='72) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='01 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='86) w/o cross-attention 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='57 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='05) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='56 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='41) w/o confidence weighting 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='02 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='50) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='23 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='08) Ours 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='52 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='15 Matcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Following (Sarlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2020), we take precision and recall as the the evaluation metric of the matcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' With an identical matcher architecture, we investigate the impact of training strategy on the matcher performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The results are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' It can be seen that compared with joint training (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022), our cascaded training achieves not only high precision and recall in the training set, but also stronger generalization on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The re- sults demonstrate that the cascade training strategy is able to mitigate the multi-task optimization difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Regressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The regressor predicts the target location based on the the matching results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We study the effects of cas- caded training, cross-attention based cross-modal fusion and confidence weighting for final location prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We use regression error as evaluation metric and compare different versions on both KITTI360Pose training and validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The results are shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Without cascaded training strategy, the regressor achieves an error of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='24 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='01 on the training and validation set, respectively, which is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='72 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='86 higher than that with cascaded training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' This re- sult suggests that our cascaded training strategy also allevi- ates the optimization difficulty of the regressor, which was caused by the noisy intermediate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Furthermore, with- out cross-attention mechanism, the regression error also in- creases by a considerable margin, showing that cross-modal collaboration is important for precise location prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Fi- nally, with confidence-based weighting, we can further re- duce the regression error on both the training and validation set, suggesting this information from the trained matcher can be further utilized to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Visualizations Embedding Space Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We visualize the learned embedding space via T-SNE (Van der Maaten and Hin- ton 2008) in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' It can be observed that the base- line method Text2Pos (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022) results in a less discriminative space, where positive cells are relatively far away from the query and sometimes separated across the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In contrast, our method draw positive cell and query representations closer in the embedding space, re- sulting in a more informative embedding space for retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Ground Truth Top-1 Top-2 Top-3 Ground Truth Top-1 Top-2 Top-3 (a) (b) (c) (e) (d) (f) 557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='85 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='00 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='00 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='00 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='99 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='0 819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='08 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='00 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='03 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='14 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='90 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='36 455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='41 1150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='00 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='40 Building Pole Traffic Light Traffic Sign Parking Sidewalk Vegetation Terrain Road Wall Garage Figure 4: Qualitative retrieval results on KITTI360Pose validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The red dot in the ground truth cell indicates the target location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In each retrieved cell, the number in the lower right indicates the center distance between this cell and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Green box indicates positive cell which contains the target location, while red box indicates negative cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Text2Pos Ours Textual Query Negative Cell Positive Cell Figure 5: T-SNE visualization of embedding space for the coarse stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' A cell is considered as positive if it contains the location described by the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Compared with baseline method (Kolmet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 2022), our method can produce better representation where positive cells are closer to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Qualitative Cell Retrieval Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We show some exam- ple text to point cloud retrieval results in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For a given query, we visualize the top-3 retrieved cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' A re- trieved cell is defined as positive if it contains the target lo- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' It can be observed that, our method can retrieve the ground truth cell or those close in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Sometimes, negative cells can also be retrieved, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=', top-1 in (a) and top-3 in (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' It can be seen that these retrieved negative cells exhibit high semantic similarity with the ground truth cell, even though far away from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' We also show a failure case (f), where the retrieved cells are all negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' It can be seen that even though far away from the target location, all these neg- ative cells have instances similar to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' These observations suggest that outdoor scenes are indeed of low diversity, indicating that successful retrieval requires highly discriminative representations to disambiguate the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Conclusion In this work, we proposed a novel method for precise text-based localization from large-scale point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Our method employs a coarse-to-fine principle and pipelines this process into two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For the coarse stage which is formu- lated as a textual query based cell retrieval task, we aim to improve representation discriminability for both point cloud and query representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' This is achieved through explicit modeling of instance relations and implemented via a newly proposed Relation-Enhanced Transformer (RET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' The core of RET is a novel Relation-enhanced Self-Attention (RSA) mechanism, whereby the instance relations for the two modalities are explicitly incorporated into the value com- putation process in a unified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' For the fine stage, our method performs description-instance matching and position refinement in a cascaded way, whereby cross- modal information collaboration is enhanced through the cross-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Extensive experiments on the KITTI360Pose dataset validated the effectiveness of the pro- posed method, which achieves new state-of-the-art perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Additional ablation studies further corroborate the effectiveness of each component in the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Acknowledgement This research is supported by the National Research Foun- dation, Singapore under its Strategic Capability Research Centres Funding Initiative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Any opinions, findings and con- clusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' tReferences Achlioptas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' Abdelreheem, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} 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End-to- End Object Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} +page_content=' In ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQf_g7i/content/2301.05372v1.pdf'} diff --git a/U9E3T4oBgHgl3EQf0QsH/content/tmp_files/2301.04735v1.pdf.txt b/U9E3T4oBgHgl3EQf0QsH/content/tmp_files/2301.04735v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fad7dba6c75c05855f488aab59acfab98ff1b492 --- /dev/null +++ b/U9E3T4oBgHgl3EQf0QsH/content/tmp_files/2301.04735v1.pdf.txt @@ -0,0 +1,3532 @@ +Revisiting Pure State Transformations with Zero Communication +Ian George and Eric Chitambar +Electrical and Computer Engineering Department, University of Illinois at Urbana-Champaign +(Dated: January 13, 2023) +It is known that general convertibility of bipartite entangled states is not possible to arbitrary error +without some classical communication. While some trade-offs between communication cost and con- +version error have been proven, these bounds can be very loose. In particular, there are many cases +in which tolerable error might be achievable using zero-communication protocols. In this work we +address these cases by deriving the optimal fidelity of pure state conversions under local unitaries as +well as local operations and shared randomness (LOSR). We also use these results to explore catalytic +conversions between pure states using zero communication. +I. +INTRODUCTION +The theory of quantum mechanics through the +lens of information and vice versa [1–3] has af- +forded the physicist and the information scientist +alike with a new way to view the objects and long- +term goals of their study. +No better example of +this can be found than quantum resource theo- +ries. Quantum resource theories specify the rele- +vant physical property in such a manner as to better +tease apart the complexities of quantum mechanics +while also establishing what tasks may be achieved +with said resource [4]. Perhaps the earliest example +of such a resource theory is the resource theory of +entanglement. Entanglement may be viewed as a +form of correlation that does not exist in the classi- +cal world [5]. Roughly speaking, the resource the- +ory of entanglement asks (1) what tasks may be per- +formed better using entangled states and (2) how +entangled states may be converted from one to an- +other under some class of free operations. +The most standard view of the resource theory +of entanglement considers the set of free operations +to be local operations and classical communication +(LOCC) which captures the ‘distant lab’ paradigm +where two (or more) parties share an entangled +state in spatially separated labs and they can only +perform operations on their respective portions and +exchange classical information (See Fig. +1). +Not +only is this the most standard set of free operations, +but in some respect it seems minimal. Indeed, Hay- +den and Winter showed that to convert one (pure) +entangled state to another to sufficiently small pre- +cision requires a certain amount of communication +between labs, regardless of how many auxiliary +EPR pairs they share [6] (see also [7]). This is dis- +tinct not only from the classical setting [8], but also +from quantum states that are not entangled [9, 10]. +However, the results of Hayden and Winter, while +fundamental, do not give us a complete picture of +the tradeoff between communication and achiev- +able tolerated error in pure state conversions. In- +(a) +(b) +FIG. 1: Conversion of pure states in distant labs. (a) The +LOCC model where communication is exchanged. (b) +The embezzling of quantum states where an auxiliary +entangled state is used. This may be seen as a special +case of catalytic conversion. +deed, it is easy to find examples of state conversions +which, according to the best known lower bounds, +still may be possible to perform with a tolerated er- +ror of 1% using no communication (see Example 1 +of Section III). This show that a relatively large gap +in our understanding of zero-communication en- +tanglement transformations still persists, and one +we aim to address in this work. +Moreover, the tools we develop to address this +problem will also allow us to study pure state trans- +formations using shared auxiliary entanglement. +The operational paradigm in which parties are al- +lowed to use arbitrary pre-shared entanglement but +no communication is known as local operations and +shared entanglement (LOSE) [11]. +By itself, the +problem of pure state convertibility |ψ⟩AB → |φ⟩AB +under LOSE is trivial since Alice and Bob could al- +ways just demand |φ⟩ as their pre-shared entangle- +ment and then throw away |ψ⟩ when it is given. +However, if one demands that the pre-shared en- +tanglement is also returned in addition to the target +state |φ⟩, then the problem becomes quite interest- +ing, i.e. |ψ⟩AB ⊗ |ω⟩A′B′ → |φ⟩AB ⊗ |ω⟩A′B′ for aux- +iliary pre-shared entanglement |ω⟩A′B′. Transfor- +mations of this form are known as catalytic trans- +formations with |ω⟩A′B being the catalyst. Remark- +ably, van Dam and Hayden have shown that there +exists a family of entangled catalysts, known as +arXiv:2301.04735v1 [quant-ph] 11 Jan 2023 + +A +BA +A +A' +A' +B' +B' +B +B2 +FIG. 2: Comparison of [12] (dark pink),[17] (dark +green), and this work’s results (blue). [17] finds lower +bounds on the classical communication necessary to +convert one state to another, but in the zero +communication setting these are too loose. We find +methods for solving this exactly (Section IV), which +establishes that communication is necessary for larger +tolerated errors. [12] establishes a method for pure state +transformations with zero communication with massive +amounts of entanglement, but it scales inversely with +the error, which we find can be too strong for a relevant +error range, even if ultimately it is optimal (Section VI). +universal embezzling states [12], such that for any +tolerated non-zero error one can always prepare a +pure state using a member of this family and zero +communication. More amazingly, they showed that +as the error tends to zero, it is roughly optimal since +it scales nearly the same as if you add LOCC and +allow the catalyst to be state dependent. This near +optimality along with Hayden and Winter’s result +has, understandably, largely ceased the study of en- +tanglement transformations with zero communica- +tion, because when one needs entanglement trans- +formations without communication, one uses em- +bezzlement [13, 14].1 It is however not clear what +is the necessary error for embezzlement to become +near optimal, which could be relevant in practical +settings. Indeed, for any tolerated error, it is easy to +find sufficient conditions on pure states to be con- +verted with no catalyst at all (Example 2 of Section +III). This is an indication that we also do not under- +stand embezzling and catalytic convertibility suffi- +ciently well. +1 The notable exceptions to this halted topic of research has been +the consideration of special embezzling families [15] and the +correlated sampling lemma [16], which may be viewed as a +variation of embezzling. +A. +Summary of Results +The primary aim of this work is to provide tighter +lower bounds on the error in pure state entangle- +ment convertibility with zero communication. +A +high level comparison of our results to the afore- +mentioned work on this topic are presented in Fig. +2. This depicts a ‘one-shot resource tradeoff’ region +that must contain the ‘true’ one-shot resource trade- +off surface for a given pure state conversion. Hay- +den and Winter’s result provides a lower bound +on the achievability independent of the amount of +shared maximally entangled states, but their result +can be too loose when considering zero communi- +cation. van Dam and Hayden’s result provides an +outer bound on the achievability surface on the face +pertaining to LOSE, but their result in fact can be +too loose when the error is not sufficiently small. +In this work, our results allow one to exactly solve +the minimal error in the zero communication set- +ting and also provide significantly tighter bounds +than quantum embezzling for a relevant region on +the LOSE face (See Fig. 2). +To formally establish our results, we reduce the +class of questions regarding optimal pure state con- +version to optimization problems that only concern +probability distributions. This is because of a bijec- +tion between the equivalence classes of pure states +under local unitaries— which are defined solely +by their Schmidt coefficients— and the probability +simplex. We do this by showing the optimal fidelity +of pure state transformations with local unitaries +is efficiently computable. +Of course, in general +one would not expect local unitaries to be the op- +timal strategy and we build on this result to present +a non-convex optimization program over an opti- +mization variable with bounded dimension. An im- +mediate corollary of this result is the impossibility +of pure state conversions with zero communication +for negligible error. We also present efficient com- +putable upper bounds on the achievable error using +a semidefinite programming (SDP) relaxation. We +also show that in the case where either the seed (i.e. +initial) or target state is a two-qubit state, the local +unitary strategy is optimal. However, we can show +for larger dimensions this is not the case. +Having established general properties in the sin- +gle copy case, we move to the multiple copy case, +i.e. where the seed and/or target state is of inde- +pendent and identically distributed (i.i.d.) +form. +This is standard in determining the rate of con- +verting one state to another. In particular, we con- +sider dilution and distillation where the seed state +or target state respectively is many copies of a max- +imally entangled state and show these are convex +optimization programs and may be seen as involv- + +Entanglement +LOSE +1 +Tolerated +0 +Error ε +Gap +LOCC +Classical +Communication3 +ing the Ky-Fan norms when extended to the regime +where they are not a norm. Lastly, in a sense ex- +tending our earlier two-qubit results, we establish +that if the target state is an n−fold copy of a two- +qubit entangled state and the seed state’s Schmidt +rank is less than the target state, then local unitaries +are the optimal strategy. +Finally, given these results, we turn our atten- +tion to quantum embezzlement. We begin by not- +ing that the correspondence between Schmidt co- +efficients and probability distributions means that +quantum embezzlement implies a classical equiv- +alent we call randomness embezzlement. We then +proceed to use our new tools to consider the prob- +lem of catalyzed pure state conversion under local +unitaries, in effect a generalization of embezzling, +and compare it to embezzling. +We show in par- +ticular that at least in general the optimality of the +embezzling states is only for very small errors. In- +deed, we show for reasonable tolerable errors, the +embezzling state may have a Schmidt rank of many +orders of magnitude larger than a simple catalyst. +This may have practical relevance and strongly re- +fines our understanding of pure state transforma- +tions under LOSE. +Organization of the Paper +The rest of the paper +is organized as follows. In Sections II and III we +present the necessary notation and background re- +spectively to understand the rest of the paper. In +Section IV, we +• Make explicit the correspondence between +pure states under LU and the probability sim- +plex and note this implies the existence of a +classical variation of embezzlement (Theorem +2) +• Prove our equation for fidelity of state con- +version under local unitaries (Theorem 5) and +our optimization for fidelity of state conver- +sion under local operations and shared ran- +domness (Theorem 6) +• Establish computable upper bounds on the fi- +delity of state conversion under LOSR (Theo- +rem 9). +In Section V we present the results where the tar- +get or seed state is of i.i.d. form. In Section VI we +discuss catalysts under local unitaries, the general +frameworks that includes quantum embezzlement,. +In Section VII we discuss why our theory does not +generalize beyond bipartite pure states. +II. +NOTATION +Our notation largely aligns with standard texts +[18, 19]. +In this paper we consider finite dimen- +sional quantum systems. Given n ∈ N, we define +[n] := {1, ..., n}. A finite dimensional Hilbert space +will be labeled with a capital roman letter, e.g. A, B, +etc. As they are finite dimensional, these Hilbert +spaces may be identified by the isomorphism A ∼= +Cd where d ∈ N. The space of linear maps from a +Hilbert space A into itself, i.e. the space of endo- +morphisms, is denoted L(A). The space of quan- +tum states, or density matrices, with respect to a +Hilbert space A, is the space of positive semidefi- +nite operators with unit trace, i.e. D(A) := {ρ ∈ +L(A) : ρ ⪰ 0 & Tr(ρ) = 1} where ⪰ is the L¨owner +order. If a quantum state is a joint state over multi- +ple Hilbert spaces, we will use a subscript to specify +this, e.g. ρAB ∈ D(A ⊗ B). We say a quantum state +ρA ∈ D(A) is pure if Tr +� +ρ2 +A +� = 1 which is equiva- +lent to there being a unit vector |ψ⟩ ∈ A such that +ρA = |ψ⟩⟨ψ|, where we are using bra-ket notation. +For this previous reason, we generally just specify +a pure state by |ψ⟩A, or ψ if we are considering its +density matrix representation. We denote the space +of pure states S(A), where S stands for unit sphere. +A state is classical if it is diagonal in a specific +choice of basis for L(A). We call this the computa- +tional basis. The space of classical probability dis- +tributions over d elements, the probability simplex +which we denote P(d), may be viewed as the set +of non-negative d−dimensional vectors that sum to +one or the set of diagonal density matrices in the +computational basis. +To distinguish between the +two, we write P for the matrix version and p for +the vector version. We also define the set of entry- +wise decreasing probability distributions over d el- +ements, i.e. elements of the form p↓(1) ≥ p↓(2) ≥ +... ≥ p↓(d), by P↓(d). +A quantum channel E +∈ +C(A, B) is a (lin- +ear) completely positive, trace preserving map E : +L(A) → L(B). Any quantum channel admits an +isometric representation, e.g. given E ∈ C(A, B), +there exists a Hilbert space E such that |E| ≤ |A||B| +and isometry V : A → B ⊗ E such that Φ(X) = +TrE(VXV†) where TrE is the partial trace on the E +space and X† is the Hermitian conjugate. +Given the space of linear operators from A ∼= Cd +to B ∼= Cd′, L(A, B), the vec mapping vec : L(A ⊗ +B) → A ⊗ B is defined by vec(|i⟩ ⟨j|) = |j⟩ ⊗ |i⟩ +where {|i⟩}i∈[d] and {|j⟩}j∈[d′] are the computa- +tional bases for A and B respectively. This choice +of vec mapping satisfies the identity +(XT +1 ⊗ X0) vec(Y) = vec(X0YX1) , +(1) +where X0 ∈ L(A0, B0), X1 ∈ L(A1, B1), and Y ∈ +L(B1, B0). The vec mapping is also an isometry in +the sense that for all X, Y ∈ L(A, B), +⟨X, Y⟩ = ⟨vec(X), vec(Y)⟩ , + +4 +where ⟨·, ·⟩ on the L.H.S. is the inner product on +L(A, B) defined by ⟨X, Y⟩ += +Tr +� +X†Y +� +and the +R.H.S. is the inner product on vectors A ⊗ B defined +by ⟨ψ|φ⟩ = ∑i ψ(i)φ(i) where · is the conjugate. +III. +BACKGROUND & MOTIVATION +Throughout this section we fix A ∼= Cd, B ∼= Cd′ +for clarity. +a. +Fidelity +The fidelity is a standard measure of +similarity between two positive semidefinite oper- +ators R, S ≥ 0. +F(R, S) = +��� +√ +R +√ +S +��� +2 +1 = Tr +��√ +SR +√ +S +�2 +, +(2) +where the square root of a positive semidefinite +operator is defined in the standard fashion on its +spectral decomposition and ∥ · ∥1 is the Schatten +1−norm. It satisfies various properties that will be +relevant for this work which we summarize here. +All of these may be verified by direct calculation or +by referring to standard texts. +Proposition 1 (Summary of Fidelity Properties). +Let ρ, σ ∈ D(A). The following hold: +1. 0 ≤ F(ρ, σ) ≤ 1 where the upper bound is +saturated if and only if ρ = σ and the lower +bound saturates if and only if their images are +orthogonal. +2. The fidelity is isometrically invariant, i.e. +given isometry V : A → B, +F(VρV†, VσV†) = F(ρ, σ) . +3. The fidelity satisfies data-processing. That is, +for any quantum channel E ∈ C(A, B), +F(ρ, σ) ≤ F(E(ρ), E(σ)) . +4. If both states are pure, +F(|φ⟩⟨φ| , |ψ⟩⟨ψ|) = | ⟨ψ|φ⟩ |2 , +and if one state is pure +F(|φ⟩⟨φ| , σ) = ⟨φ| σ |φ⟩ . +5. If both states are classical, P, Q ∈ P(d), then +the fidelity reduces to the square of the Bhat- +tacharyya coefficient: +F(P, Q) = +� +� ∑ +i∈[d] +� +p(i)q(i) +� +� +2 += BC(p, q)2 , +where p(i) = P(i, i) and likewise for Q. +6. Given pure states with the same eigenbasis +and real amplitudes, |ψ⟩ = ∑x +� +p(x) |x⟩, +|φ⟩ = ∑x +� +q(x) |x⟩ , the fidelity reduces to +the square of the Bhattacharyya coefficient of +the probability distributions defined by the +amplitudes: +F(|φ⟩⟨φ| , |ψ⟩⟨ψ|) = BC(p, q)2 . +We also note that in all of these definitions there +is a pesky squaring that effectively we don’t care +about. For this reason we could define the square +root fidelity: +√ +F(R, S) := +� +F(R, S) . +Note the square root fidelity could be viewed as +the quantum extension of the Bhattacharyya coef- +ficient. +b. +Norms +In defining the fidelity we used the +Schatten 1−norm. +More generally, there are the +Schatten p−norms which for X ∈ L(A, B) may be +defined as ∥X∥p := ∥σ(X)∥p where σ(X) is the +ordered vector of singular values of X, σ1(X) ≥ +σ2(X) ≥ ... ≥ σrank(X)(X) and it is being evalu- +ated under the Lp−norm where p ≥ 1. +The in- +finity norm, ∞−norm, is limp→∞ ∥X∥p = ∥X∥∞ = +maxi σi(X). The infinity norm was generalized to +the Ky Fan k−norms ∥X∥(k) := ∑ σi(X) for 1 ≤ k ≤ +min{d, d′}. The Ky Fan norms have relevance in +measuring entanglement [20]. A generalization of +the Ky Fan and Schatten norms together is given by +the (k, p)−norms [21] +∥X∥(k,p) := +� +� ∑ +i∈[k] +σi(X)p +� +� +1/p +, +(3) +which also have use in measuring entanglement +of pure states [22]. +Much like is common to +do for the Schatten p−norms, we can extend the +(k, p)−norms to p > 0 with the caveat they won’t +be norms as they won’t in general satisfy subaddi- +tivity (the triangle inequality) for p ∈ [0, 1). +c. +Entanglement Theory +A bipartite quantum +state ρAB is separable if there exists n ∈ N, p ∈ +P(n), {σi +A}i∈[n] ⊂ D(A), and {τi +B}i∈[n] such that +ρAB = ∑ +i∈[n] +p(i)σi +A ⊗ τi +B . +Otherwise the state is entangled. As a pure state +|ψ⟩⟨ψ|AB is defined by a unit vector, this reduces to +a pure state is separable, referred to product in this +setting, if and only if there exists |φ⟩A , |ϕ⟩B such + +5 +that |ψ⟩ = |φ⟩A ⊗ |ϕ⟩B. While this is sufficient for +determining if a bipartite pure state is entangled, +there is also a notion of ‘how’ entangled a state is +in terms of Schmidt rank. Every bipartite pure state +|ψ⟩AB admits a unique (up to re-ordering) decom- +position of the form +|ψ⟩AB = ∑ +i∈[k] +� +p(i) |ui⟩A ⊗ |vi⟩B , +(4) +where +k += +max{d, d′}, +p +∈ +P(k) +and +{|ui⟩}i∈[k], {|vi⟩}i∈[k] are orthonormal bases of A +and B respectively. The +� +p(i) > 0 terms are re- +ferred to as the Schmidt coefficients. The Schmidt +rank of |ψ⟩AB, SR(|ψ⟩) = supp(p), i.e. the num- +ber of Schmidt coefficients. +This may be viewed +as a measure of entanglement in the sense that the +Schmidt rank of a product state is 1 and the maxi- +mally entangled state |Φ+⟩CdCd = +1 +√ +d ∑i |i⟩Cd |i⟩Cd +has Schmidt rank d. We define the set SR(d) := +{|ψ⟩ : SR(|ψ⟩) ≤ d}, where we note this set is in- +dependent of the dimension the state is embedded +in. +Lastly we note a particularly nice property of +pure states, known as Uhlmann’s theorem. +Lemma 1 (Uhlmann’s Theorem). Given ρ, σ +∈ +D(A) and |ψ⟩ ∈ A ⊗ B such that TrB(ψ) = ρ, then +F(ρ, σ) = max{| ⟨ψ|φ⟩ |2 : |φ⟩ ∈ A ⊗ B , TrB(φ) = σ} . +d. +No-Go Theorems, Embezzling, & Motivation +With the established background, we now present +the previous results related to zero communication +pure state transformations which we will discuss +our results in relation to. The first is a lower bound +on the number of qubits or classical bits necessary +to convert between pure states [6]. +Proposition 2. ([6, Theorem 8]) Consider a state +transformation via channel E ∈ C(A ⊗ B, A ⊗ B) +from seed state |φ⟩AB to target state |ψ⟩AB such +that F(E(φ), ψ) ≥ 1 − ε. Then, independent of any +amount of entanglement assistance, for δ = +8√ε, in +the implementation of E, q qubits were exchanged +where +q ≥1 +2 [∆δ(TrB(|φ⟩⟨φ|)) − ∆0(TrB(|ψ⟩⟨ψ|))] ++ log(1 − δ) , +(5) +where +exp(∆ε(P)) = min rank( �P) · λmax( �P) +s.t. Tr +� +�P +� +≥ 1 − ε +�P = ΠPΠ +[P, Π] = 0 +Π2 = Π . +Moreover, the bound given in (5) holds for a neces- +sary amount of classical communication by multi- +plying the R.H.S. by two. +While the above proposition is very powerful +and implies two states with different Schmidt de- +compositions cannot be perfectly converted with +zero communication, it is not sufficient in every sce- +nario. In particular, the following example shows +that in certain cases Proposition 2 cannot eliminate +any state from being able to be converted to a given +target state with relatively high fidelities. +Example 1 (On the necessity of communication). +Up to local unitaries, let the target state be |ψ⟩ = +0.54 |00⟩ + 0.02 |11⟩ + 0.44 |22⟩, the seed state be any +state |φ⟩ = ∑i∈[k] +� +p(i) |vi⟩ |ui⟩, and assume we +are interested in a state transformation E such that +F(E(φ), ψ) = 0.99. Then ε = 0.01, so δ > 0.56. +One may verify ∆δ(P) = log(|1| · 0.44) < −1.18, +by removing the 0.02 and 0.54 eigenvalues of P. It +may be shown [6] that ∆0(TrB(|ψ⟩⟨ψ|)) ≥ 0, and +log(1 − δ) < 0. It follows that in this setting the +R.H.S. of (5) is negative. +Therefore, we have no +proof from this bound that any transformation for +any seed state which achieves this relatively high +fidelity of 99% requires any communication. +While the above example shows there are reason- +ably small tolerated errors ε where Proposition 2 is +not helpful, when the tolerated error is sufficiently +small, it will imply the need for communication. +This sort of structure for sufficiently small ε also +appears when considering quantum embezzlement +[12], which may be seen as a solution to Proposition +2 implying communication is necessary. Quantum +embezzlement in effect shows one can make pure +state transformations with zero communication to +any non-zero error if they have the right sufficiently +large entangled catalyst. +Proposition 3. ([12]) Consider the family of catalyst +states |µ(n)⟩A′B′ = +1 +√Hn ∑n +j=1 +1√ +j |j⟩A′ |j⟩B′ where +Hn := ∑n +i=1 n−1 is the Harmonic number. For any +ε > 0 and target bipartite pure state |ψ⟩AB with +Schmidt rank m, for n > m1/ε there exist unitaries +UAA′, WBB′ such that +F(UAA′ ⊗ WBB′(|µ(n)⟩A′B′ |0⟩A |0⟩B), +|µ(n)⟩A′B′ ⊗ |ψ⟩AB) ≥ 1 − ε . +Moreover, U, W are in effect permutations on the +joint Schmidt bases. +One can see quantum embezzlement implies a +way to convert one pure state to another to non- +zero error by picking a large enough catalyst and + +6 +then first ‘embezzling out’ the original state (un- +computing |φ⟩ to |0⟩ |0⟩ via embezzling) and then +‘embezzling in’ the target state |ψ⟩. +What is perhaps most remarkable about the +above approach is that it was shown in the original +work that even if we allow LOCC and a state de- +pendent catalyst, the error scales as Ω(1/ log(n)) +whereas for the above result the errors scales as +O(1/ log(n)), so as the error is driven down, em- +bezzling is near optimal. That is, as ε → 0, this strat- +egy is effectively optimal. However, just as with the +discussion pertaining to Proposition 2, it’s clear em- +bezzling isn’t necessary for reasonable error levels +in general. In fact, we show in the following ex- +ample that for any non-zero error there exist states +which can be converted without any catalyst. +Example 2 (On the necessity of embezzling). As +noted, as ε → 0, embezzling is necessary. However, +it is not in general clear at what point embezzling +becomes necessary. +This can be seen as follows. +Consider ε ∈ (0, 1) and two probability distribu- +tions p, q ∈ P(m) such that the BC(p, q)2 ≥ 1 − ε. +Define the seed state as |φ⟩ = ∑i∈[m] +� +p(i) |i⟩A |i⟩B +and the target state as |ψ⟩ = ∑i∈[m] +� +q(i) |i⟩A |i⟩B. +Then we have +F(|φ⟩⟨φ| , |ψ⟩⟨ψ|) = BC(p, q)2 ≥ 1 − ε , +where we have used Item 5 of Proposition 1. There- +fore, given |φ⟩, it requires no communication or en- +tanglement to generate |ψ⟩ to error ε. In fact, as we +show later (Proposition 4), this will be true for con- +verting the set of states with Schmidt coefficients +defined via p to the set of states with Schmidt coef- +ficients defined via q in general. +Given these two examples, we see that while +these results give strong characterizations of pure +state transformations with zero communication, +neither the need for communication by Proposition +2 nor the optimality of Proposition 3 when the error +tends to zero give us a full understanding of this +setting. It would therefore be of value to better un- +derstand this task, and this is what the rest of this +work addresses. +IV. +SINGLE COPY PURE STATE CONVERSION +WITH ZERO COMMUNICATION +Our primary goal of this section is to deter- +mine the minimal error of conversion between pure +states with zero communication, which would re- +solve the gap presented in Example 1. To do this, +we will use the correspondence between the prob- +ability simplex and Schmidt coefficients under lo- +cal unitaries (LU), which we establish in the follow- +ing subsection. We also note that this implies the +existence of a classical equivalent of embezzling, +which we call randomness embezzling (Theorem +2). This correspondence motivates the idea that the +optimal fidelity of pure state conversion under local +unitaries is simply re-ordering the Schmidt coeffi- +cients, which we in fact prove (Theorem 5). We then +use the local unitary result to establish a bounded +but non-linear optimization program that deter- +mines the optimal achievable fidelity under conver- +sion via local operations and shared randomness +(LOSR), which does not require shared randomness +(Theorem 6). We end the section by discussing the +relationship between the LU and LOSR strategies +and introducing an SDP relaxation for efficiently es- +tablishing upper bounds on the achievable fidelity +of pure state conversions under LOSR. +A. +Correspondence Under Local Unitaries between +Schmidt Coefficients and the Probability Simplex +In this subsection we establish the bijection be- +tween Schmidt coefficients, which define the equiv- +alence classes of bipartite pure states under local +unitaries, and the probability simplex. One reason +for this is because the rest of the results of this work +might be best seen as verifying that in the zero com- +munication setting this correspondence is all that +matters. Indeed, we will see this in the subsequent +subsections which show that the minimal fidelity +error of pure state transformations under zero com- +munication will always be functions of only the +Schmidt coefficients. +Proposition 4. Up to local unitaries, any pure quan- +tum state is of the form +|ψ⟩AB = ∑ +i∈[k] +� +p↓(i) |i⟩A ⊗ |i⟩B , +where p↓(i) ≥ p↓(i + 1) for all i ∈ [k − 1], k = +max{d, d′}, p↓ ∈ P↓(k), and {|i⟩} is the computa- +tional basis in both cases. In other words, there exist +both equivalence classes on pure states under local +unitary operations in terms of Schmidt coefficients +and ordered Schmidt coefficients. +Proof. Consider |ψ⟩AB = ∑j∈[k] +� +p′(j) +��uj +� ⊗ +��vj +� +as +decomposed in (4). Now fix the permutation π on +[k] such that p′(π−1(i)) ≥ p′(π−1(i + 1)) for all +i ∈ [k − 1], i.e. π re-labels p′ so that it is decreasing. +Define the unitaries UA = ∑j∈[k] |π(j)⟩ +� +uj +��, WB = +∑j∈[k] |π(j)⟩ +� +vj +��, which may be verified to be uni- +taries by direct calculation. Then (UA ⊗ WB) |ψ⟩AB + +7 +will be of the form given in the proposition state- +ment. Finally, we could make this argument for any +pure state without ordering the Schmidt coefficients +to get one set of equivalence classes. As such, under +local unitaries, we can define equivalence classes +of pure states in terms of ordered or non-ordered +Schmidt coefficients. This completes the proof. +Definition 1. The space of (representatives of the +equivalence class of) ordered Schmidt coefficient +pure states with Schmidt rank bounded by d is +given by SR↓(d). +That is, if |ψ⟩ ∈ SR↓(d), then +|ψ⟩ = ∑i∈[d] +� +p↓(i) |ui⟩ |i⟩ |i⟩ where p↓ ∈ P↓(d). +We can use the previous proposition to relate the +(ordered) probability simplex over d elements to +to the equivalence classes of (ordered) Schmidt de- +compositions with Schmidt rank bounded by d. +Proposition 5. Consider the functions vec(√·) : +L(Cd) → Cd ⊗ Cd and vec−1(·⊙2) : Cd ⊗ Cd → +L(Cd) where ·⊙2 is the entry-wise square of a vec- +tor. These functions define a bijection between P(d) +(resp. P↓(d)) and the space of equivalence classes +of Schmidt decompositions under local unitaries +with Schmidt rank bounded by d (resp. the space +SR↓(d).) +Proof. We prove it via direct calculation for P(d) +and the space of Schmidt decompositions. +The +proof in the other case works the same. Let C ∼= Cd. +First, consider p ∈ P(d) which we write in its den- +sity matrix form, e.g. P = ∑i∈[d] p(i) |i⟩⟨i|. Then +vec( +√ +P) = vec +� +� ∑ +i∈[d] +� +p(i) |i⟩⟨i| +� +� += ∑ +i∈[d] +� +p(i) |i⟩C ⊗ |i⟩C′ , +which is in the specified equivalence class by ap- +plying an isometries that take the computational +bases from C, C′ to A, B. In the other direction, take +the Schmidt decomposition in the purified basis, +|ψ⟩AB = ∑i∈[d] +� +q(i) |i⟩A ⊗ |i⟩B. We can convert +the A space to C via the channel +FA→C(·) := V† · V + (1 − V†V) · (1 − V†V) , +where V = ∑i∈[d] |i⟩A ⟨i|C is the isometry that takes +the C space to the A space as |A| ≥ |C| by assump- +tion. The same type of conversion holds for the B +and C′. Therefore, we have (up to equivalences) +|ψ⟩AB = ∑i∈[d] +� +q(i) |i⟩C |i⟩C′. Then, +vec−1(|ψ⟩·2) = vec−1( ∑ +i∈[d] +q(i) |i⟩C |i⟩C′) += ∑ +i∈[d] +q(i) |i⟩⟨i|C , +where in the last line we used that C′ ∼= C so that +L(C, C′) ∼= L(C). This completes the proof. +The reason this is useful is it draws equivalence +between the equivalence classes of entangled states +in terms of Schmidt coefficients and probability dis- +tributions under fidelity. +Proposition +6. +Consider +|φ⟩ += +∑i∈[d] +� +p(i) |i⟩A |i⟩B, |ψ⟩ = ∑i∈[d] +� +q(i) |i⟩A |i⟩B. +Then F(|φ⟩⟨φ| , |ψ⟩⟨ψ|) = BC(p, q)2. +Proof. First note V : |i⟩A → |i⟩A |i⟩B is an isom- +etry. +Thus by isometric equivalence of fidelity +(Item 2 of Proposition 1), we have F(|φ⟩ , |ψ⟩) = +F(V |φ′⟩ V†, V |ψ′⟩ V†) where the primed versions +just remove the B register. Then using Item 6 of +Proposition 1 completes the proof. +Randomness Embezzling +Before moving forward, +we note that independent of the focus of this work, +this equivalence between Schmidt coefficients and +the probability simplex means that the proof of +quantum embezzlement also proves the existence +of a classical version. +Specifically, if one looked +at the proof of quantum embezzlement [12], one +would only need to note the starting and ending +state they bound the fidelity between are in the +computational basis locally and use +F(|ψ⟩ , |φ⟩) = |⟨ψ, φ⟩|2 = +���⟨ +√ +P, +� +Q⟩ +��� +2 += +� +∑ +i +� +p(i)q(i) +�2 +=BC(p, q)2 , +which follows the same argument as the previous +few propositions, to ultimately conclude the same +proof bounds a classical equivalent (Theorem 2). As +we did not present the proof for embezzlement of +quantum states, we present the proof of embezzle- +ment of probability distributions in full for clarity +in Appendix A. +Theorem 2. For any ε > 0 and target probabil- +ity distribution P ∈ P(m), the catalyst distribution +Rn := +1 +Hn ∑n +j=1 +1 +j |j⟩⟨j| is such that for n > m1/ε there +exists a unitary representation of a basis relabeling +Uf of the joint distribution such that +F(Uf (Rn ⊗ |0⟩⟨0|)U† +f , Rn ⊗ P) ≥ 1 − ε . + +8 +(a) +(b) +FIG. 3: Comparison between embezzlement of classical +distributions and quantum states. (a) The embezzlement +of classical distributions happens within one lab and a +local permutation of the joint computational basis. (b) +The embezzling of quantum states happens across two +labs where each party applies the permutation of the +joint computational basis on their local halves. +We note the major difference between random- +ness and quantum embezzlement is the role of lo- +cality. In the classical case there is a single party +and the distribution is not bipartite, both of which +remove the notion of locality. These differences are +non-trivial: one cannot construct a non-local classi- +cal equivalent of embezzling that at the same time +demands that the catalyst remains decoupled as +in Proposition 3, and one cannot find a quantum +equivalent of the non-local classical variation that +one can implement as follows from Proposition 2. +As it is not central to the rest of this work, we pro- +vide an extended discussion of this nuance for the +interested reader in Appendix A after the proof of +Theorem 2. +B. +Pure State Conversion under Local Unitaries +Having established the relationship between the +equivalence classes of pure states in terms of +Schmidt coefficients and the probability simplex, +we now show the optimal strategy for converting +one pure state to another under local unitaries is +simply re-labeling the Schmidt basis so the order- +ing of the Schmidt coeffficients is the same. This is +not necessarily surprising. It is not clear what more +one could do, and indeed this is the strategy that +is used to implement quantum embezzlement [12]. +For intuition, we quickly show the equivalent result +in the classical setting first. +Proposition 7. Let p↓, q↓ ∈ P↓(d) Then for any i ∈ +[d] and d − k ≥ k ≥ 1, +1 ≥ +� +p↓(i) +� +q↓(i) ≥ +� +p↓(i) +� +q↓(i + k) . +Proof. This just follows from the fact if 1 ≥ p↓(i) ≥ +p↓(i + 1) and the same for q↓. +Corollary 1. Given p, q ∈ P(d), +max +π∈Sd +BC(p, πq) = BC(p↓, q↓) , +where Sd is the set of permutations on d elements. +Proof. All we are looking for is the permutation of +the elements of q such that BC(p, πq) is maximized. +We can apply the permutation σ such that σPσ† = +P↓, the matrix representation of p↓, to both sides. By +the isometric equivalence of fidelity and that per- +mutations are group, the problem is the same. That +is, we can consider +max +π∈Sd +BC(p↓, πq) = max +π∈Sd ∑ +i∈[d] +� +p↓(i) +� +q(π(i)) . +It immediately follows from Proposition 7 that the +optimal π is the one that takes Q to Q↓. This com- +pletes the proof. +The idea is then to lift this result to quantum +states optimized over unitaries and then use this +with Uhlmann’s theorem to lift to the bipartite set- +ting with local unitaries. +The main challenge is +minimizing over unitaries in the lift of the pre- +vious result as now we have to deal with non- +commutivity. This is done by reducing optimizing +over unitaries to optimizing over permutations us- +ing the Birkhoff-von Neumann Theorem. +Lemma 3 (Birkhoff-von Neumann Theorem). Let +d ∈ N. Given a linear operator X ∈ L(Rd), X is bis- +tochastic (non-negative entries such that each col- +umn and each row sums to one) if and only if there +exists a probability distribution p ∈ P(|Sd|) such +that +X = ∑ +π∈Sd +p(π)Vπ , +where Vπ(i, j) := δi,π(j) are permutation matrices +and δi,j is the Kronecker delta. That is, a linear oper- +ator is bistochastic if and only if it is a convex com- +bination of permutation matrices. +Lemma 4. Let ρ, σ ∈ D(Cd). Then +max +U +F(ρ, UσU†) = F(P↓, Q↓) , +where P↓ = ∑i νi(ρ) |i⟩⟨i| and likewise for Q↓ but +with respect to σ. In other words, the fidelity be- +tween ρ and σ maximized over unitaries is equal to +the fidelity of their ordered eigenvalues. +Proof. First, by the isometric invariance of fidelity +(Item 2 of Proposition 1), F(ρ, σ) = F(P↓, VσV†) + +P +10)(0| +Ur +RnA +A +UAA' +A' +A' +B' +B' +M +BB' +B +B +89 +where V is the unitary such that VρV† = P↓ = +∑i νi(ρ) |i⟩⟨i|. As unitaries are closed under multi- +plication and conjugate transpose, +max +U +F(ρ, UσU†) = max +U′ F(P↓, U′VσV†U′†) +as the optimal U′ = U⋆V† where U⋆ is the opti- +mizer for the L.H.S. of the equality. Therefore we +just define Q ≡ VσV† and focus on solving the +R.H.S. for clarity. Therefore, we are interested in +maxU F(P↓, UQU†). +Denote the spectral decomposition of Q += +∑j q(j) +��φj +�� +φj +��. +Note that without loss of gener- +ality, we may write U = ∑j +��ψj +� � +φj +�� for some +orthonormal basis { +��ψj +�}. +Therefore, UQU† = +∑j q(j) +��ψj +�� +ψj +��. +Furthermore, define P(X) +:= +∑i |i⟩⟨i| X |i⟩⟨i|, which is the pinching, or dephasing, +channel onto the computational basis. Then +P(UQU†) =∑ +i,j +|i⟩⟨i| q(j) +��ψj +�� +ψj +�� |i⟩⟨i| +=∑ +i,j +q(j)| +� +i +��ψj +� |2 |i⟩⟨i| . +Note that in contrast, P↓ is invariant under this +pinching. Combining these points, +max +U +F(P↓, UQU†) +≤ max +U +F(P↓, P(UQU†)) += max +U +Tr +��√ +P↓P(UQU†) +√ +P↓ +�1/2�2 += max +{|ψj⟩} +Tr +�� +∑ +j,i,i′,�i +q(j) +� +p↓(i′)p↓(�i) +��� +� +i +��ψj +� ��� +2 +· +��i′� � +i′��i +� � +i +����i +� � +�i +��� +�1/2 +�2 += max +{|ψj⟩} +Tr +� +� +� +∑ +j,i +q(j)p↓(i)| +� +i +��ψj +� |2 |i⟩⟨i| +�1/2� +� +2 +, +where the inequality is the data-processing inequal- +ity (Item 3 of Proposition 1) with the pinching chan- +nel along with the invariance of P↓ under this chan- +nel, the first equality is using the definition of fi- +delity (2), the second is just expanding everything, +and the third is collapsing the implicit Kronecker +deltas. +Now note the following trick. +We can define +the square matrix A via its elements: A(j, i) := +| +� +i +��ψj +� |2. +We know 0 +≤ +| +� +i +��ψj +� |2 +≤ +1, +∑i | +� +i +��ψj +� |2 = 1, and ∑j | ⟨i|j⟩ |2 = 1 as {|i⟩} and +{ +��ψj +�} are orthonormal bases. It follows that A is +a bistochastic matrix by definition. Therefore, by +the Birkhoff-von Neumann Theorem (Lemma 3), +A = ∑π∈Sd r(π)Wπ where Wπ is the permutation +matrix for π and r is a probability distribution over +the permutations. Thus, plugging this back in to +what we started with, +max +{|ψj⟩} +Tr +� +� +� +∑ +j,i +q(j)p↓(i)| +� +i +��ψj +� |2 |i⟩⟨i| +�1/2� +� +2 += max +r +Tr +� +� +� +∑ +j,i,π +q(j)p↓(i)r(π)Wπ(j, i) |i⟩⟨i| +�1/2� +� +2 +. +Now note every permutation matrix is the iden- +tity matrix with columns permuted, i.e. +π += +� +eT +π(0) eT +π(1) . . . eT +π(d−1) +�T +, where ei := |i⟩. It fol- +lows that +∑ +π∈Sd +r(π)Wπ(j, i) +=∑ +π +r(π)1{Wπ(j, i) = 1} +=∑ +π +r(π)1{π(j) = i} =: Pr +r [π(j) = i′] , +where 1{A} is the indicator function for an event +and the second equality is because W(j, i) = 1 if +and only if π(j) = i. We stress the final definition +is a function of the choice of r and j, i and form a +joint probability over (j, i) as ∑j Prr[π(j) = i] = 1 = +∑i Prr[π(j) = i] and every element is non-negative. +This simplifies the problem to +max +r +Tr +� +� +� +∑ +j,i,π +q(j)p↓(i)r(π)Wπ(j, i) |i⟩⟨i| +�1/2� +� +2 += max +r +� +∑ +i +� +p↓(i) +� +∑ +j +� +q(j) Pr +r [Π(j) = i] +��2 +, +where we have just grouped terms and used that +the operator is diagonal, so we can apply the square +root entry-wise and take the sum to compute the +trace. +So we want to determine the maximal distri- +bution r, but we can show this is achieved by +element-wise optimizing the sum. +Note +� +p↓(1) +is the largest element and bounded above by 1, +so we want to multiply it by the largest value +∑j +� +q(j) Prr[π(j) = i] can take. +� +q(j) ≤ 1 for +all j and ∑j Prr[π(j) = i] = 1 so the largest value + +10 +this sum can take is maxj +� +q(j). Note if we pick +a different distribution each term will be smaller +than it could be by Proposition 7. This means we +choose r such that all non-zero probability permu- +tations map argmaxj q(j) to 1. We then have the +same problem as initially but with +� +p↓(2) serving +the largest element and q not containing its largest +element. Doing the argument recursively, we con- +clude the optimal distribution r has unit probability +on permutation σ such that ∑i q(σ(i)) = q↓. Thus, +max +r +� +∑ +i +� +p↓(i) +� +∑ +j +� +q(j) Pr +r [Π(j) = i] +��2 += +� +∑ +i +� +p↓(i) +� +q↓(i) +�2 +=BC(p↓, q↓)2 +=F(P↓, Q↓), +where the first equality is by the preceding explana- +tion and the last two are using Item 5 of Proposition +1. Note this means we have established an upper +bound as we used the data processing inequality at +the beginning. However, this is clearly achievable +by picking by the permutation unitary that maps σ +to Q↓. Thus this completes the proof. +We now can use the above lemma to establish the +pure state property we are actually interested in. +For notational simplicity, we define the following +notation: +FLU(ρ, σ) := max +U,V F(ρ, (U ⊗ V)(σ)) , +which is without loss of generality unitaries as we +can just trivially embed the smaller dimensional +state. +Theorem 5. +FLU(|ψ⟩ , |φ⟩) = F(P↓, Q↓) , +where P↓ is the distribution defined by the decreas- +ing Schmidt coefficients of |ψ⟩ and likewise for Q↓ +and |φ⟩. In other words, the optimal fidelity of con- +verting |φ⟩ to |ψ⟩ via local unitaries is given by the +fidelity of their ordered Schmidt coefficients. +Proof. Up to local unitaries, |ψ⟩ = ∑i +� +p↓ |i⟩ |i⟩. +Therefore without loss of generality, that can be +taken as our target state by allowing free local uni- +taries on the seed state. We can take the seed state +to be of the form |φ⟩ = ∑i +� +q(i) |i⟩ |i⟩ by the same +argument. Then by assumption, we are interested +in maxU,V F(|ψ⟩ , (U ⊗ V) |φ⟩) with the specified +forms. Note +TrB((U ⊗ V) |φ⟩⟨φ| (U ⊗ V)†) +=∑ +i,i′ +� +q(i)q(i′)U |i⟩ +� +i′�� U† Tr +� +V |i⟩ +� +i′�� V†� +=∑ +i +q(i)U |i⟩⟨i| U† =: UQU†. +Now for any unitary U we define the following pu- +rification +���w|U� +:= vec( +� +UQU†) +=(U ⊗ U) vec( +� +Q) = (U ⊗ U) |φ⟩ , +where we have used +� +UQU† = U√QU† and the +vec map identity (1). Now we have +F(P↓, UQU†) = max +|w′⟩ F(|ψ⟩ , +��w′�) += max +V +F +� +|ψ⟩ , (1 ⊗ V) +���w|U�� += max +V +F(ψ, (U ⊗ VU) |φ⟩) , +(6) +where the first equality is by Uhlmann’s theorem +(Lemma 1), the second is because all purifications +of a given operator are unitarily equivalent on the +purifying space, so there exists a V such that (1 ⊗ +V) +���w|U� += |w′⟩. The final line is just expanding +the definition of +���w|U� +. +It follows, +max +W,V F(|ψ⟩ , (W ⊗ V) |φ⟩) += max +U,V′ F(|ψ⟩ , (U ⊗ V′U) |φ⟩) += max +U,V′ F(|ψ⟩ , (1 ⊗ V) +���w|U� +) += max +U +F(P↓, UQU†) +=F(P↓, Q↓) , +where the first equality is because unitaries are +closed under multiplication and the optimizations +are independent, the second and third are both +by (6) for clarity, the third is because unitaries are +closed under conjugation and then the final equal- +ity is by applying Lemma 4. +This completes the +proof. +This means under local unitaries, it is efficient to +compute the optimal fidelity and that in fact the op- +timal strategy is simply Alice and Bob re-ordering +the basis so that the Schmidt coefficients are in the + +11 +same relative ordering. It also follows from Item +1 of Proposition 1 that unless all the Schmidt coef- +ficients are equal, the fidelity cannot be one under +local unitary strategies. +C. +Pure State Conversions under Local Operations +and Shared Randomness +While the previous section is nice in that it finds +an efficient way of calculating the optimal conver- +sion strategy under local unitaries, it would be nat- +ural to ask if local operations can do better than lo- +cal unitaries as it is a much more general class of +operations. In fact, we can see that it must do bet- +ter in some cases in a trivial manner. Consider the +target state |ψ⟩ and the seed state |φ⟩ = |ψ⟩ ⊗ |ζ⟩ +where |ζ⟩ is not product. Under local unitaries this +transformation isn’t possible to arbitrary precision +because of |ζ⟩, but of course in reality the parties +could trace out whichever portion(s) of |ζ⟩ they +hold. Thus, we need a theory of transformations +under local operations. +Note that this trivial example we have given +would not be resolved by local mixed unitary +strategies. Indeed, we begin by noting that local +mixed unitary strategies cannot ever outperform lo- +cal unitary strategies. +Corollary 2. Let |ψ⟩ be the target state and |φ⟩ be +the seed state and only optimize over Alice and Bob +using mixed unitary channels. Then the optimal is +the same as in Theorem 5. +Proof. Letting EU, FW be local mixed unitary maps, +max +EU,FW +F(ψ, (EU ⊗ FW)(φ)) += ⟨ψ| (EU ⊗ FW)(φ) |ψ⟩ += +� +U,W ⟨ψ| (U ⊗ W)(φ) |ψ⟩ dU dW +≤ +� +U,W max +U,W ⟨ψ| (U ⊗ W)(φ) |ψ⟩ += max +U,W ⟨ψ| (U ⊗ W)(φ) |ψ⟩ +=F(P↓, Q↓) , +where the first equality is by Item 4 of Proposition 1, +the second is letting the mixed unitary map be for +any probability measures dU,dW over the unitary +group. The inequality is because the inner product +is real and so it is lower bounded by the maximum. +The second to last equality is by linearity, and the +final equality is by Theorem 5. Noting that a specific +choice of local unitaries is a special case of mixed +unitary channels completes the proof. +The above tells us that we must escape the use +of unitaries to improve our bounds. Note however +that in general the only maps that preserve pure +states are isometries, and our results so far have +been in terms of pure states, so we need to main- +tain this structure to build on them. For this reason, +the following proof will make use of the isometric +representation of quantum channels. +For notational simplicity, we define the optimal +fidelity of conversion under local operations and +shared randomness (LOSR) fidelity +FLOSR(ρ, σ) := max +µ,Eλ,Fλ +F(ρ, +� +(Eλ ⊗ Fλ)(σ)dµ(λ)) , +where µ is a probability measure over an index set +for sets of local channels {Eλ} and {Fλ}. Similarly, +we can define optimal fidelity of conversion under +local operations (LO) as +FLO(ρ, σ) := max +E,F F(ρ, E ⊗ F)(σ)) . +With these defined, we prove the following. +Theorem 6. +FLOSR(|ψ⟩ , |φ⟩) = FLO(|ψ⟩ , |φ⟩) += +max +P′∈P(Σ) F((P ⊗ P′)↓, Q↓ +embed) , +where |Σ| ≤ SR(|φ⟩) · SR(|ψ⟩), P is the probability +distribution defined by |ψ⟩’s Schmidt coefficients +and likewise for Qembed with the Schmidt coeffi- +cients of |φ⟩ except the distribution is embedded +into the joint space. +Proof. The first equivalence follows similarly to the +mixed unitary case. +Clearly the class of LOSR +strategies is more general than the class of LO +strategies, so we just need to show LOSR is only +as strong as LO here. +FLOSR(φ, ψ) =F +� +ψ, +� +(Eλ ⊗ Fλ)(φ)dµ(λ) +� += +� +⟨ψ| (Eλ ⊗ Fλ)(φ) |ψ⟩ dµ(λ) +≤ +� +max +E,F [⟨ψ| (E ⊗ F)(φ) |ψ⟩] dµ(λ) += max +E,F ⟨ψ| E ⊗ F)(φ) |ψ⟩ +=FLO(φ, ψ) , +where the first equality is by definition and denot- +ing the optimizers by µ, {Eλ}, {Fλ}, the second is +by linearity of the Lebesgue integral, the inequal- +ity is because ⟨ψ| (E ⊗ F)(φ) |ψ⟩ is a real number +for any choice of local channels, the third equality + +12 +is because µ is a probability measure that is now +independent of the argumenbt of the integral, and +the final equality is by definition. This proves the +reduction of LOSR to LO if the target state is pure. +Next, we bound the dimension of Σ. We want +to consider maxE,F F(ψ, (E ⊗ F)(φ)). Without loss +of generality, we assume the local spaces are ‘com- +pressed’ such that din := SR(|φ⟩) so that E, F both +act on L(Cdin). We now show that without loss of +generality we may restrict the output dimension of +E, F to be dout := SR(|ψ⟩). +This is just because +we can project onto the support of the marginal +of |ψ⟩ on both local spaces, so we can restrict the +local maps to this space. +Formally, this can be +seen as follows. +Consider arbitrary E, F and let +|ψ⟩ = ∑i +� +p(i) |i⟩ |i⟩. Define ΠP := ∑i:p(i)>0, i.e. +the projector onto the support of TrB(ψ) = TrA(ψ), +where the equality is up to the change in space. +Note rank(ΠP) = Schmidt(ψ). +By construction, +(ΠP ⊗ ΠP) |ψ⟩ = |ψ⟩. Therefore, +F(ψ, (E ⊗ F)(φ)) += ⟨ψ| (E ⊗ F)(φ) |ψ⟩ += Tr[|ψ⟩⟨ψ| (E ⊗ F)(φ)] += Tr +� +ψΠ⊗2 +P (E ⊗ F)(φ)Π⊗2 +P +� +, +where in the first equality we have used Item 4 +of Proposition 1 and the other two use cyclicity of +trace along with invariance of ψ under the projec- +tor. Now we can expand, +Π⊗2 +P (E ⊗ F)(φ)Π⊗2 +P +=∑ +k,l +ΠPAk ⊗ ΠPBkφA† +kΠP ⊗ B† +l ΠP +≡(EΠ ⊗ FΠ)(ψ) , +where {Ak}, {Bl} are the Kraus operators of E, F +respectively and EΠ, FΠ are CPTNI maps defined +by {ΠPAk}, {ΠPBl} respectively. Note this equiva- +lence holds as (ΠAk)† = A† +kΠP since Π† +P = ΠP so +it is CP and it is TNI because +∑ +k +(ΠPAk)†(ΠPAk) =∑ +k +A† +kΠPAk +≤∑ +k +A† +k1Ak = 1 , +where we used Π2 +P = ΠP in the first equality, +ΠP ≤ 1 and that E is CP in the inequality, and +that E is TP in the last inequality. An identical ar- +gument holds for FP. This proves the optimizer +is achieved with CPTNI maps T(L(Cdin), L(Cdout)). +Finally, we can lift EP, FP to being CPTP, denoted +�E, �F ∈ T(L(Cdin), L(Cdout)) by adding one Kraus +operator, e.g. for EP add the Kraus operator Z ∈ +L(Cdin, Cdout) where Z†Z = (1 − ∑k A† +kΠAk) ≥ 0 +which always exists by definition of the space of +positive semidefinite operators. By linearity, +F(ψ, (E ⊗ F)φ) = Tr[ψ(EΠ ⊗ FΠ)(φ)] +≤ Tr +� +ψ( �E ⊗ �F)(φ) +� +. +Therefore without loss of generality the optimal +channels are E, F ∈ C(Cdin, Cdout). Note this means +that Rank(JE) ≤ dindout and likewise for JF. +We now derive the equation using the isometric +representation of the channel. +max +E,F F(ψ, (E ⊗ F)(φ)) +=⟨ψ, (E ⊗ F)(φ)⟩ += max +V1,V2,|ζ⟩ |⟨ψ| ⟨ζ| (V1 ⊗ V2) |φ⟩|2 += +max +U1,U2,|ζ⟩ +���⟨ψ| ⟨ζ| (U1 ⊗ U2) |φ⟩ |0⟩E1 |0⟩E2 +��� +2 += +max +U′ +1,U′ +2, +���ζp′ +� +���⟨ψ| +� +ζp′ +��� (U′ +1 ⊗ U′ +2) |φ⟩ |0⟩E1 |0⟩E2 +��� += max +P′ +F((P ⊗ P′)↓, Q↓ +embed) , +where the second line is because there exists an iso- +metric representation of each channel which means +(V1 ⊗ V2)(φ) is a pure state, so we can apply Uhlm- +man’s theorem to find a purification of |ψ⟩ that sat- +urates the bound, but as |ψ⟩ is already pure, any +purification will be a product state. The third line +is because we can always convert an isometry into +a unitary on the appropriately large space. +The +fourth line means that ζp′ = ∑i′ +� +p′(i) |i⟩ |i⟩, which +can always be achieved by local unitaries on the +E1 and E2 spaces, which result on new unitaries +on the other side but the same maximum. The fi- +nal equality is just using Theorem 5 and we write +Qembed to stress it is defined over the whole alpha- +bet. Lastly, as we established bounds on the ranks +of the local maps Choi matrices, we have bounds +E1, E2 ≤ dindout, which justifies the maximum and +tells us how large of a system we have to consider +in the statement of the theorem. +It is useful to see how this result works. It in ef- +fect shows the following equivalence of conversion +when measured under fidelity +|φ⟩ −→ +LO |ψ⟩ = max +|ζ⟩ +� +|φ⟩ −→ +LU |ψ⟩ ⊗ |ζ⟩ +� +, +which can be viewed both by proof and via intu- +ition as a special case of the isometric representa- +tion of a channel. Moreover, it is easy to see in this + +13 +form how it handles our motivating example. In- +deed, if the target state is |ψ⟩ and the seed state is +|ψ⟩ ⊗ |ζ⟩, then clearly the maximizer is chosen by +the ancillary state being |ζ⟩ and the local unitaries +being trivial. +D. +Relation between LO and LU Strategies +The natural question given the previous theo- +rems is if we can better understand the relationship +between LO and LU strategies. We first show that +LU and LO strategies are equivalent when either +the target or the seed state is a two qubit state. +1. +LU and LO Equivalence for Two-Qubit Seed or Target +State +Proposition 8. Consider entangled two qubit seed +state |φ⟩ ∈ C2 ⊗ C2. +Let the target entangled +state be |ψ⟩ ∈ Cd ⊗ Cd′. +Then the optimal non- +communicative strategy is the local unitary strat- +egy. +Proof. Without loss of generality, q↓ = (q, 1 − q) +where q ≥ 1/2 and p↓ = (p(1), p(2), ...). +Then +the optimal local unitary strategy is +� +qp(1) + +� +(1 − q)p(2). +For any P′ we can write (p′)↓ = +(p′(1), p′(2), ...). The optimal CPTP strategy (up to +a square) is of the form +� +qp(1)p′(1) + +� +(1 − q) max{p(1)p′(2), p(2)p′(1)} . +These values can only increase by assuming p′ has +two outcomes, so let us assume so without loss +of generality and parameterize the distribution by +p′ ∈ [1/2, 1] to obtain +� +qp(1)p′ + +� +(1 − q) max{p(1)(1 − p′), p(2)p′} . +Moreover note p(2)p′ < p(2) unless p′ = 1, which +is equivalent to the LU strategy, so the second entry +in the maximization would be lower than the LU +setting. Therefore, we focus on the remaining case. +We are specifically interested in when the following +strict inequality holds: +� +qp(1)p′ + +� +(1 − q)p(1)(1 − p′) +> +� +qp(1) + +� +(1 − q)p(2) +⇔ g(p′) := +� +qp(1)( +� +p′ − 1) ++ +� +1 − q( +� +p(1)(1 − p′) +− +� +p(2)) > 0 . +Then +d +dp′ g(p′) = +√ +qp(1) +2√ +p′ + +√ +p(1)(1−q) +2√ +1−p′ +. It follows, +� +qp(1) +� +1 − p′ +2 +� +p′� +1 − p′ ++ +� +p′� +p(1)(1 − q) +2 +� +1 − p′� +p′ +≥ 0 +⇔ +� +qp(1) +� +1 − p′ + +� +p′ +� +p(1)(1 − q) ≥ 0 +⇔√q +� +1 − p′ + +� +p′ +� +(1 − q) ≥ 0 +⇔ +√ +F(Q↓, P′↓) ≥ 0 , +where the first line is multiplying to get identical +denominators, the second line is multiplying by the +denominator, the third is dividing out p(1), and +the final is by the definition of square root fidelity. +Note the final inequality will always hold strictly +unless q ∈ {0, 1}, i.e. the state is a product state, +by Item 1 of Proposition 1. +If q ∈ {0, 1}, then +the state is a product state which would contradict +that we assume the state is entangled. Therefore, +in our setting, g(p′) only increases over its inter- +val, p′ ∈ [0, 1]. Thus, the optimal choice of p′ is +p′ = 1, but in this case the value is +� +qp(1) ≤ +� +qp(1) + +� +(1 − q)p(2), i.e. the optimal choice is +lower bounding the optimal local unitary strategy. +It follows this is never optimal. This completes the +proof. +Proposition 9. Consider entangled two qubit target +state |ψ⟩ ∈ C2 ⊗ C2 and any seed state |φ⟩ ∈ Cd ⊗ +Cd′. The optimal non-communicative strategy is the +local unitary strategy. +Proof. The proof is basically the same as for the two +qubit seed case. Without loss of generality, p↓ = +(p, 1 − p). We can re-order |φ⟩ such that it is q↓ = +(q(1), q(2), ...). The optimal CPTP strategy (up to a +square) is of the form +� +q(1)p′(1)p + +� +q(2) max{p′(1)(1 − p), p′(2)p} . +This sum can only increase if p′(1) + p′(2) = 1, +so we can parameterize the distribution by p′ ∈ +[1/2, 1] to obtain +� +q(1)p′p + +� +q(2) max{p′(1 − p), (1 − p′)p} . +Note that p′(1 − p) < (1 − p) unless p′ = 1. If +p′ = 1, this is the LU strategy, if p′ < 1, then this is +worse than an LU strategy. Therefore, we only care +about the other maximization case. That is, we are + +14 +interested in when p ∈ [1/2, 1) and the following +strict inequality holds: +� +q(1)p′p + +� +q(2)p(1 − p′) +> +� +q(1)p + +� +q(2)(1 − p) . +However, +� +q(1)p′p +< +� +q(1)p +and +� +q(2)p(1 − p′) < +� +q(2)(1 − p) as p′ ∈ [1/2, 1). +Therefore this strict inequality can never hold. +Therefore the optimal strategy is always the LU +strategy. This completes the proof. +2. +LU and LO Inequivalence for States with Schmidt Rank +Greater than Two +If there is equivalence for two qubit seed or tar- +get states, it is natural to ask if this property per- +sists. One might expect that this is a special prop- +erty of qubit systems as are found throughout quan- +tum information science results. Indeed, generally +this property does not hold, which we will prove +via example. +Theorem 7. For seed and target state with Schmidt +rank ≥ 3, the optimal LO strategy may be better +than the optimal LU strategy. +Proof. We construct an example for Schmidt rank 3. +By continuity of the fidelity, one can embed the tar- +get and seed in bigger spaces with arbitrarily small +perturbations for it to hold in higher dimensions, +which is why this is sufficient. Consider target state +|ψ⟩ = 0.85 |00⟩ + 0.08 |11⟩ + 0.07 |22⟩ and seed state +|φ⟩ = 0.45(|00⟩ + |11⟩) + 0.1 |22⟩. Then, the optimal +LU strategy fidelity is +F(P↓, Q↓) += +�√ +0.45( +√ +0.85 + +√ +0.08) + +� +0.1(0.07) +�2 +<0.796 . +In contrast, if we consider P′ = [0.55, 0.28, 0.17], +then +F((P ⊗ P′)↓, Q↓) += +�√ +0.45 +√ +0.4675 + +√ +0.45 +√ +0.238 + +√ +0.1 +√ +0.1445 +�2 +>0.82 . +As we maximize over P′, the optimal LO strategy +achieves a value that is strictly above the LU strat- +egy. This completes the proof. +E. +Inefficiency of Optimal LOSR Fidelity and +Computable Upper Bounds +In the above we have constructed an example +where the local operations strategy outperforms the +local unitary strategy (though we have not shown +what the strategy itself is). +A natural question +would then be how easy it is to solve for the op- +timal fidelity value or even a bound. By Theorem +5, we can conclude the optimal local unitary strat- +egy is polynomial time to solve as all one needs to +do is sort the Schmidt coefficients and calculate the +fidelity. Indeed, one could solve for the ordering of +the Schmidt coefficients using the linear program +for sorting a vector. +In contrast, for optimizing LO strategies, we have +no such luck. In effect this is because there are two +things to optimize over at once. Indeed, recall +FLO(|ψ⟩ , |φ⟩) = +max +P′∈P(Σ) F((P ⊗ P′)↓, Q↓) . +Then the problem is that one must first tensor P +onto variable P′ and then re-order the vector. One +cannot even in general order an optimization vari- +able, which we will refer to as ‘sorting,’ as sorting is +in general non-convex. In sorting a vector using a +linear program, one relaxes to bistochastic channels +and considers a linear function so that the optimizer +is an extreme point which by the Birkhoff von Neu- +mann theorem is a specific permutation. However, +we are many levels of involvement above that: we +want the distribution P′ such that its product dis- +tribution P ⊗ P′ when sorted optimizes the fidelity +with Q↓. Therefore, we need to optimize over P′ +and the permutation at the same time. It’s not clear +that we can actually relax to bistochastic strategies +because of the joint concavity of fidelity. That is to +say, for any bistochastic channel E, +F(E(P ⊗ P′), Q↓) =F(∑ +π +r(π)Vπ(P ⊗ P′), Q↓) +≥∑ +π +F(r(π)Vπ(P ⊗ P′), r(π)Q↓) +=∑ +π +r(π)F(Vπ(P ⊗ P′), Q↓) , +where the first line is Birkhoff-von Neumann the- +orem, the second is joint concavity using Q↓ = +∑π r(π)Q↓ as r is a probability distribution, and +the last line is because F(λP, Q) = λF(P, Q) = +F(P, λQ). +Thus any bistochastic channel may +strictly do better than the average of its extreme +points. Moreover, even if we could optimize over +bistochastic channels, we would have a non-convex +objective function as the bistochastic channel, an +optimization variable, would be applied to P ⊗ P′ +which is also partially an optimization variable. + +15 +Given the above, it seems likely the best option if +one were to try and find a (near) optimum would be +to use gradient descent from random initial P′, real- +izing it will only work locally and will break down +at ‘kinks’ where the ordering changes. Otherwise +more sophisticated non-convex optimization tech- +niques might be used. +Computable Upper Bound Methods +Perhaps even +worse than our inability to calculate the exact fi- +delity, is that it is not clear in general how to de- +termine good bounds. Certainly we have the fol- +lowing result. +Theorem 8. Unless the target state is |ψ⟩ = |φ⟩ ⊗ +|ζ⟩ where |φ⟩ is the seed state, there exists ε > 0 +such that there does not exist local operations that +will take |φ⟩ to |ψ⟩. +Proof. This follows from Theorem 6 along with Item +1 of Proposition 1. +The above theorem, while derived from a very +different strategy than Proposition 2, does not seem +to give us much more information as to at what +point communication is necessary. What we would +want to efficiently improve this would be to estab- +lish upper bounds on the equation given in Theo- +rem 6 that have a closed form that does not depend +on P′. One option is to use the data processing in- +equality for fidelity. This can be seen in the follow- +ing proposition. +Proposition 10. Consider target state |ψ⟩ and seed +state |φ⟩ with corresponding Schmidt distributions +p, q respectively. If pmax ≤ qmax, then +FLO(|ψ⟩ , |φ⟩) ≤ F(p, q) , +where p = pmax |0⟩⟨0| + (1 − pmax) |1⟩⟨1| and like- +wise for q. +Proof. Without loss of generality let d be the maxi- +mum local dimension. Let E(·) = |0⟩⟨0| · |0⟩⟨0| + +∑i∈{1,...,d−1} |1⟩ ⟨i| · |i⟩ ⟨1|. That is, E coarse-grains +a probability distribution to the Bernoulli distribu- +tion with its first element untouched and the sum +of all the others as the other outcome. Then using +data processing of fidelity (Item 3 of Proposition 1), +max +P′∈P(Σ) F((P ⊗ P′)↓, Q↓) +≤ max +P′∈P(Σ) F(E((P ⊗ P′)↓), E(Q↓)) += max +p′∈[0,1] F +� +�P(p′), E(Q↓) +� +, +where �P(p′) := pmaxp′ |0⟩⟨0| + (1 − pmaxp′) |1⟩⟨1| +and E(Q↓) = qmax |0⟩⟨0| − (1 − qmax) |1⟩⟨1|. Now +note that by assumption pmax ≤ qmax. As the fi- +delity will only decrease as pmaxp′ moves away +from qmax, the optimal choice is p′ = 1. This com- +pletes the proof. +The problem with the above bound is that there +will be cases where pmax > qmax. +Why the in- +equality in the other direction was required was to +know for a fact what element of p was relevant, +namely pmax and that any choice of p′ ̸= 1 would be +sub-optimal. In general this strategy would require +q↓(j) is sufficiently large relative to p↓(j). This can +be determined in some cases. Here we provide a +simple example. +Example 3. Let +p↓ = [3/4, 1/8, 1/8]T +q↓ = [1/2, 1/2]T . +Then (p ⊗ p′)↓[1 : 2] = p′(1)[3/4, 1/8]T, and so +we can coarse-grain on the second element to ob- +tain P(p′) = 1/8p′ |0⟩⟨0| + (1 − 1/8p′) |1⟩⟨1| and +Q = Q↓. Then as 1/8p′ < 1/2, the upper bound +is F( 1 +8 |0⟩⟨0| + 7 +8 |1⟩⟨1| , 1 +21) ≈ 0.83. +The above shows that while data processing can +be sufficient in certain cases, it does not provide +an easy general method. Another common alter- +native in quantum information theory is semidefi- +nite relaxations of optimization problems because +semidefinite programs are efficient to evaluate. +In Appendix B, we establish the following upper +bound and show it may be expressed as a semidef- +inite program, which, as everything is in terms of +probability distributions, is due to the non-linearity +of fidelity and nothing particularly quantum. +Theorem 9. Consider target state |ψ⟩ and seed state +|φ⟩. Let SR(ψ) = d and SR(φ) = d′. Define A = Cd, +B = Cd·d′. Then, +FLOSR(|ψ⟩ , |φ⟩) ≤ max F(R, Q↓ +embed) +s.t. TrB[R] = P↓ +R ∈ P↓(d2 · d′) , +(7) +where P and Q are the distributions defined by |ψ⟩ +and |φ⟩’s Schmidt coefficients respectively. More- +over, this admits the following simple semidefinite +program over the reals: +max ∑ +i∈[d2·d′] +x(i) +s.t. +�diag(r) +diag(x) +diag(x) diag(q↓ +embed) +� +⪰ 0 +TrB[diag(r)] = P↓ +r ∈ P↓([d2 · d]) +x ∈ Rd2·d′ , +(8) + +16 +Physically, this relaxation may be seen as relaxing +the isometric representation of the optimal LOSR +strategy to one where one allows the ancillary en- +vironment start off entangled with the local system. +Mathematically, this is not too loose because we re- +quire this entangled pure state has a notion of “lo- +cal Schmidt coefficients” that pertain to the original +target state, although this physically does not seem +to have a clean interpretation. Nonetheless, we can +see that (7) will not achieve unity unless there exists +a joint distribution Q = R, which would require +Q↓ +embed to have P↓ as it’s marginal, which seems +highly restrictive. Therefore, (7) should provide an +upper bound that is non-trivial. +V. +MANY COPY PURE STATE CONVERSION +WITH ZERO COMMUNICATION +Having established what happens for single +copies, +we consider many copies. +We pro- +vide two motivations for doing this. +First, we +note that it’s not clear what the limiting be- +haviour will be even in the LU setting. +A +reader may recall from other works that the fi- +delity is multiplicative so if F(P, Q) < 1, then +limn→∞ F(P⊗n, Q⊗n) += +limn→∞ F(P, Q)n +→ +0. +However, we lose the multiplicativity as we are +considering limn→∞ F((P⊗n)↓, (Q⊗n)↓). This issue +is further aggravated if we consider local opera- +tions and the ancillary variable. +The second motivation is that what was initially +considered in the literature, albeit with LOCC [23], +was the conversion of many copies of states. A par- +ticular focus in the referenced work and subsequent +ones is the case where either the target or seed state +is the maximally entangled state, known as distil- +lation and dilution respectively. With LOCC, we +know there are ‘rates’ in the conversions. By [6] +along with previous results in this work, we would +not expect there to be non-negative rates without +the communication assuming the error is required +to be vanishing, i.e. ε → 0. +In this section we establish convex optimiza- +tion problems for dilution and distillation in the +zero communication setting. +These results are +established in terms of the not-actually-a-norm +∥ · ∥(k,1/2), which we remind the reader is the +(k, p)−norms extended to p < 1 introduced in Sec- +tion III with the choice of p = 1/2. We also look +at the limiting behaviour as the number of copies +grows. In particular, we find a closed form when +trying to convert n−fold two qubit states to a dif- +ferent n−fold two qubit state. Moreover, we prove +the fidelity goes to zero in this case. We discuss +the extension of this to entangled states with larger +Schmidt rank. +1. +Dilution Under Local Operations +We begin by determining the limits of dilution. +For intuition, we begin with local unitaries where +there is no optimization. Recall that the Schmidt +coefficients of the maximally entangled state are all +√ +d−1, so they correspond to the maximally mixed +distribution under our bijection between Schmidt +coefficients and probability distributions. +Proposition 11. For local unitary strategies the op- +timal dilution fidelity is given by +FLU +� +|ψ⟩ , +��Φ+ +d +�⊗n� += d−n ∥P∥(dn,1/2) . +Proof. Generally, if |ψ⟩ ̸= +��Φ+ +d +� +, +1 >F(P↓, π⊗n +d )↓) +=F(P↓, π⊗n +d ) += +� +�d−n/2 ∑ +i∈[dn] +� +P↓(i) +� +� +2 +=d−n +� +� ∑ +i∈[dn] +� +P↓(i) +� +� +2 +=d−n∥P∥(dn,1/2) , +where the first equality is because π⊗n +d +is invari- +ant under ordering, the second is using the defi- +nition of fidelity and that π⊗n +d +has uniform coeffi- +cients, and the final equality is the definition of the +(k, p)−norms. In particular note we have dropped +the sorting. +We remark we could have set |φ⟩ = |φ′⟩⊗m to get +a tradeoff, but this does not seem to provide any +insight. +Just as in the one-shot setting, we know the above +result isn’t as useful in general because it can’t +throw out resources, so we now present the general +result. +Proposition 12. The optimal fidelity of converting +n d−local dimensional EPR pairs to |ψ⟩ under local +operations is given by +FLO(|ψ⟩ , +��Φ+ +d +�⊗n) = d−n +max +P′∈P(Σ) ∥(P ⊗ P′)∥(dn,1/2) , +where ∥ · ∥(k,p) is (k, p)−norm generalized to p ≥ 0. +Moreover, for fixed n, this is a convex optimization +problem. + +17 +Proof. Starting from the result of Theorem 6, +max +P′∈P(Σ) F((P ⊗ P′)↓, (π⊗n +d )↓) += max +P′∈P(Σ) F((P ⊗ P′)↓, π⊗n +d ) += +� +� +1 +dn/2 +max +P′∈P(Σ) ∑ +i∈[dn] +� +(P ⊗ P′)↓(i) +� +� +2 +(⋆) += 1 +dn +max +P′∈P(Σ) ∥P ⊗ P′∥(dn,1/2) , +the first inequality is invariance of π⊗n +d +under sort- +ing, the second is definition of fidelity and that each +element of π⊗n +d +is the same, the last is the definition +of (k, p)-norm extended to p ≥ 0. +To show this is a convex optimization problem, +note that ΦP(·) := P ⊗ · is linear, −√· is opera- +tor convex, and the sum of the k largest eigenvalues +of a PSD P, which we will denote Σk(P) is convex. +Thus, starting from (⋆), +� +�d−n/2 +max +P′∈P(Σ) ∑ +i∈[dn] +√ +P ⊗ P′↓(i) +� +� +2 += +� +−d−n/2 +min +P′∈P(Σ)Σdn +� +− +� +ΦP(P′) +��2 +, +where +we +have +used +maxx∈C f (x) += +− minx∈C − f (x) and our definitions. +Then ig- +noring the −d−n/2 factor and the square, the +optimization +problem +is +over +the +probability +simplex, which is a convex subset of the positive +semidefinite matrices, and the objective function +is convex over the positive semidefinite cone as +− +� +ΦP(·) is operator convex and Σdn is a convex +function over the space of Hermitian operators. +this completes the proof. +Unfortunately, +while +this +gives +computable +bounds, it is not clear how one could determine the +optimal value analytically. +2. +Distillation Under Local Operations +We now present the same results in the distilla- +tion case, where we take some state to many EPR +states. +For completeness, we state the local uni- +taries case. +Proposition 13. The fidelity of distillation under lo- +cal unitaries and zero communication is given by +FLU( +��Φ+ +d +�⊗m , |ψ⟩⊗n) = d−m∥P⊗n∥|S|,1/2 , +where S = [min{dm, rank(P)n}]. +Proof. The proof is effectively identical to the dilu- +tion case by symmetry of the fidelity. +In contrast to the local unitary case, the symmetry +is broken when one considers local operations. +Theorem 10. For fixed d, m, n the optimal fidelity +for dilution under local operations is given by +FLO( +��Φ+ +d +�⊗m , |ψ⟩⊗n) +=d−m +� +min +P′∈P↓(Σ) − ∑ +i∈I +αi +� +p′(i) +�2 +, +where P↓(Σ) is the set of decreasing distributions +as defined in Section III, I ≡ [⌈rank(P)n/dm⌉], and +αi := ∑j∈[(i−1)dm:min{i·dm,rank(P)n}] +� +p↓ +n(i). Note the +minimization is a convex optimization program. +Proof. Yet again, we use the square root fidelity and +then take the square at the end. Then, using Theo- +rem 6, we have +FLO((Φ+ +d )⊗m, ψ⊗n) += max +P′∈P(Σ) F((π⊗m +d +⊗ P′)↓, (P⊗n)↓) += +� +max +P′∈P(Σ) ∑ +i∈S +� +(π⊗m +d +⊗ p′)↓(i) +� +p↓ +n(i) +�2 +. +Next, note +(π⊗m +d +⊗ P′)↓ = d−m/2 ∑ +i′∈Σ +p↓(i′)1Cdm , +where we have just used that π⊗m +d +is invariant +under ordering. +It follows that if we let I +≡ +[⌈rank(P)n/dm⌉], we can rewrite, +FLO((Φ+ +d )⊗m, ψ⊗n) +=d−m +� +max +P′∈P(Σ) ∑ +i∈I +� +(p′)↓(i) +· +∑ +j∈[(i−1)dm:min{i·dm,rank(P)n}] +� +p↓ +n(i) +�2 +. +Now +first +define +αi +:= +∑j∈[(i−1)dm:min{i·dm,rank(P)n}] +� +p↓ +n(i) +as +these +co- +efficients may be pre-computed. +Second, note +that the probability simplex restricted to de- +scending distributions, P↓(Σ) is itself convex as +r↓ +λ := λp↓ + (1 − λ)q↓ satisfies +λp↓(i) + (1 − λ)q↓(i) ≥ λp↓(i + 1) + (1 − λ)q↓(i) , + +18 +for all i ∈ [|r|]. Thus we have, +FLO( +��Φ+ +d +�⊗m , |ψ⟩⊗n) += +� +− d−m +min +P′∈P↓(Σ) − ∑ +i∈I +αi +� +p′(i) +�2 +. +The minimization is a convex optimization problem +because if we consider f (p′) := − ∑i αi +� +p′(i), then +its Hessian is ∇2 f = ∑i[αi/4p′(i)−3/2] |i⟩⟨i|, which +is positive semidefinite on the interior of the prob- +ability simplex (i.e. when p′(i) > 0 for all i). This +completes the proof. +3. +Two Qubit Setting +We have now seen that even in the basic dilu- +tion and distillation setting, while we can deter- +mine convex optimization programs, we can’t seem +to get clean analytic results. In this section we con- +sider an even more tractable setting to attempt to +resolve this: many copy two-qubit seed and target +states. We show in this setting under certain as- +sumptions the local unitary strategy is optimal and +lobby this to show in particular that the optimal fi- +delity of converting n copies of |φ⟩ to n copies |ψ⟩ +goes to zero as n goes to infinity. We note that this +setting is more manageable because we effectively +only have to reason about Bernoulli distributions. +Lemma 11. Given Bernoulli distribution P += +p |0⟩⟨0| + (1 − p) |1⟩⟨1|, then P⊗n is such that the +sequence xn with (n − k) zeros has probability +pn−k(1 − p)k. +Moreover, there are (n +k) sequences +with probability pk(1 − p)n−k and the same for +pn−k(1 − p)k. +Proof. The claim that xn with (n − k) zeros has prob- +ability pn−k(1 − p)k is straightforward. The second +point actually just follows from the fact there are (n +k) +sequences with k zeros, which could be proven by +induction in a straightforward manner. +We can now use the above lemma along with +Theorem 5 to get the optimal LU fidelity as a func- +tion of the number of copies n. +Corollary 3. Consider entangled states |ψ⟩ , |φ⟩ ∈ +C2 ⊗ C2. Then, +FLU(ψ⊗n, φ⊗n) += ∑ +k∈[n] +�n +k +� +(pq)(n−k)/2((1 − p)(1 − q))k/2 . +Proof. By +Theorem +5 +we +can +reduce +to +the +Bernoulli distributions from the Schmidt coeffi- +cients, |ψ⟩⊗n �→ P⊗n, |φ⟩⊗n �→ Q⊗n. Since these are +Bernoulli distributions, if we assume without loss +of generality p ≥ (1 − p), we can order the proba- +bilities simply by the exponent, e.g. pj−k(1 − p)k ≥ +pj−k−k′(1 − p)k+k′ for any 0 ≤ k′ ≤ j − k. More- +over, the cardinality of each set of sequences will be +the same for both P⊗n and Q⊗n because |ψ⟩ , |φ⟩ are +only entangled if their Schmidt rank is two. There- +fore, +F((P⊗n)↓, (Q⊗n)↓) += ∑ +k∈[n] +�n +k +� +(pq)(n−k)/2((1 − p)(1 − q))k/2 +where the sum is over the number of zeros in the +string, the cardinality was proven in the previous +lemma, and the last term is just a re-writing of +� +pn−k(1 − p)k +� +qn−k(1 − q)k. +We note it is straightforward to generalize the +above result to the case where you have the num- +ber of states differs between the seed and the target, +but the form would be ugly as one would need to +count how many sequences of a given probability +there are and keep track of this in the sum. Indeed +at this point the problem is elaborate enough that +there is no advantage with dealing with two-qubit +states as it’s a question of the type classes [24]. We +state this as a remark. +Remark 1. Consider states |ψ⟩ , |φ⟩ respectively +with ordered probability distributions correspond- +ing to their Schmidt coefficients, P and Q respec- +tively. FLU(|ψ⟩⊗n , |φ⟩⊗m) can be computed. This is +because the probability of a given sequence drawn +in i.i.d. form from a distribution has a closed form +[24, Theorem 11.1.2]. It follows that as long as one +determines the type classes exactly and takes into +account that the sizes of the type classes may dif- +fer between P and Q, the computation is possible, +albeit tedious. +Rather than dealing with the computational +nightmare of generalizing beyond two qubit states, +we now show that the term in Corollary 3 always +goes to zero as n goes to infinity. +Proposition +14. +Consider +entangled +states +|ψ⟩ , |φ⟩ ∈ C2 ⊗ C2. +lim +n→∞ FLU(|φ⟩⊗n , |ψ⟩⊗n) = 0 . +Proof. Let the probability distributions correspond- +ing to their Schmidt coefficients be parameterized + +19 +by p and q = p + ε where ε ∈ [−1/2, 1/2]. Then, +That way, +FLU(|ψ⟩⊗n , |φ⟩⊗n) += ∑ +k∈[n] +�n +k +� +(p2 + ε)(n−k)/2 +· [(1 − p)2 − ε(1 − p)]k/2 . +Now note p2 + ε < 1 as otherwise |ψ⟩ is product. +Define α := (p2 + ε)1/2 < 1. Then we have +�n +k +� +(p2 + ε)(n−k)/2 · [(1 − p)2 − ε(1 − p)]k/2 +≤ +�n · e +k +�k +αn−k[(1 − p)2 − ε(1 − p)]k/2 += +� e +k α−1�k +[(1 − p)2 − ε(1 − p)]k/2nk · αn +=O(poly(n))O(exp(−n)) +→0 , +where in the inequality we have used an upper +bound on the binomial coefficient, in the first equal- +ity we have grouped terms by scaling, in the next +equality we have used that the first portion is a +polynomial in n and that α < 1, so αn scales in- +verse exponentially in n. The limiting factor is then +because an inverse exponential times a polynomial +goes to zero. We also remark that the term where +k = n will also go to zero as [(1 − p)2 − ε(1 − p)]k/2 +will go to zero as k goes to infinity as its magnitude +will be bounded by 1. +Therefore, each term in the sum goes to zero as +n goes to infinity, so the entire sum will go to zero. +This completes the proof. +We note our proof tells us nothing about the scal- +ing as a function of the difference between p and q +nor does it tell us how fast it goes to zero compared +to F(P⊗n, Q⊗n). These are shown numerically for +specific cases in Fig. 4. +It is then natural to ask if what we have seen so +far is something special to local unitaries. We show +that under sufficient conditions, just like in the sin- +gle copy case, when two-qubit seed states are in- +volved, local unitary strategies are optimal. +Theorem 12. Let |ψ⟩ ∈ C2 ⊗ C2 and the target state +be |ψ⟩⊗n. Let the seed state |φ⟩ satisfy SR(|φ⟩) ≤ +nSR(|ψ⟩). Then the optimal local operations strat- +egy is the optimal local unitary strategy. +Proof. By Theorem 6, +FLO(|ψ⟩⊗n , |φ⟩) +(a) Fidelity under local unitaries as n grows for +various choices of q = p + ε. +(b) Fidelity under local unitaries as n grows for +various choices of q = p + ε compared to F(P, Q)⊗n. +FIG. 4: Degradation of fidelity of trying to convert n +copies of one pure two-qubit entangle state to another +for various differences in Schmidt coefficients, q = p + ε +where we choose p = 0.55. (a) Shows the rate that the +local unitary strategy degrades is a nonlinear function of +the size of ε. (b) Compares to the case where one does +not re-order the Schmidt coefficients, i.e. compares to +F(P, Q)n. += max +P′∈P(Σ) F((P⊗n ⊗ P′)↓, Q↓) += ∑ +i∈|Q| +� +Q↓(i) +� +(P⊗n ⊗ P′)↓(i) . +We will show that P′ should be the delta distribu- +tion. If p ̸= 1/2, p′(1) < 1, then for any 0 ≤ k ≤ n, +we have the inequalities +pn−k(1 − p)k >pn−k(1 − p)kp′(1) +>pn−k(1 − p)kp′(2) +and +pn−k(1 − p)k >pn−k(1 − p)kp′(1) +>pn−(k+1)(1 − p)k+1p′(1) +>pn−(k+1)(1 − p)k+1p′(2) +As square root is a monotone, this holds when +we take the square root. Note that by assumption + +ConvergenceComparison +1.0 +0.8 +UnorderedE=0.4 +OrderedE=0.4 +0.6 +idelity +Unordered E=0.1 +0.4 +OrderedE=0.1 +UnorderedE=0.05 +0.2 +OrderedE=0.05 +0.0 +0 +50 +100 +150 +200 +250 +Numberof CopiesnConvergenceComparison +1.0 +0.8 +UnorderedE=0.4 +OrderedE=0.4 +0.6 +idelity +Unordered E=0.1 +0.4 +OrderedE=0.1 +UnorderedE=0.05 +0.2 +OrderedE=0.05 +0.0 +0 +50 +100 +150 +200 +250 +Numberof Copiesn20 +P⊗n has enough entries by itself for there to be +one corresponding to each q↓. +Therefore, given +the inequalities above, it follows if p′(1) ̸= 1, each +term in the sum only decreases. Therefore, p′(1) is +optimal for every n and k. Thus, when p ̸= 1/2, +the optimal value is obtain by P′ being a delta +distribution, which means it’s equivalent to the +local unitary strategy. +Finally, if p = 1/2, then pn−k(1 − p)k = 2−n for +all k. Therefore, if p′(1) < 1, the inequalities simpli- +fies for all 0 ≤ k ≤ n: +pn−k(1 − p)kp′(1) =pn−(k+1)(1 − p)k+1p′(1) +>pn−(k+1)(1 − p)k+1p′(2) +and +pn−k(1 − p)kp′(1) > pn−k(1 − p)kp′(2) . +Again because each q term is paired up already, this +means if p′(1) ̸= 1, the value decreases. Therefore, +we again conclude the optimal strategy is the LU +strategy. This completes the proof. +We note that a trivial example of why we need +the Schmidt rank constraint in the previous theo- +rem is our original example for the advantage of +LO strategies: if |φ⟩⊗n+ℓ where ℓ ≥ 1, then there +is a better LO strategy than an LU strategy. Finally, +we note it immediately follows from these previous +results that +Corollary 4. If |φ⟩ , |ψ⟩ ∈ C2 ⊗ C2 are both entan- +gled, then +lim +n→∞ FLO(|ψ⟩⊗n , |φ⟩⊗n) = 0 . +VI. +ON CATALYTIC CONVERSION +We now have established a rather robust theory +of pure state transformations under local opera- +tions. It is natural to return to the topic of conver- +sion of one state to another using an ancillary en- +tanglement, i.e. cataltyic transformations, which is +a special case of the setting, and includes quantum +embezzlement. Of course, it is immediate from our +results so far that we know the optimization pro- +gram that determines the optimal pure state cata- +lyst, as we state in the following proposition. +Proposition 15. For any Schmidt rank d, the opti- +mal pure state catalyst for state conversion |φ⟩ to +|ψ⟩ is the quantum state |ζ⟩ = vec( +√ +R) that is de- +termined via the optimization +max +R∈P(d),P′∈P(Σ) F((P ⊗ P′)↓, (Q ⊗ R)↓) . +Proof. This immediately follows from the input be- +ing |φ⟩ ⊗ vec( +√ +R) and then applying Theorem 6. +Note this means |Σ| scales as function of d. +However, as we have already addressed, even +without a free variable for the catalyst, the opti- +mization in Theorem 6 seems unmanageable di- +rectly. While in principle one could use the relax- +ation in Theorem 9 to obtain efficient upper bounds, +it is less obvious how often these will be non-trivial +given that R is a free variable. +The next most natural setting would be that of +catalytic state conversion under local unitaries, i.e. +we consider transformations of the form +|φ⟩ |ζ⟩ +LU +←→ ≈ε |ψ⟩ |ζ⟩ , +where |ζ⟩ is the catalytic resource and the arrow +going in both directions is because local unitaries +are reversible. This may be seen as a generaliza- +tion of embezzlement where |φ⟩ = |0⟩A |0⟩B and +|ζ⟩ = |µ(n)⟩.2 +Now as noted in the background, embezzling is +known to be in effect optimal for sufficiently small +ε. It follows for sufficiently small error ε > 0, the +strategy that embezzles out the seed state and then +embezzles in the target state is roughly optimal, i.e. +|φ⟩ |µ(n)⟩ +LU +←→ |0⟩ |0⟩ |µ(n)⟩ +LU +←→ |ψ⟩ |µ(n)⟩ +is effectively optimal. Nonetheless, we may explore +at what point this becomes necessary. +Using Theorem 5, we know the optimal strategy +is given by3 +max +R∈P(d) F((P ⊗ R)↓, (Q ⊗ R)↓) . +Even in the case P, Q, R ∈ P(2) this technically +can’t be solved using gradient methods as one has +to sort the p(1 − r) and (1 − p)r terms of p ⊗ r and +likewise for q ⊗ r. Nonetheless, it is hopefully clear +that r ∈ [min{p, q}, max{p, q}], as it is trying to +make the distributions be more similar. Nonethe- +less, this issue will only grow in difficulty with the +dimension and it is unclear how one would prove +an ansatz is optimal in general. Therefore, we pro- +vide two-qubit examples which characterizes the +general insights. +2 We refer the reader to Proposition 3 if the notation has been +forgotten. +3 We stress that by the correspondence of Schmidt coefficients to +probability distributions as discussed at the start of the work, +even without Theorem 5, this would be a legitimate strategy, +we simply wouldn’t know analytically it was optimal. + +21 +Example 4 (Resource Gap Between Embezzling +and Optimal Catalyst). Consider Bernoulli distri- +butions P, Q, R parameterized by p = 0.5, q = 0.7 +and we leave r unspecified for now. In other words, +one of the states is the maximally entangled states +and the other is, up to local unitaries, +√ +0.7 |00⟩ + +√ +0.3 |11⟩. Therefore, depending on which way one +runs the transformation, we are considering entan- +glement dilution or distillation with a catalytic re- +source. Without the resource, +FLO(|ψ⟩ , |φ⟩) = F(P↓, Q↓) ≈ 0.958. +One can verify that the optimal choice of r⋆ ≈ 0.6 +in this case. For this choice +FLU(|ψ⟩ |ζ⟩ , |φ⟩ |ζ⟩) =F((P ⊗ R⋆)↓, (Q ⊗ R⋆)↓) +>0.979 . +The first problem is that 0.979 is not an accept- +ably high fidelity even by contemporary standards. +Nonetheless, note that to get this state via embez- +zling (and ignoring that embezzling out the initial +state introduces error), it would require generating +|µ(n)⟩ where n > m1/(1−0.979) = 2 · 1014. That is, +even to embezzle a two-qubit pure state would re- +quire generating an inconceivable amount of entan- +glement. For this reason, specially engineered cat- +alysts seems a significant improvement up to any +error that can be achieved. +On the other hand, one might note that if we +could generate R where r = 0.55, then we may as +well have just used this state to begin with as +F(P↓, R↓) =0.98989 +>F((P ⊗ R⋆)↓, (Q ⊗ R⋆)) . +From a practical perspective we agree with this cri- +tique. Nonetheless, from a basic science perspec- +tive, if we are interested in local unitary conversions +under catalysts, then the above tells us there are bet- +ter choices in general than embezzlement, although +embezzling has the special property of being uni- +versal and optimal for sufficiently small ε. +We close this consideration with two final re- +marks. First, if one picks two states that are more +similar to begin with, then the scaling of the embez- +zling state will be even larger. Second, we have not +presented how the fidelity for this example scales as +the local dimension of |ζ⟩ grows. Both the dimen- +sion scaling and two states that are more similar are +considered in Fig. 5 where the near-optimal fideli- +ties are found via brute force numerical search. +(a) Maximum achievable fidelity of transformation +under local unitaries as a function of the Schmidt +rank of the catalyst. +(b) Order of the Schmidt Rank of embezzling state +|µ(n)⟩ to achieve same fidelity. +FIG. 5: Plots regarding dimension scaling in Example 4. +(a) The achievable fidelity of converting one two-qubit +entangled state to another parameterized by p and q +under local unitaries using a catalyst with a given local +dimension (equivalently, Schmidt rank) using brute +force search. (b) The order (i.e. the power of 10) in the +Schmidt rank of the embezzling state |µ(n)⟩ to obtain +the same maximum fidelity. This is calculated using +21/(1−Fmax) following Proposition 3. All optimizer +catalysts provided in an appendix for verification. +VII. +ON EXTENSIONS OF THE THEORY +As a final consideration, we discuss the applica- +tion of our results beyond bipartite pure states. First +we remark upon extensions to multipartite pure +states. In this case the problem is that in establish- +ing all of the results, we have used that local uni- +taries can take the Schmidt decomposition of the +state to one of a canonical form. However, in the +multipartite case, the Schmidt decomposition does +not even exist in general [25]. As such this argu- +ment immediately breaks down. Furthermore, in +the proof of Theorem 5 we used Uhlmann’s theo- +rem, which requires partitioning the state into two +pieces, one of which is the purification. Therefore, it + +MaximumFidelityasaFunctionofCatalystDimension +1.00 +0.99 +L +lity +Fideli +0.98 +p=0.6,q=0.65 +p=0.5,q=0.7 +Max +0.97 +0.96 +0 +2 +4 +6 +8 +AllowedCatalvstSchmidtRankOrderofEmbezzlingStateDimforsameMaxFidelity +Schmidt Rank +500 +100 +p=0.6,q=0.65 +OrderofEmbezzling +50 +p=0.5,q=0.7 +10 +0 +2 +4 +6 +8 +AllowedCatalvstSchmidtRank22 +seems no multipartite extension of this work holds. +Similarly, +there are issues with approaching +mixed states. One issue is to note that all relation- +ships we have been able to establish have stemmed +from the fidelity under local unitaries of pure states. +Even in the case where local operations made a +pure state no longer pure, we purified operations so +that the states were pure. We simply cannot do this +if we start with mixed states in both arguments of +the fidelity. 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Tan, Common information, noise +stability, and their extensions, Foundations and +Trends® in Communications and Information The- +ory 19, 107 (2022). +Appendix A: Randomness Embezzling Proof and +Discussion on Locality +In this section we provide the proof of Theorem +2 and then briefly discuss how it differs from quan- +tum embezzlement. +Proof. The proof is largely the same as for embezzle- +ment of quantum states [12]. Let P = ∑i p(i) |i⟩⟨i|. +Define Wn as Rn ⊗ P except with probabilities in de- + +23 +creasing order. Note +Rn ⊗ P = +1 +Hn ∑ +i,j +p(i) +j +|i⟩⟨i| ⊗ |j⟩⟨j| , +so there exists a relabeling on {(i, j)} that will take +this to Wn. In particular, letting f : [m] × [n] → [m · +n] be a bijection, we have |i⟩ |j⟩ → | f (i, j)⟩ ≡ |i′⟩ |j′⟩ +such that +� +z f (i,j) := p(i) +jHn +� +(i,j) satisfy zk ≥ zk+1 for all +k ∈ [m · n]. Therefore it suffices to approximate Wn, +which means we want to bound the overlap of this +with Rn ⊗ P. +For fixed t and i, we let +Nt +i := +���� +� +(i, j) : p(i) +jHn +> +1 +tHn +����� . +The inequality may be manipulated to imply 1 ≤ +j < p(i)t. It follows that Nt +i < p(i)t. From this we +obtain ∑m +i=1 Nt +i < ∑m +i=1 p(i)t < t, where we have +used ∑i p(i) = 1. +As z1 ≥ z2 ≥ ..., it follows +zj ≤ +1 +jHn for all 1 ≥ j ≥ n. We may restate this as for +1 ≤ j ≤ n, there are at most t′ − 1 pairs (i, j) such +that p(i)/(jHn) > 1/(t′Hn). Recalling z1 ≥ z2 ≥ ..., +this means that z1 < 1/Hn and that there is at most +one pair (i, j) pair such that p(i)/(jHn) < 1/(2Hn), +which, since z1 ≥ z2, means if such a pair exists, +it is z1. By applying this argument in effect recur- +sively, we see that for t′, there are at most t′ − 1 +(i, j) pairs such that p(i)/(jHn) > 1/(t′Hn) and +since zk ≥ zk+1, if all of these pairs exist, then it +must be z1, ..., zt′−1. Therefore, zj ≤ 1/(jHn) for all +1 ≤ j ≤ n. We can now use this to bound the fi- +delity. +F(Rn ⊗ |0⟩⟨0| , Wn) = +� +n +∑ +j=1 +� +zj +jHn +�2 +≥ +� +n +∑ +j=1 +�zj +�2 +≥ +n +∑ +j=1 +zj, +where in the equality we have used the definition +of fidelity, in the second we used our established in- +equality, and in the third we have used √x + √y ≥ +√x + y for x, y ≥ 0 to pull the square root out +around the sum and cancel with the square. +Now we want to lower bound this sum, which +requires managing the zj terms. +We consider +Tn = Rn ⊗ πm with probabilities t(j) where πm := +1 +m ∑m +i=1 |i⟩⟨i|. +Now note that zk ≥ tk for all k ∈ +[m · n], and this is independent of what the distri- +bution P is. We can then bound the relevant sum by +the sum for Tn. It follows +n +∑ +j=1 +tj = +⌊n/m⌋ +∑ +j=1 +m +∑ +i=1 +1 +jHnm = +⌊n/m⌋ +∑ +j=1 +1 +jHn += +H⌊n/m⌋ +Hm +≥ln(n/m) +ln(n) += 1 − log(m) +log(n) , +where the second inequality is using Hn ≥ ln(n) +and the final form is converting from ln to log in +both the numerator and denominator so it cancels. +Finally, leting 1 − log(m)/ log(n) > 1 − ε will result +in n > m1/ε, which completes the proof. +With the proof established, we expand upon the +distinction between the entangled and classical dis- +tribution cases of embezzlement in terms of local- +ity briefly mentioned in the main text. In the clas- +sical case, one party embezzles a distribution lo- +cally by themselves, whereas in the entangled case +two parties act locally on a non-local distribution. +Mathematically, this simply follows from the fact +the vec(·) map and its inverse converts between bi- +partite states and a probability distribution. How- +ever, it is also physically interesting that these are +the two cases that align as it is clear other varia- +tions are either classically or quantumly impossible +as we now explain. +The first reasonable variation would be if there +is a non-local classical case where two parties try +and construct some joint distribution pXY using cat- +alyst rX′Y′. It is easy to see that they cannot in gen- +eral satisfy the decoupling condition that is satisfied +in quantum embezzlement, i.e. they cannot satisfy +pXY ⊗ rX′Y′ in this setting. This is because without +loss of generality the state will be of the form +qXYX′Y′ = ∑ +x,x′,y,y′ +q(x|x′)q′(y|y′)r(x, y) +· +��x, y, x′, y′�� +x, y, x′, y′�� . +This form means that X will be correlated to X′ and +Y to Y′ unless qXY may be generated non-locally +without a seed state to correlate the two which +means they are (up to the allowed error) indepen- +dent, i.e. qXY ≈ε qX ⊗ qY. In this sense, there cannot +be a classical non-local equivalent of quantum em- +bezzlement. +On the other hand, if one does not require the +decoupling, then this is a task that is possible in +the classical setting and is known as distributed +source simulation, where the question is the min- +imal needed shared randomness as the seed state to +generate the target state up to an (arbitrary) er- +ror [26]. +This was determined asymptotically in +the classical case by Wyner [8], extended to sepa- +rable states by Hayashi [9], and recently general- +ized to the one-shot setting for separable states in +[10]. However, as in this setting variation there is no + +24 +communication between the acting parties and the +catalyst acts as the seed state, it follows from Propo- +sition 2 that distributed source simulation cannot +admit an entangled state equivalent. For these rea- +sons, not only does the vec bijection specify the cor- +respondence of embezzlement in the classical and +quantum setting, but deviating from it makes either +a quantum or classical version impossible. +Appendix B: Semidefinite Program Relaxation of Max +Fidelity of Pure State Transformation Under LOSR +In this section we prove Theorem 9. We begin by +establishing (7) is true. +Lemma 13. Consider target state |ψ⟩ and seed state +|φ⟩. Let SR(ψ) = d and SR(φ) = d′. Define A = Cd, +B = Cd·d′. Then, +FLOSR(|ψ⟩ , |φ⟩) ≤ max F(R, Q↓ +embed) +s.t. TrB[R] = P↓ +R ∈ P↓(d2 · d′) , +where P and Q are the distributions defined by |ψ⟩ +and |φ⟩’s Schmidt coefficients respectively. +Proof. The above seems intuitively true from The- +orem 6 as we have just relaxed the tensor product +structure with the partial trace constraint. The tech- +nical issue is the ordering operation ·↓ is defined in +terms of a permutation of a fixed basis, so we need +to make sure this works with the partial trace. +Note the feasible set, the set we can optimizer +over, in Theorem 6 is S1(P) := {(P ⊗ P′)↓ : P′ ∈ +P(Σ)}. Now note this is the same as the set +S2(P) := {(P↓ ⊗ P′↓)↓ : P′ ∈ P(Σ)} , +because the ordering applied to the tensor product +will result in the same thing regardless of whether +or not P, P′ were ordered. Therefore, we can focus +on P↓ ⊗ P′↓ to make the explanation clearer. +In general, in terms of vectors, +(p↓ ⊗ p′↓)↓ = +� +� +� +� +� +� +p↓(1)p′↓ +p↓(2)p′↓ +... +p↓(d)p′↓ +� +� +� +� +� +� +, +where p(i) ≥ p(i + k) for k ≥ 0. Formally, we also +have +p↓(i)p′↓(1) ≥ p↓(i + k)p′↓(j) +for all i ∈ [d], k ∈ {0, ..., d − i}, and j ∈ Σ. In partic- +ular what this means is that without loss of general- +ity for any i ∈ [d], p↓(i)p′↓(1) appears before any el- +ement that is not of the form p↓(i − ℓ)p′↓(j) for some +0 < ℓ ≤ i − 1. It follows that under the ordering of +(p↓ ⊗ p′↓)↓, when the partial trace marginalizes to +the A space, the induced ordering on the local space +will be the ordering based on p↓. Formally, this can +be expressed as +TrC|Σ|[(P↓ ⊗ P′↓)↓] += ∑ +j∈Σ +1A ⊗ ⟨j| (P↓ ⊗ P′↓)↓ |j⟩ += ∑ +i∈[d] +p↓(i) |i⟩⟨i| , +where the first equality is a representation of the +partial trace and the second is using the property +noted of the ordering on the joint ordered distribu- +tion. +Thus, if X ∈ S2(P), TrC|Σ|(X) = P↓ and X ∈ +P↓(d · |Σ|). Noting that |Σ| = d · d′, this is the fea- +sible set we have defined in the proposition. This +completes the proof. +The remaining point is to prove this is the +semidefinite program given in (8). There is much to +the theory of semidefinite programs for quantum +information [19], but for our purposes all we will +need is the following definition. +Definition 2. A semidefinite program may be ex- +pressed as +max Tr(AX) +s.t. Φ(X) = B +XCd ⪯ 0 , +where Φ ∈ T(Cd, Cd′) is a Hermitian-preserving +map, A ∈ Herm(Cd), B ∈ Herm(Cd′), and Herm(·) +is the space of Hermitian operators on a given +Hilbert space. +The fidelity is known to be a semidefinite pro- +gram [19], so we are really just verifying all of our +constraints work and that we can write the SDP +simply by making use of that. +Lemma 14. The optimization program in the pre- +vious lemma, may be expressed as the following + +25 +semidefinite program over the reals. +max ∑ +i∈[d2·d′] +x(i) +s.t. +�diag(r) +diag(x) +diag(x) diag(q↓ +embed) +� +⪰ 0 +TrB[diag(r)] = P↓ +r ∈ P↓([d2 · d]) +x ∈ Rd2·d′ , +where d, d′ are defined in the previous lemma. +Proof. We begin by expressing the objective func- +tion of the previous lemma, which is in terms of +fidelity, using the primal problem for the SDP for +fidelity from [19, Theorem 3.17]: +max1 +2 +� +Tr(X) + Tr +� +X†�� +� R +X +X† Q↓ +embed +� +≥ 0 +X ∈ L(C[d2·d′]) . +Now our goal is to reduce X to the diagonal of a +real vector. +Note that R, Q↓ +embed are always invariant under +pinching onto the computational basis of C[d2·d′], +which we can denote ∆. Note that this pinching is +a CPTP, so by the CP property, +(idC2 ⊗ ∆) +� R +X +X† Q↓ +embed +� += +� +R +∆(X) +∆(X†) Q↓ +embed +� +≥ 0 . +It also then follows as a positive semidefinite oper- +ator is always Hermitian that +� +R +∆(X†) +∆(X) Q↓ +embed +� +≥ 0 . +Thus by taking these two cases and averaging them, +we have that +� +R +1 +2 +� +∆(X + X†) +� +1 +2 +� +∆(X + X†) +� +Q↓ +embed +� +≥ 0 . +Define X := 1 +2 +� +∆(X + X†) +� +. Then note +1 +2 +� +Tr(X) + Tr +� +X†�� +=1 +2 +� +Tr(∆(X)) + Tr +� +∆(X†) +�� +=1 +2 +� +Tr +� +X +� + Tr +� +X†�� += Tr +� +X +� +, +where the first equality is because the pinching is +trace preserving, the second is by definition of X, +as is the final equality. Thus, for any X that sat- +isfies the positivity constraint, we could replace it +with X without loss of generality as we are con- +sidering a maximization. Finally, note that X is a +real diagonal matrix by the pinching along with the +fact a + a∗ = 2 Re{a}. Thus X = diag(x) for some +x ∈ Rd2·d′. Combining all these points and using +Tr +� +X +� = ∑i∈[d2·d′] x(i), we have reduced to consid- +ering +max ∑ +i∈[d2·d′] +x(i) +�diag(r) +diag(x) +diag(x) diag(q↓ +embed) +� +≥ 0 +x ∈ Rd2·d′ . +This argument works for any choice of diagonal r, +so this is the major reduction. +What remains is to prove all the constraints are +Hermitian maps. One can write the constraints for +r ∈ P↓ as r(i) ≥ r(i + 1) for all i, which are semidef- +inite constraints and can be written as Hermitian +preserving maps on the variables r, x. +diag is a +Hermitian preserving map as is the partial trace, so +TrC[diag(r)] is a Hermitian preserving map. Like- +wise is the block matrix mapping if one allows for +the complex conjugate in the lower left block, but +noting diag(x)† = diag(x), we can leave it as writ- +ten. Thus all the maps are Hermitian-preserving. +The conversion to actual standard form we then +omit as it provides no insight. This completes the +proof. +The above two proofs establish Theorem 9. +Appendix C: Data for Catalyst Figure +For p = 0.5, q = 0.7: +Dimension Optimal distribution r +1 n/a +2 +1 +100[4, 6] +3 +1 +100[21, 32, 47] +4 +1 +100[12, 18.5, 0.28, 0.415] +5 +1 +100[7, 11, 17, 26, 39] +6 +1 +100[5, 8, 11, 16, 24, 35] +7 +1 +100[3, 5, 8, 11, 16, 23, 24] +8 +1 +100[5, 6, 9, 9, 13, 14, 19, 25] + +26 +For p = 0.6, q = 0.65: +Dimension Optimal distribution r +1 n/a +2 +1 +100[32, 63] +3 +1 +100[18, 31, 51] +4 +1 +100[10, 17, 28, 45] +5 +1 +100[6, 10, 16, 26, 42] +6 +1 +100[7, 11, 12, 18, 20, 32] +7 +1 +100[0, 7, 11, 12, 18, 20, 32] +8 +1 +100[0, 0, 7, 11, 12, 18, 20, 32] + diff --git a/U9E3T4oBgHgl3EQf0QsH/content/tmp_files/load_file.txt b/U9E3T4oBgHgl3EQf0QsH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c06225a65e7b127b0ed49cd78af643e151e899a8 --- /dev/null +++ b/U9E3T4oBgHgl3EQf0QsH/content/tmp_files/load_file.txt @@ -0,0 +1,1151 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf,len=1150 +page_content='Revisiting Pure State Transformations with Zero Communication Ian George and Eric Chitambar Electrical and Computer Engineering Department, University of Illinois at Urbana-Champaign (Dated: January 13, 2023) It is known that general convertibility of bipartite entangled states is not possible to arbitrary error without some classical communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' While some trade-offs between communication cost and con- version error have been proven, these bounds can be very loose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In particular, there are many cases in which tolerable error might be achievable using zero-communication protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In this work we address these cases by deriving the optimal fidelity of pure state conversions under local unitaries as well as local operations and shared randomness (LOSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We also use these results to explore catalytic conversions between pure states using zero communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' INTRODUCTION The theory of quantum mechanics through the lens of information and vice versa [1–3] has af- forded the physicist and the information scientist alike with a new way to view the objects and long- term goals of their study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' No better example of this can be found than quantum resource theo- ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Quantum resource theories specify the rele- vant physical property in such a manner as to better tease apart the complexities of quantum mechanics while also establishing what tasks may be achieved with said resource [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Perhaps the earliest example of such a resource theory is the resource theory of entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Entanglement may be viewed as a form of correlation that does not exist in the classi- cal world [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Roughly speaking, the resource the- ory of entanglement asks (1) what tasks may be per- formed better using entangled states and (2) how entangled states may be converted from one to an- other under some class of free operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The most standard view of the resource theory of entanglement considers the set of free operations to be local operations and classical communication (LOCC) which captures the ‘distant lab’ paradigm where two (or more) parties share an entangled state in spatially separated labs and they can only perform operations on their respective portions and exchange classical information (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Not only is this the most standard set of free operations, but in some respect it seems minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Indeed, Hay- den and Winter showed that to convert one (pure) entangled state to another to sufficiently small pre- cision requires a certain amount of communication between labs, regardless of how many auxiliary EPR pairs they share [6] (see also [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This is dis- tinct not only from the classical setting [8], but also from quantum states that are not entangled [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' However, the results of Hayden and Winter, while fundamental, do not give us a complete picture of the tradeoff between communication and achiev- able tolerated error in pure state conversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In- (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 1: Conversion of pure states in distant labs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' (a) The LOCC model where communication is exchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' (b) The embezzling of quantum states where an auxiliary entangled state is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This may be seen as a special case of catalytic conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' deed, it is easy to find examples of state conversions which, according to the best known lower bounds, still may be possible to perform with a tolerated er- ror of 1% using no communication (see Example 1 of Section III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This show that a relatively large gap in our understanding of zero-communication en- tanglement transformations still persists, and one we aim to address in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Moreover, the tools we develop to address this problem will also allow us to study pure state trans- formations using shared auxiliary entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The operational paradigm in which parties are al- lowed to use arbitrary pre-shared entanglement but no communication is known as local operations and shared entanglement (LOSE) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' By itself, the problem of pure state convertibility |ψ⟩AB → |φ⟩AB under LOSE is trivial since Alice and Bob could al- ways just demand |φ⟩ as their pre-shared entangle- ment and then throw away |ψ⟩ when it is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' However, if one demands that the pre-shared en- tanglement is also returned in addition to the target state |φ⟩, then the problem becomes quite interest- ing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' |ψ⟩AB ⊗ |ω⟩A′B′ → |φ⟩AB ⊗ |ω⟩A′B′ for aux- iliary pre-shared entanglement |ω⟩A′B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Transfor- mations of this form are known as catalytic trans- formations with |ω⟩A′B being the catalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Remark- ably, van Dam and Hayden have shown that there exists a family of entangled catalysts, known as arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content="04735v1 [quant-ph] 11 Jan 2023 A BA A A' A' B' B' B B2 FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 2: Comparison of [12] (dark pink),[17] (dark green), and this work’s results (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' [17] finds lower bounds on the classical communication necessary to convert one state to another, but in the zero communication setting these are too loose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We find methods for solving this exactly (Section IV), which establishes that communication is necessary for larger tolerated errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' [12] establishes a method for pure state transformations with zero communication with massive amounts of entanglement, but it scales inversely with the error, which we find can be too strong for a relevant error range, even if ultimately it is optimal (Section VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' universal embezzling states [12], such that for any tolerated non-zero error one can always prepare a pure state using a member of this family and zero communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' More amazingly, they showed that as the error tends to zero, it is roughly optimal since it scales nearly the same as if you add LOCC and allow the catalyst to be state dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This near optimality along with Hayden and Winter’s result has, understandably, largely ceased the study of en- tanglement transformations with zero communica- tion, because when one needs entanglement trans- formations without communication, one uses em- bezzlement [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='1 It is however not clear what is the necessary error for embezzlement to become near optimal, which could be relevant in practical settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Indeed, for any tolerated error, it is easy to find sufficient conditions on pure states to be con- verted with no catalyst at all (Example 2 of Section III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This is an indication that we also do not under- stand embezzling and catalytic convertibility suffi- ciently well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 1 The notable exceptions to this halted topic of research has been the consideration of special embezzling families [15] and the correlated sampling lemma [16], which may be viewed as a variation of embezzling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Summary of Results The primary aim of this work is to provide tighter lower bounds on the error in pure state entangle- ment convertibility with zero communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' A high level comparison of our results to the afore- mentioned work on this topic are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This depicts a ‘one-shot resource tradeoff’ region that must contain the ‘true’ one-shot resource trade- off surface for a given pure state conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Hay- den and Winter’s result provides a lower bound on the achievability independent of the amount of shared maximally entangled states, but their result can be too loose when considering zero communi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' van Dam and Hayden’s result provides an outer bound on the achievability surface on the face pertaining to LOSE, but their result in fact can be too loose when the error is not sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In this work, our results allow one to exactly solve the minimal error in the zero communication set- ting and also provide significantly tighter bounds than quantum embezzling for a relevant region on the LOSE face (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' To formally establish our results, we reduce the class of questions regarding optimal pure state con- version to optimization problems that only concern probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This is because of a bijec- tion between the equivalence classes of pure states under local unitaries— which are defined solely by their Schmidt coefficients— and the probability simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We do this by showing the optimal fidelity of pure state transformations with local unitaries is efficiently computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Of course, in general one would not expect local unitaries to be the op- timal strategy and we build on this result to present a non-convex optimization program over an opti- mization variable with bounded dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' An im- mediate corollary of this result is the impossibility of pure state conversions with zero communication for negligible error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We also present efficient com- putable upper bounds on the achievable error using a semidefinite programming (SDP) relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We also show that in the case where either the seed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' initial) or target state is a two-qubit state, the local unitary strategy is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' However, we can show for larger dimensions this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Having established general properties in the sin- gle copy case, we move to the multiple copy case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' where the seed and/or target state is of inde- pendent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=') form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This is standard in determining the rate of con- verting one state to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In particular, we con- sider dilution and distillation where the seed state or target state respectively is many copies of a max- imally entangled state and show these are convex optimization programs and may be seen as involv- Entanglement LOSE 1 Tolerated 0 Error ε Gap LOCC Classical Communication3 ing the Ky-Fan norms when extended to the regime where they are not a norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Lastly, in a sense ex- tending our earlier two-qubit results, we establish that if the target state is an n−fold copy of a two- qubit entangled state and the seed state’s Schmidt rank is less than the target state, then local unitaries are the optimal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Finally, given these results, we turn our atten- tion to quantum embezzlement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We begin by not- ing that the correspondence between Schmidt co- efficients and probability distributions means that quantum embezzlement implies a classical equiv- alent we call randomness embezzlement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We then proceed to use our new tools to consider the prob- lem of catalyzed pure state conversion under local unitaries, in effect a generalization of embezzling, and compare it to embezzling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We show in par- ticular that at least in general the optimality of the embezzling states is only for very small errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In- deed, we show for reasonable tolerable errors, the embezzling state may have a Schmidt rank of many orders of magnitude larger than a simple catalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This may have practical relevance and strongly re- fines our understanding of pure state transforma- tions under LOSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Organization of the Paper The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In Sections II and III we present the necessary notation and background re- spectively to understand the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In Section IV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' we Make explicit the correspondence between pure states under LU and the probability sim- plex and note this implies the existence of a classical variation of embezzlement (Theorem 2) Prove our equation for fidelity of state con- version under local unitaries (Theorem 5) and our optimization for fidelity of state conver- sion under local operations and shared ran- domness (Theorem 6) Establish computable upper bounds on the fi- delity of state conversion under LOSR (Theo- rem 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In Section V we present the results where the tar- get or seed state is of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In Section VI we discuss catalysts under local unitaries, the general frameworks that includes quantum embezzlement,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In Section VII we discuss why our theory does not generalize beyond bipartite pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' NOTATION Our notation largely aligns with standard texts [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In this paper we consider finite dimen- sional quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Given n ∈ N, we define [n] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' A finite dimensional Hilbert space will be labeled with a capital roman letter, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' A, B, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' As they are finite dimensional, these Hilbert spaces may be identified by the isomorphism A ∼= Cd where d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The space of linear maps from a Hilbert space A into itself, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' the space of endo- morphisms, is denoted L(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The space of quan- tum states, or density matrices, with respect to a Hilbert space A, is the space of positive semidefi- nite operators with unit trace, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' D(A) := {ρ ∈ L(A) : ρ ⪰ 0 & Tr(ρ) = 1} where ⪰ is the L¨owner order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' If a quantum state is a joint state over multi- ple Hilbert spaces, we will use a subscript to specify this, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' ρAB ∈ D(A ⊗ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We say a quantum state ρA ∈ D(A) is pure if Tr � ρ2 A � = 1 which is equiva- lent to there being a unit vector |ψ⟩ ∈ A such that ρA = |ψ⟩⟨ψ|, where we are using bra-ket notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For this previous reason, we generally just specify a pure state by |ψ⟩A, or ψ if we are considering its density matrix representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We denote the space of pure states S(A), where S stands for unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' A state is classical if it is diagonal in a specific choice of basis for L(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We call this the computa- tional basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The space of classical probability dis- tributions over d elements, the probability simplex which we denote P(d), may be viewed as the set of non-negative d−dimensional vectors that sum to one or the set of diagonal density matrices in the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' To distinguish between the two, we write P for the matrix version and p for the vector version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We also define the set of entry- wise decreasing probability distributions over d el- ements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' elements of the form p↓(1) ≥ p↓(2) ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' ≥ p↓(d), by P↓(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' A quantum channel E ∈ C(A, B) is a (lin- ear) completely positive, trace preserving map E : L(A) → L(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Any quantum channel admits an isometric representation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' given E ∈ C(A, B), there exists a Hilbert space E such that |E| ≤ |A||B| and isometry V : A → B ⊗ E such that Φ(X) = TrE(VXV†) where TrE is the partial trace on the E space and X† is the Hermitian conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Given the space of linear operators from A ∼= Cd to B ∼= Cd′, L(A, B), the vec mapping vec : L(A ⊗ B) → A ⊗ B is defined by vec(|i⟩ ⟨j|) = |j⟩ ⊗ |i⟩ where {|i⟩}i∈[d] and {|j⟩}j∈[d′] are the computa- tional bases for A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This choice of vec mapping satisfies the identity (XT 1 ⊗ X0) vec(Y) = vec(X0YX1) , (1) where X0 ∈ L(A0, B0), X1 ∈ L(A1, B1), and Y ∈ L(B1, B0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The vec mapping is also an isometry in the sense that for all X, Y ∈ L(A, B), ⟨X, Y⟩ = ⟨vec(X), vec(Y)⟩ , 4 where ⟨·, ·⟩ on the L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' is the inner product on L(A, B) defined by ⟨X, Y⟩ = Tr � X†Y � and the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' is the inner product on vectors A ⊗ B defined by ⟨ψ|φ⟩ = ∑i ψ(i)φ(i) where · is the conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' BACKGROUND & MOTIVATION Throughout this section we fix A ∼= Cd, B ∼= Cd′ for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Fidelity The fidelity is a standard measure of similarity between two positive semidefinite oper- ators R, S ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' F(R, S) = ��� √ R √ S ��� 2 1 = Tr ��√ SR √ S �2 , (2) where the square root of a positive semidefinite operator is defined in the standard fashion on its spectral decomposition and ∥ · ∥1 is the Schatten 1−norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It satisfies various properties that will be relevant for this work which we summarize here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' All of these may be verified by direct calculation or by referring to standard texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proposition 1 (Summary of Fidelity Properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let ρ, σ ∈ D(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The following hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 0 ≤ F(ρ, σ) ≤ 1 where the upper bound is saturated if and only if ρ = σ and the lower bound saturates if and only if their images are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The fidelity is isometrically invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' given isometry V : A → B, F(VρV†, VσV†) = F(ρ, σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The fidelity satisfies data-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' That is, for any quantum channel E ∈ C(A, B), F(ρ, σ) ≤ F(E(ρ), E(σ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' If both states are pure, F(|φ⟩⟨φ| , |ψ⟩⟨ψ|) = | ⟨ψ|φ⟩ |2 , and if one state is pure F(|φ⟩⟨φ| , σ) = ⟨φ| σ |φ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' If both states are classical, P, Q ∈ P(d), then the fidelity reduces to the square of the Bhat- tacharyya coefficient: F(P, Q) = � � ∑ i∈[d] � p(i)q(i) � � 2 = BC(p, q)2 , where p(i) = P(i, i) and likewise for Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Given pure states with the same eigenbasis and real amplitudes, |ψ⟩ = ∑x � p(x) |x⟩, |φ⟩ = ∑x � q(x) |x⟩ , the fidelity reduces to the square of the Bhattacharyya coefficient of the probability distributions defined by the amplitudes: F(|φ⟩⟨φ| , |ψ⟩⟨ψ|) = BC(p, q)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We also note that in all of these definitions there is a pesky squaring that effectively we don’t care about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For this reason we could define the square root fidelity: √ F(R, S) := � F(R, S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note the square root fidelity could be viewed as the quantum extension of the Bhattacharyya coef- ficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Norms In defining the fidelity we used the Schatten 1−norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' More generally, there are the Schatten p−norms which for X ∈ L(A, B) may be defined as ∥X∥p := ∥σ(X)∥p where σ(X) is the ordered vector of singular values of X, σ1(X) ≥ σ2(X) ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' ≥ σrank(X)(X) and it is being evalu- ated under the Lp−norm where p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The in- finity norm, ∞−norm, is limp→∞ ∥X∥p = ∥X∥∞ = maxi σi(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The infinity norm was generalized to the Ky Fan k−norms ∥X∥(k) := ∑ σi(X) for 1 ≤ k ≤ min{d, d′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The Ky Fan norms have relevance in measuring entanglement [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' A generalization of the Ky Fan and Schatten norms together is given by the (k, p)−norms [21] ∥X∥(k,p) := � � ∑ i∈[k] σi(X)p � � 1/p , (3) which also have use in measuring entanglement of pure states [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Much like is common to do for the Schatten p−norms, we can extend the (k, p)−norms to p > 0 with the caveat they won’t be norms as they won’t in general satisfy subaddi- tivity (the triangle inequality) for p ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Entanglement Theory A bipartite quantum state ρAB is separable if there exists n ∈ N, p ∈ P(n), {σi A}i∈[n] ⊂ D(A), and {τi B}i∈[n] such that ρAB = ∑ i∈[n] p(i)σi A ⊗ τi B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Otherwise the state is entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' As a pure state |ψ⟩⟨ψ|AB is defined by a unit vector, this reduces to a pure state is separable, referred to product in this setting, if and only if there exists |φ⟩A , |ϕ⟩B such 5 that |ψ⟩ = |φ⟩A ⊗ |ϕ⟩B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' While this is sufficient for determining if a bipartite pure state is entangled, there is also a notion of ‘how’ entangled a state is in terms of Schmidt rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Every bipartite pure state |ψ⟩AB admits a unique (up to re-ordering) decom- position of the form |ψ⟩AB = ∑ i∈[k] � p(i) |ui⟩A ⊗ |vi⟩B , (4) where k = max{d, d′}, p ∈ P(k) and {|ui⟩}i∈[k], {|vi⟩}i∈[k] are orthonormal bases of A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The � p(i) > 0 terms are re- ferred to as the Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The Schmidt rank of |ψ⟩AB, SR(|ψ⟩) = supp(p), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' the num- ber of Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This may be viewed as a measure of entanglement in the sense that the Schmidt rank of a product state is 1 and the maxi- mally entangled state |Φ+⟩CdCd = 1 √ d ∑i |i⟩Cd |i⟩Cd has Schmidt rank d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We define the set SR(d) := {|ψ⟩ : SR(|ψ⟩) ≤ d}, where we note this set is in- dependent of the dimension the state is embedded in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Lastly we note a particularly nice property of pure states, known as Uhlmann’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Lemma 1 (Uhlmann’s Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Given ρ, σ ∈ D(A) and |ψ⟩ ∈ A ⊗ B such that TrB(ψ) = ρ, then F(ρ, σ) = max{| ⟨ψ|φ⟩ |2 : |φ⟩ ∈ A ⊗ B , TrB(φ) = σ} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' No-Go Theorems, Embezzling, & Motivation With the established background, we now present the previous results related to zero communication pure state transformations which we will discuss our results in relation to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The first is a lower bound on the number of qubits or classical bits necessary to convert between pure states [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' ([6, Theorem 8]) Consider a state transformation via channel E ∈ C(A ⊗ B, A ⊗ B) from seed state |φ⟩AB to target state |ψ⟩AB such that F(E(φ), ψ) ≥ 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then, independent of any amount of entanglement assistance, for δ = 8√ε, in the implementation of E, q qubits were exchanged where q ≥1 2 [∆δ(TrB(|φ⟩⟨φ|)) − ∆0(TrB(|ψ⟩⟨ψ|))] + log(1 − δ) , (5) where exp(∆ε(P)) = min rank( �P) · λmax( �P) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Tr � �P � ≥ 1 − ε �P = ΠPΠ [P, Π] = 0 Π2 = Π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Moreover, the bound given in (5) holds for a neces- sary amount of classical communication by multi- plying the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' by two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' While the above proposition is very powerful and implies two states with different Schmidt de- compositions cannot be perfectly converted with zero communication, it is not sufficient in every sce- nario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In particular, the following example shows that in certain cases Proposition 2 cannot eliminate any state from being able to be converted to a given target state with relatively high fidelities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Example 1 (On the necessity of communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Up to local unitaries, let the target state be |ψ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='54 |00⟩ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='02 |11⟩ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='44 |22⟩, the seed state be any state |φ⟩ = ∑i∈[k] � p(i) |vi⟩ |ui⟩, and assume we are interested in a state transformation E such that F(E(φ), ψ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='01, so δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' One may verify ∆δ(P) = log(|1| · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='44) < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='18, by removing the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='02 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='54 eigenvalues of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It may be shown [6] that ∆0(TrB(|ψ⟩⟨ψ|)) ≥ 0, and log(1 − δ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It follows that in this setting the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' of (5) is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, we have no proof from this bound that any transformation for any seed state which achieves this relatively high fidelity of 99% requires any communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' While the above example shows there are reason- ably small tolerated errors ε where Proposition 2 is not helpful, when the tolerated error is sufficiently small, it will imply the need for communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This sort of structure for sufficiently small ε also appears when considering quantum embezzlement [12], which may be seen as a solution to Proposition 2 implying communication is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Quantum embezzlement in effect shows one can make pure state transformations with zero communication to any non-zero error if they have the right sufficiently large entangled catalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' ([12]) Consider the family of catalyst states |µ(n)⟩A′B′ = 1 √Hn ∑n j=1 1√ j |j⟩A′ |j⟩B′ where Hn := ∑n i=1 n−1 is the Harmonic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For any ε > 0 and target bipartite pure state |ψ⟩AB with Schmidt rank m, for n > m1/ε there exist unitaries UAA′, WBB′ such that F(UAA′ ⊗ WBB′(|µ(n)⟩A′B′ |0⟩A |0⟩B), |µ(n)⟩A′B′ ⊗ |ψ⟩AB) ≥ 1 − ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Moreover, U, W are in effect permutations on the joint Schmidt bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' One can see quantum embezzlement implies a way to convert one pure state to another to non- zero error by picking a large enough catalyst and 6 then first ‘embezzling out’ the original state (un- computing |φ⟩ to |0⟩ |0⟩ via embezzling) and then ‘embezzling in’ the target state |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' What is perhaps most remarkable about the above approach is that it was shown in the original work that even if we allow LOCC and a state de- pendent catalyst, the error scales as Ω(1/ log(n)) whereas for the above result the errors scales as O(1/ log(n)), so as the error is driven down, em- bezzling is near optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' That is, as ε → 0, this strat- egy is effectively optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' However, just as with the discussion pertaining to Proposition 2, it’s clear em- bezzling isn’t necessary for reasonable error levels in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In fact, we show in the following ex- ample that for any non-zero error there exist states which can be converted without any catalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Example 2 (On the necessity of embezzling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' As noted, as ε → 0, embezzling is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' However, it is not in general clear at what point embezzling becomes necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This can be seen as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider ε ∈ (0, 1) and two probability distribu- tions p, q ∈ P(m) such that the BC(p, q)2 ≥ 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Define the seed state as |φ⟩ = ∑i∈[m] � p(i) |i⟩A |i⟩B and the target state as |ψ⟩ = ∑i∈[m] � q(i) |i⟩A |i⟩B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then we have F(|φ⟩⟨φ| , |ψ⟩⟨ψ|) = BC(p, q)2 ≥ 1 − ε , where we have used Item 5 of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' There- fore, given |φ⟩, it requires no communication or en- tanglement to generate |ψ⟩ to error ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In fact, as we show later (Proposition 4), this will be true for con- verting the set of states with Schmidt coefficients defined via p to the set of states with Schmidt coef- ficients defined via q in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Given these two examples, we see that while these results give strong characterizations of pure state transformations with zero communication, neither the need for communication by Proposition 2 nor the optimality of Proposition 3 when the error tends to zero give us a full understanding of this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It would therefore be of value to better un- derstand this task, and this is what the rest of this work addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' SINGLE COPY PURE STATE CONVERSION WITH ZERO COMMUNICATION Our primary goal of this section is to deter- mine the minimal error of conversion between pure states with zero communication, which would re- solve the gap presented in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' To do this, we will use the correspondence between the prob- ability simplex and Schmidt coefficients under lo- cal unitaries (LU), which we establish in the follow- ing subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We also note that this implies the existence of a classical equivalent of embezzling, which we call randomness embezzling (Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This correspondence motivates the idea that the optimal fidelity of pure state conversion under local unitaries is simply re-ordering the Schmidt coeffi- cients, which we in fact prove (Theorem 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We then use the local unitary result to establish a bounded but non-linear optimization program that deter- mines the optimal achievable fidelity under conver- sion via local operations and shared randomness (LOSR), which does not require shared randomness (Theorem 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We end the section by discussing the relationship between the LU and LOSR strategies and introducing an SDP relaxation for efficiently es- tablishing upper bounds on the achievable fidelity of pure state conversions under LOSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Correspondence Under Local Unitaries between Schmidt Coefficients and the Probability Simplex In this subsection we establish the bijection be- tween Schmidt coefficients, which define the equiv- alence classes of bipartite pure states under local unitaries, and the probability simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' One reason for this is because the rest of the results of this work might be best seen as verifying that in the zero com- munication setting this correspondence is all that matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Indeed, we will see this in the subsequent subsections which show that the minimal fidelity error of pure state transformations under zero com- munication will always be functions of only the Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Up to local unitaries, any pure quan- tum state is of the form |ψ⟩AB = ∑ i∈[k] � p↓(i) |i⟩A ⊗ |i⟩B , where p↓(i) ≥ p↓(i + 1) for all i ∈ [k − 1], k = max{d, d′}, p↓ ∈ P↓(k), and {|i⟩} is the computa- tional basis in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In other words, there exist both equivalence classes on pure states under local unitary operations in terms of Schmidt coefficients and ordered Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider |ψ⟩AB = ∑j∈[k] � p′(j) ��uj � ⊗ ��vj � as decomposed in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Now fix the permutation π on [k] such that p′(π−1(i)) ≥ p′(π−1(i + 1)) for all i ∈ [k − 1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' π re-labels p′ so that it is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Define the unitaries UA = ∑j∈[k] |π(j)⟩ � uj ��, WB = ∑j∈[k] |π(j)⟩ � vj ��, which may be verified to be uni- taries by direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then (UA ⊗ WB) |ψ⟩AB 7 will be of the form given in the proposition state- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Finally, we could make this argument for any pure state without ordering the Schmidt coefficients to get one set of equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' As such, under local unitaries, we can define equivalence classes of pure states in terms of ordered or non-ordered Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The space of (representatives of the equivalence class of) ordered Schmidt coefficient pure states with Schmidt rank bounded by d is given by SR↓(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' That is, if |ψ⟩ ∈ SR↓(d), then |ψ⟩ = ∑i∈[d] � p↓(i) |ui⟩ |i⟩ |i⟩ where p↓ ∈ P↓(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We can use the previous proposition to relate the (ordered) probability simplex over d elements to to the equivalence classes of (ordered) Schmidt de- compositions with Schmidt rank bounded by d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider the functions vec(√·) : L(Cd) → Cd ⊗ Cd and vec−1(·⊙2) : Cd ⊗ Cd → L(Cd) where ·⊙2 is the entry-wise square of a vec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' These functions define a bijection between P(d) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' P↓(d)) and the space of equivalence classes of Schmidt decompositions under local unitaries with Schmidt rank bounded by d (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' the space SR↓(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We prove it via direct calculation for P(d) and the space of Schmidt decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The proof in the other case works the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let C ∼= Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' First, consider p ∈ P(d) which we write in its den- sity matrix form, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' P = ∑i∈[d] p(i) |i⟩⟨i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then vec( √ P) = vec � � ∑ i∈[d] � p(i) |i⟩⟨i| � � = ∑ i∈[d] � p(i) |i⟩C ⊗ |i⟩C′ , which is in the specified equivalence class by ap- plying an isometries that take the computational bases from C, C′ to A, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In the other direction, take the Schmidt decomposition in the purified basis, |ψ⟩AB = ∑i∈[d] � q(i) |i⟩A ⊗ |i⟩B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We can convert the A space to C via the channel FA→C(·) := V† · V + (1 − V†V) · (1 − V†V) , where V = ∑i∈[d] |i⟩A ⟨i|C is the isometry that takes the C space to the A space as |A| ≥ |C| by assump- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The same type of conversion holds for the B and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, we have (up to equivalences) |ψ⟩AB = ∑i∈[d] � q(i) |i⟩C |i⟩C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then, vec−1(|ψ⟩·2) = vec−1( ∑ i∈[d] q(i) |i⟩C |i⟩C′) = ∑ i∈[d] q(i) |i⟩⟨i|C , where in the last line we used that C′ ∼= C so that L(C, C′) ∼= L(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The reason this is useful is it draws equivalence between the equivalence classes of entangled states in terms of Schmidt coefficients and probability dis- tributions under fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider |φ⟩ = ∑i∈[d] � p(i) |i⟩A |i⟩B, |ψ⟩ = ∑i∈[d] � q(i) |i⟩A |i⟩B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then F(|φ⟩⟨φ| , |ψ⟩⟨ψ|) = BC(p, q)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' First note V : |i⟩A → |i⟩A |i⟩B is an isom- etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus by isometric equivalence of fidelity (Item 2 of Proposition 1), we have F(|φ⟩ , |ψ⟩) = F(V |φ′⟩ V†, V |ψ′⟩ V†) where the primed versions just remove the B register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then using Item 6 of Proposition 1 completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Randomness Embezzling Before moving forward, we note that independent of the focus of this work, this equivalence between Schmidt coefficients and the probability simplex means that the proof of quantum embezzlement also proves the existence of a classical version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Specifically, if one looked at the proof of quantum embezzlement [12], one would only need to note the starting and ending state they bound the fidelity between are in the computational basis locally and use F(|ψ⟩ , |φ⟩) = |⟨ψ, φ⟩|2 = ���⟨ √ P, � Q⟩ ��� 2 = � ∑ i � p(i)q(i) �2 =BC(p, q)2 , which follows the same argument as the previous few propositions, to ultimately conclude the same proof bounds a classical equivalent (Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' As we did not present the proof for embezzlement of quantum states, we present the proof of embezzle- ment of probability distributions in full for clarity in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For any ε > 0 and target probabil- ity distribution P ∈ P(m), the catalyst distribution Rn := 1 Hn ∑n j=1 1 j |j⟩⟨j| is such that for n > m1/ε there exists a unitary representation of a basis relabeling Uf of the joint distribution such that F(Uf (Rn ⊗ |0⟩⟨0|)U† f , Rn ⊗ P) ≥ 1 − ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 8 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 3: Comparison between embezzlement of classical distributions and quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' (a) The embezzlement of classical distributions happens within one lab and a local permutation of the joint computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' (b) The embezzling of quantum states happens across two labs where each party applies the permutation of the joint computational basis on their local halves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We note the major difference between random- ness and quantum embezzlement is the role of lo- cality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In the classical case there is a single party and the distribution is not bipartite, both of which remove the notion of locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' These differences are non-trivial: one cannot construct a non-local classi- cal equivalent of embezzling that at the same time demands that the catalyst remains decoupled as in Proposition 3, and one cannot find a quantum equivalent of the non-local classical variation that one can implement as follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' As it is not central to the rest of this work, we pro- vide an extended discussion of this nuance for the interested reader in Appendix A after the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Pure State Conversion under Local Unitaries Having established the relationship between the equivalence classes of pure states in terms of Schmidt coefficients and the probability simplex, we now show the optimal strategy for converting one pure state to another under local unitaries is simply re-labeling the Schmidt basis so the order- ing of the Schmidt coeffficients is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This is not necessarily surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It is not clear what more one could do, and indeed this is the strategy that is used to implement quantum embezzlement [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For intuition, we quickly show the equivalent result in the classical setting first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let p↓, q↓ ∈ P↓(d) Then for any i ∈ [d] and d − k ≥ k ≥ 1, 1 ≥ � p↓(i) � q↓(i) ≥ � p↓(i) � q↓(i + k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This just follows from the fact if 1 ≥ p↓(i) ≥ p↓(i + 1) and the same for q↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Given p, q ∈ P(d), max π∈Sd BC(p, πq) = BC(p↓, q↓) , where Sd is the set of permutations on d elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' All we are looking for is the permutation of the elements of q such that BC(p, πq) is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We can apply the permutation σ such that σPσ† = P↓, the matrix representation of p↓, to both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' By the isometric equivalence of fidelity and that per- mutations are group, the problem is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' That is, we can consider max π∈Sd BC(p↓, πq) = max π∈Sd ∑ i∈[d] � p↓(i) � q(π(i)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It immediately follows from Proposition 7 that the optimal π is the one that takes Q to Q↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This com- pletes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The idea is then to lift this result to quantum states optimized over unitaries and then use this with Uhlmann’s theorem to lift to the bipartite set- ting with local unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The main challenge is minimizing over unitaries in the lift of the pre- vious result as now we have to deal with non- commutivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This is done by reducing optimizing over unitaries to optimizing over permutations us- ing the Birkhoff-von Neumann Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Lemma 3 (Birkhoff-von Neumann Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Given a linear operator X ∈ L(Rd), X is bis- tochastic (non-negative entries such that each col- umn and each row sums to one) if and only if there exists a probability distribution p ∈ P(|Sd|) such that X = ∑ π∈Sd p(π)Vπ , where Vπ(i, j) := δi,π(j) are permutation matrices and δi,j is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' That is, a linear oper- ator is bistochastic if and only if it is a convex com- bination of permutation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let ρ, σ ∈ D(Cd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then max U F(ρ, UσU†) = F(P↓, Q↓) , where P↓ = ∑i νi(ρ) |i⟩⟨i| and likewise for Q↓ but with respect to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In other words, the fidelity be- tween ρ and σ maximized over unitaries is equal to the fidelity of their ordered eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=" First, by the isometric invariance of fidelity (Item 2 of Proposition 1), F(ρ, σ) = F(P↓, VσV†) P 10)(0| Ur RnA A UAA' A' A' B' B' M BB' B B 89 where V is the unitary such that VρV† = P↓ = ∑i νi(ρ) |i⟩⟨i|." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' As unitaries are closed under multi- plication and conjugate transpose, max U F(ρ, UσU†) = max U′ F(P↓, U′VσV†U′†) as the optimal U′ = U⋆V† where U⋆ is the opti- mizer for the L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' of the equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore we just define Q ≡ VσV† and focus on solving the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, we are interested in maxU F(P↓, UQU†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Denote the spectral decomposition of Q = ∑j q(j) ��φj �� φj ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note that without loss of gener- ality, we may write U = ∑j ��ψj � � φj �� for some orthonormal basis { ��ψj �}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, UQU† = ∑j q(j) ��ψj �� ψj ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Furthermore, define P(X) := ∑i |i⟩⟨i| X |i⟩⟨i|, which is the pinching, or dephasing, channel onto the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then P(UQU†) =∑ i,j |i⟩⟨i| q(j) ��ψj �� ψj �� |i⟩⟨i| =∑ i,j q(j)| � i ��ψj � |2 |i⟩⟨i| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note that in contrast, P↓ is invariant under this pinching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Combining these points,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' max U F(P↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' UQU†) ≤ max U F(P↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' P(UQU†)) = max U Tr ��√ P↓P(UQU†) √ P↓ �1/2�2 = max {|ψj⟩} Tr �� ∑ j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='�i q(j) � p↓(i′)p↓(�i) ��� � i ��ψj � ��� 2 ��i′� � i′��i � � i ����i � � �i ��� �1/2 �2 = max {|ψj⟩} Tr � � � ∑ j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='i q(j)p↓(i)| � i ��ψj � |2 |i⟩⟨i| �1/2� � 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' where the inequality is the data-processing inequal- ity (Item 3 of Proposition 1) with the pinching chan- nel along with the invariance of P↓ under this chan- nel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' the first equality is using the definition of fi- delity (2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' the second is just expanding everything,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' and the third is collapsing the implicit Kronecker deltas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Now note the following trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We can define the square matrix A via its elements: A(j, i) := | � i ��ψj � |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We know 0 ≤ | � i ��ψj � |2 ≤ 1, ∑i | � i ��ψj � |2 = 1, and ∑j | ⟨i|j⟩ |2 = 1 as {|i⟩} and { ��ψj �} are orthonormal bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It follows that A is a bistochastic matrix by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, by the Birkhoff-von Neumann Theorem (Lemma 3), A = ∑π∈Sd r(π)Wπ where Wπ is the permutation matrix for π and r is a probability distribution over the permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus, plugging this back in to what we started with, max {|ψj⟩} Tr � � � ∑ j,i q(j)p↓(i)| � i ��ψj � |2 |i⟩⟨i| �1/2� � 2 = max r Tr � � � ∑ j,i,π q(j)p↓(i)r(π)Wπ(j, i) |i⟩⟨i| �1/2� � 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Now note every permutation matrix is the iden- tity matrix with columns permuted, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' π = � eT π(0) eT π(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' eT π(d−1) �T , where ei := |i⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It fol- lows that ∑ π∈Sd r(π)Wπ(j, i) =∑ π r(π)1{Wπ(j, i) = 1} =∑ π r(π)1{π(j) = i} =: Pr r [π(j) = i′] , where 1{A} is the indicator function for an event and the second equality is because W(j, i) = 1 if and only if π(j) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We stress the final definition is a function of the choice of r and j, i and form a joint probability over (j, i) as ∑j Prr[π(j) = i] = 1 = ∑i Prr[π(j) = i] and every element is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This simplifies the problem to max r Tr � � � ∑ j,i,π q(j)p↓(i)r(π)Wπ(j, i) |i⟩⟨i| �1/2� � 2 = max r � ∑ i � p↓(i) � ∑ j � q(j) Pr r [Π(j) = i] ��2 , where we have just grouped terms and used that the operator is diagonal, so we can apply the square root entry-wise and take the sum to compute the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' So we want to determine the maximal distri- bution r, but we can show this is achieved by element-wise optimizing the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note � p↓(1) is the largest element and bounded above by 1, so we want to multiply it by the largest value ∑j � q(j) Prr[π(j) = i] can take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' � q(j) ≤ 1 for all j and ∑j Prr[π(j) = i] = 1 so the largest value 10 this sum can take is maxj � q(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note if we pick a different distribution each term will be smaller than it could be by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This means we choose r such that all non-zero probability permu- tations map argmaxj q(j) to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We then have the same problem as initially but with � p↓(2) serving the largest element and q not containing its largest element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Doing the argument recursively, we con- clude the optimal distribution r has unit probability on permutation σ such that ∑i q(σ(i)) = q↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus, max r � ∑ i � p↓(i) � ∑ j � q(j) Pr r [Π(j) = i] ��2 = � ∑ i � p↓(i) � q↓(i) �2 =BC(p↓, q↓)2 =F(P↓, Q↓), where the first equality is by the preceding explana- tion and the last two are using Item 5 of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note this means we have established an upper bound as we used the data processing inequality at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' However, this is clearly achievable by picking by the permutation unitary that maps σ to Q↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We now can use the above lemma to establish the pure state property we are actually interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For notational simplicity, we define the following notation: FLU(ρ, σ) := max U,V F(ρ, (U ⊗ V)(σ)) , which is without loss of generality unitaries as we can just trivially embed the smaller dimensional state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' FLU(|ψ⟩ , |φ⟩) = F(P↓, Q↓) , where P↓ is the distribution defined by the decreas- ing Schmidt coefficients of |ψ⟩ and likewise for Q↓ and |φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In other words, the optimal fidelity of con- verting |φ⟩ to |ψ⟩ via local unitaries is given by the fidelity of their ordered Schmidt coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Up to local unitaries, |ψ⟩ = ∑i � p↓ |i⟩ |i⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore without loss of generality, that can be taken as our target state by allowing free local uni- taries on the seed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We can take the seed state to be of the form |φ⟩ = ∑i � q(i) |i⟩ |i⟩ by the same argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then by assumption, we are interested in maxU,V F(|ψ⟩ , (U ⊗ V) |φ⟩) with the specified forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note TrB((U ⊗ V) |φ⟩⟨φ| (U ⊗ V)†) =∑ i,i′ � q(i)q(i′)U |i⟩ � i′�� U† Tr � V |i⟩ � i′�� V†� =∑ i q(i)U |i⟩⟨i| U† =: UQU†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Now for any unitary U we define the following pu- rification ���w|U� := vec( � UQU†) =(U ⊗ U) vec( � Q) = (U ⊗ U) |φ⟩ , where we have used � UQU† = U√QU† and the vec map identity (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Now we have F(P↓, UQU†) = max |w′⟩ F(|ψ⟩ , ��w′�) = max V F � |ψ⟩ , (1 ⊗ V) ���w|U�� = max V F(ψ, (U ⊗ VU) |φ⟩) , (6) where the first equality is by Uhlmann’s theorem (Lemma 1), the second is because all purifications of a given operator are unitarily equivalent on the purifying space, so there exists a V such that (1 ⊗ V) ���w|U� = |w′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The final line is just expanding the definition of ���w|U� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It follows, max W,V F(|ψ⟩ , (W ⊗ V) |φ⟩) = max U,V′ F(|ψ⟩ , (U ⊗ V′U) |φ⟩) = max U,V′ F(|ψ⟩ , (1 ⊗ V) ���w|U� ) = max U F(P↓, UQU†) =F(P↓, Q↓) , where the first equality is because unitaries are closed under multiplication and the optimizations are independent, the second and third are both by (6) for clarity, the third is because unitaries are closed under conjugation and then the final equal- ity is by applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This means under local unitaries, it is efficient to compute the optimal fidelity and that in fact the op- timal strategy is simply Alice and Bob re-ordering the basis so that the Schmidt coefficients are in the 11 same relative ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It also follows from Item 1 of Proposition 1 that unless all the Schmidt coef- ficients are equal, the fidelity cannot be one under local unitary strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Pure State Conversions under Local Operations and Shared Randomness While the previous section is nice in that it finds an efficient way of calculating the optimal conver- sion strategy under local unitaries, it would be nat- ural to ask if local operations can do better than lo- cal unitaries as it is a much more general class of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In fact, we can see that it must do bet- ter in some cases in a trivial manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider the target state |ψ⟩ and the seed state |φ⟩ = |ψ⟩ ⊗ |ζ⟩ where |ζ⟩ is not product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Under local unitaries this transformation isn’t possible to arbitrary precision because of |ζ⟩, but of course in reality the parties could trace out whichever portion(s) of |ζ⟩ they hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus, we need a theory of transformations under local operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note that this trivial example we have given would not be resolved by local mixed unitary strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Indeed, we begin by noting that local mixed unitary strategies cannot ever outperform lo- cal unitary strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let |ψ⟩ be the target state and |φ⟩ be the seed state and only optimize over Alice and Bob using mixed unitary channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then the optimal is the same as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Letting EU, FW be local mixed unitary maps, max EU,FW F(ψ, (EU ⊗ FW)(φ)) = ⟨ψ| (EU ⊗ FW)(φ) |ψ⟩ = � U,W ⟨ψ| (U ⊗ W)(φ) |ψ⟩ dU dW ≤ � U,W max U,W ⟨ψ| (U ⊗ W)(φ) |ψ⟩ = max U,W ⟨ψ| (U ⊗ W)(φ) |ψ⟩ =F(P↓, Q↓) , where the first equality is by Item 4 of Proposition 1, the second is letting the mixed unitary map be for any probability measures dU,dW over the unitary group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The inequality is because the inner product is real and so it is lower bounded by the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The second to last equality is by linearity, and the final equality is by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Noting that a specific choice of local unitaries is a special case of mixed unitary channels completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The above tells us that we must escape the use of unitaries to improve our bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note however that in general the only maps that preserve pure states are isometries, and our results so far have been in terms of pure states, so we need to main- tain this structure to build on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For this reason, the following proof will make use of the isometric representation of quantum channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For notational simplicity, we define the optimal fidelity of conversion under local operations and shared randomness (LOSR) fidelity FLOSR(ρ, σ) := max µ,Eλ,Fλ F(ρ, � (Eλ ⊗ Fλ)(σ)dµ(λ)) , where µ is a probability measure over an index set for sets of local channels {Eλ} and {Fλ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Similarly, we can define optimal fidelity of conversion under local operations (LO) as FLO(ρ, σ) := max E,F F(ρ, E ⊗ F)(σ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' With these defined, we prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' FLOSR(|ψ⟩ , |φ⟩) = FLO(|ψ⟩ , |φ⟩) = max P′∈P(Σ) F((P ⊗ P′)↓, Q↓ embed) , where |Σ| ≤ SR(|φ⟩) · SR(|ψ⟩), P is the probability distribution defined by |ψ⟩’s Schmidt coefficients and likewise for Qembed with the Schmidt coeffi- cients of |φ⟩ except the distribution is embedded into the joint space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The first equivalence follows similarly to the mixed unitary case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Clearly the class of LOSR strategies is more general than the class of LO strategies, so we just need to show LOSR is only as strong as LO here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' FLOSR(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' ψ) =F � ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' � (Eλ ⊗ Fλ)(φ)dµ(λ) � = � ⟨ψ| (Eλ ⊗ Fλ)(φ) |ψ⟩ dµ(λ) ≤ � max E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='F [⟨ψ| (E ⊗ F)(φ) |ψ⟩] dµ(λ) = max E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='F ⟨ψ| E ⊗ F)(φ) |ψ⟩ =FLO(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' ψ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' where the first equality is by definition and denot- ing the optimizers by µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' {Eλ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' {Fλ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' the second is by linearity of the Lebesgue integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' the inequal- ity is because ⟨ψ| (E ⊗ F)(φ) |ψ⟩ is a real number for any choice of local channels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' the third equality 12 is because µ is a probability measure that is now independent of the argumenbt of the integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' and the final equality is by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This proves the reduction of LOSR to LO if the target state is pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Next, we bound the dimension of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We want to consider maxE,F F(ψ, (E ⊗ F)(φ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Without loss of generality, we assume the local spaces are ‘com- pressed’ such that din := SR(|φ⟩) so that E, F both act on L(Cdin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We now show that without loss of generality we may restrict the output dimension of E, F to be dout := SR(|ψ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This is just because we can project onto the support of the marginal of |ψ⟩ on both local spaces, so we can restrict the local maps to this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Formally, this can be seen as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider arbitrary E, F and let |ψ⟩ = ∑i � p(i) |i⟩ |i⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Define ΠP := ∑i:p(i)>0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' the projector onto the support of TrB(ψ) = TrA(ψ), where the equality is up to the change in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note rank(ΠP) = Schmidt(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' By construction, (ΠP ⊗ ΠP) |ψ⟩ = |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, F(ψ, (E ⊗ F)(φ)) = ⟨ψ| (E ⊗ F)(φ) |ψ⟩ = Tr[|ψ⟩⟨ψ| (E ⊗ F)(φ)] = Tr � ψΠ⊗2 P (E ⊗ F)(φ)Π⊗2 P � , where in the first equality we have used Item 4 of Proposition 1 and the other two use cyclicity of trace along with invariance of ψ under the projec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Now we can expand, Π⊗2 P (E ⊗ F)(φ)Π⊗2 P =∑ k,l ΠPAk ⊗ ΠPBkφA† kΠP ⊗ B† l ΠP ≡(EΠ ⊗ FΠ)(ψ) , where {Ak}, {Bl} are the Kraus operators of E, F respectively and EΠ, FΠ are CPTNI maps defined by {ΠPAk}, {ΠPBl} respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note this equiva- lence holds as (ΠAk)† = A† kΠP since Π† P = ΠP so it is CP and it is TNI because ∑ k (ΠPAk)†(ΠPAk) =∑ k A† kΠPAk ≤∑ k A† k1Ak = 1 , where we used Π2 P = ΠP in the first equality, ΠP ≤ 1 and that E is CP in the inequality, and that E is TP in the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' An identical ar- gument holds for FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This proves the optimizer is achieved with CPTNI maps T(L(Cdin), L(Cdout)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Finally, we can lift EP, FP to being CPTP, denoted �E, �F ∈ T(L(Cdin), L(Cdout)) by adding one Kraus operator, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' for EP add the Kraus operator Z ∈ L(Cdin, Cdout) where Z†Z = (1 − ∑k A† kΠAk) ≥ 0 which always exists by definition of the space of positive semidefinite operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' By linearity, F(ψ, (E ⊗ F)φ) = Tr[ψ(EΠ ⊗ FΠ)(φ)] ≤ Tr � ψ( �E ⊗ �F)(φ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore without loss of generality the optimal channels are E, F ∈ C(Cdin, Cdout).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note this means that Rank(JE) ≤ dindout and likewise for JF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We now derive the equation using the isometric representation of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' max E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='F F(ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' (E ⊗ F)(φ)) =⟨ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' (E ⊗ F)(φ)⟩ = max V1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='V2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='|ζ⟩ |⟨ψ| ⟨ζ| (V1 ⊗ V2) |φ⟩|2 = max U1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='U2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='|ζ⟩ ���⟨ψ| ⟨ζ| (U1 ⊗ U2) |φ⟩ |0⟩E1 |0⟩E2 ��� 2 = max U′ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='U′ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' ���ζp′ � ���⟨ψ| � ζp′ ��� (U′ 1 ⊗ U′ 2) |φ⟩ |0⟩E1 |0⟩E2 ��� = max P′ F((P ⊗ P′)↓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Q↓ embed) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' where the second line is because there exists an iso- metric representation of each channel which means (V1 ⊗ V2)(φ) is a pure state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' so we can apply Uhlm- man’s theorem to find a purification of |ψ⟩ that sat- urates the bound,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' but as |ψ⟩ is already pure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' any purification will be a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The third line is because we can always convert an isometry into a unitary on the appropriately large space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The fourth line means that ζp′ = ∑i′ � p′(i) |i⟩ |i⟩, which can always be achieved by local unitaries on the E1 and E2 spaces, which result on new unitaries on the other side but the same maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The fi- nal equality is just using Theorem 5 and we write Qembed to stress it is defined over the whole alpha- bet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Lastly, as we established bounds on the ranks of the local maps Choi matrices, we have bounds E1, E2 ≤ dindout, which justifies the maximum and tells us how large of a system we have to consider in the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It is useful to see how this result works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It in ef- fect shows the following equivalence of conversion when measured under fidelity |φ⟩ −→ LO |ψ⟩ = max |ζ⟩ � |φ⟩ −→ LU |ψ⟩ ⊗ |ζ⟩ � , which can be viewed both by proof and via intu- ition as a special case of the isometric representa- tion of a channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Moreover, it is easy to see in this 13 form how it handles our motivating example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In- deed, if the target state is |ψ⟩ and the seed state is |ψ⟩ ⊗ |ζ⟩, then clearly the maximizer is chosen by the ancillary state being |ζ⟩ and the local unitaries being trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Relation between LO and LU Strategies The natural question given the previous theo- rems is if we can better understand the relationship between LO and LU strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We first show that LU and LO strategies are equivalent when either the target or the seed state is a two qubit state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' LU and LO Equivalence for Two-Qubit Seed or Target State Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider entangled two qubit seed state |φ⟩ ∈ C2 ⊗ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let the target entangled state be |ψ⟩ ∈ Cd ⊗ Cd′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then the optimal non- communicative strategy is the local unitary strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Without loss of generality, q↓ = (q, 1 − q) where q ≥ 1/2 and p↓ = (p(1), p(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then the optimal local unitary strategy is � qp(1) + � (1 − q)p(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For any P′ we can write (p′)↓ = (p′(1), p′(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The optimal CPTP strategy (up to a square) is of the form � qp(1)p′(1) + � (1 − q) max{p(1)p′(2), p(2)p′(1)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' These values can only increase by assuming p′ has two outcomes, so let us assume so without loss of generality and parameterize the distribution by p′ ∈ [1/2, 1] to obtain � qp(1)p′ + � (1 − q) max{p(1)(1 − p′), p(2)p′} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Moreover note p(2)p′ < p(2) unless p′ = 1, which is equivalent to the LU strategy, so the second entry in the maximization would be lower than the LU setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, we focus on the remaining case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We are specifically interested in when the following strict inequality holds: � qp(1)p′ + � (1 − q)p(1)(1 − p′) > � qp(1) + � (1 − q)p(2) ⇔ g(p′) := � qp(1)( � p′ − 1) + � 1 − q( � p(1)(1 − p′) − � p(2)) > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then d dp′ g(p′) = √ qp(1) 2√ p′ + √ p(1)(1−q) 2√ 1−p′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It follows, � qp(1) � 1 − p′ 2 � p′� 1 − p′ + � p′� p(1)(1 − q) 2 � 1 − p′� p′ ≥ 0 ⇔ � qp(1) � 1 − p′ + � p′ � p(1)(1 − q) ≥ 0 ⇔√q � 1 − p′ + � p′ � (1 − q) ≥ 0 ⇔ √ F(Q↓, P′↓) ≥ 0 , where the first line is multiplying to get identical denominators, the second line is multiplying by the denominator, the third is dividing out p(1), and the final is by the definition of square root fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note the final inequality will always hold strictly unless q ∈ {0, 1}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' the state is a product state, by Item 1 of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' If q ∈ {0, 1}, then the state is a product state which would contradict that we assume the state is entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, in our setting, g(p′) only increases over its inter- val, p′ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus, the optimal choice of p′ is p′ = 1, but in this case the value is � qp(1) ≤ � qp(1) + � (1 − q)p(2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' the optimal choice is lower bounding the optimal local unitary strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It follows this is never optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider entangled two qubit target state |ψ⟩ ∈ C2 ⊗ C2 and any seed state |φ⟩ ∈ Cd ⊗ Cd′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The optimal non-communicative strategy is the local unitary strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The proof is basically the same as for the two qubit seed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Without loss of generality, p↓ = (p, 1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We can re-order |φ⟩ such that it is q↓ = (q(1), q(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The optimal CPTP strategy (up to a square) is of the form � q(1)p′(1)p + � q(2) max{p′(1)(1 − p), p′(2)p} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This sum can only increase if p′(1) + p′(2) = 1, so we can parameterize the distribution by p′ ∈ [1/2, 1] to obtain � q(1)p′p + � q(2) max{p′(1 − p), (1 − p′)p} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note that p′(1 − p) < (1 − p) unless p′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' If p′ = 1, this is the LU strategy, if p′ < 1, then this is worse than an LU strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, we only care about the other maximization case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' That is, we are 14 interested in when p ∈ [1/2, 1) and the following strict inequality holds: � q(1)p′p + � q(2)p(1 − p′) > � q(1)p + � q(2)(1 − p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' However, � q(1)p′p < � q(1)p and � q(2)p(1 − p′) < � q(2)(1 − p) as p′ ∈ [1/2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore this strict inequality can never hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore the optimal strategy is always the LU strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' LU and LO Inequivalence for States with Schmidt Rank Greater than Two If there is equivalence for two qubit seed or tar- get states, it is natural to ask if this property per- sists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' One might expect that this is a special prop- erty of qubit systems as are found throughout quan- tum information science results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Indeed, generally this property does not hold, which we will prove via example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For seed and target state with Schmidt rank ≥ 3, the optimal LO strategy may be better than the optimal LU strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We construct an example for Schmidt rank 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' By continuity of the fidelity, one can embed the tar- get and seed in bigger spaces with arbitrarily small perturbations for it to hold in higher dimensions, which is why this is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider target state |ψ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='85 |00⟩ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='08 |11⟩ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='07 |22⟩ and seed state |φ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='45(|00⟩ + |11⟩) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='1 |22⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then, the optimal LU strategy fidelity is F(P↓, Q↓) = �√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='45( √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='85 + √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='08) + � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='1(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='07) �2 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='796 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In contrast, if we consider P′ = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='28, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='17], then F((P ⊗ P′)↓, Q↓) = �√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='45 √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='4675 + √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='45 √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='238 + √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='1 √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='1445 �2 >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='82 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' As we maximize over P′, the optimal LO strategy achieves a value that is strictly above the LU strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Inefficiency of Optimal LOSR Fidelity and Computable Upper Bounds In the above we have constructed an example where the local operations strategy outperforms the local unitary strategy (though we have not shown what the strategy itself is).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' A natural question would then be how easy it is to solve for the op- timal fidelity value or even a bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' By Theorem 5, we can conclude the optimal local unitary strat- egy is polynomial time to solve as all one needs to do is sort the Schmidt coefficients and calculate the fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Indeed, one could solve for the ordering of the Schmidt coefficients using the linear program for sorting a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In contrast, for optimizing LO strategies, we have no such luck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In effect this is because there are two things to optimize over at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Indeed, recall FLO(|ψ⟩ , |φ⟩) = max P′∈P(Σ) F((P ⊗ P′)↓, Q↓) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then the problem is that one must first tensor P onto variable P′ and then re-order the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' One cannot even in general order an optimization vari- able, which we will refer to as ‘sorting,’ as sorting is in general non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In sorting a vector using a linear program, one relaxes to bistochastic channels and considers a linear function so that the optimizer is an extreme point which by the Birkhoff von Neu- mann theorem is a specific permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' However, we are many levels of involvement above that: we want the distribution P′ such that its product dis- tribution P ⊗ P′ when sorted optimizes the fidelity with Q↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, we need to optimize over P′ and the permutation at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It’s not clear that we can actually relax to bistochastic strategies because of the joint concavity of fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' That is to say, for any bistochastic channel E, F(E(P ⊗ P′), Q↓) =F(∑ π r(π)Vπ(P ⊗ P′), Q↓) ≥∑ π F(r(π)Vπ(P ⊗ P′), r(π)Q↓) =∑ π r(π)F(Vπ(P ⊗ P′), Q↓) , where the first line is Birkhoff-von Neumann the- orem, the second is joint concavity using Q↓ = ∑π r(π)Q↓ as r is a probability distribution, and the last line is because F(λP, Q) = λF(P, Q) = F(P, λQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus any bistochastic channel may strictly do better than the average of its extreme points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Moreover, even if we could optimize over bistochastic channels, we would have a non-convex objective function as the bistochastic channel, an optimization variable, would be applied to P ⊗ P′ which is also partially an optimization variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 15 Given the above, it seems likely the best option if one were to try and find a (near) optimum would be to use gradient descent from random initial P′, real- izing it will only work locally and will break down at ‘kinks’ where the ordering changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Otherwise more sophisticated non-convex optimization tech- niques might be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Computable Upper Bound Methods Perhaps even worse than our inability to calculate the exact fi- delity, is that it is not clear in general how to de- termine good bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Certainly we have the fol- lowing result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Unless the target state is |ψ⟩ = |φ⟩ ⊗ |ζ⟩ where |φ⟩ is the seed state, there exists ε > 0 such that there does not exist local operations that will take |φ⟩ to |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This follows from Theorem 6 along with Item 1 of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The above theorem, while derived from a very different strategy than Proposition 2, does not seem to give us much more information as to at what point communication is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' What we would want to efficiently improve this would be to estab- lish upper bounds on the equation given in Theo- rem 6 that have a closed form that does not depend on P′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' One option is to use the data processing in- equality for fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This can be seen in the follow- ing proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider target state |ψ⟩ and seed state |φ⟩ with corresponding Schmidt distributions p, q respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' If pmax ≤ qmax, then FLO(|ψ⟩ , |φ⟩) ≤ F(p, q) , where p = pmax |0⟩⟨0| + (1 − pmax) |1⟩⟨1| and like- wise for q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Without loss of generality let d be the maxi- mum local dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let E(·) = |0⟩⟨0| · |0⟩⟨0| + ∑i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=',d−1} |1⟩ ⟨i| · |i⟩ ⟨1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' That is, E coarse-grains a probability distribution to the Bernoulli distribu- tion with its first element untouched and the sum of all the others as the other outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then using data processing of fidelity (Item 3 of Proposition 1), max P′∈P(Σ) F((P ⊗ P′)↓, Q↓) ≤ max P′∈P(Σ) F(E((P ⊗ P′)↓), E(Q↓)) = max p′∈[0,1] F � �P(p′), E(Q↓) � , where �P(p′) := pmaxp′ |0⟩⟨0| + (1 − pmaxp′) |1⟩⟨1| and E(Q↓) = qmax |0⟩⟨0| − (1 − qmax) |1⟩⟨1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Now note that by assumption pmax ≤ qmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' As the fi- delity will only decrease as pmaxp′ moves away from qmax, the optimal choice is p′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This com- pletes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The problem with the above bound is that there will be cases where pmax > qmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Why the in- equality in the other direction was required was to know for a fact what element of p was relevant, namely pmax and that any choice of p′ ̸= 1 would be sub-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In general this strategy would require q↓(j) is sufficiently large relative to p↓(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This can be determined in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Here we provide a simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let p↓ = [3/4, 1/8, 1/8]T q↓ = [1/2, 1/2]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then (p ⊗ p′)↓[1 : 2] = p′(1)[3/4, 1/8]T, and so we can coarse-grain on the second element to ob- tain P(p′) = 1/8p′ |0⟩⟨0| + (1 − 1/8p′) |1⟩⟨1| and Q = Q↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then as 1/8p′ < 1/2, the upper bound is F( 1 8 |0⟩⟨0| + 7 8 |1⟩⟨1| , 1 21) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The above shows that while data processing can be sufficient in certain cases, it does not provide an easy general method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Another common alter- native in quantum information theory is semidefi- nite relaxations of optimization problems because semidefinite programs are efficient to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In Appendix B, we establish the following upper bound and show it may be expressed as a semidef- inite program, which, as everything is in terms of probability distributions, is due to the non-linearity of fidelity and nothing particularly quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider target state |ψ⟩ and seed state |φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let SR(ψ) = d and SR(φ) = d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Define A = Cd, B = Cd·d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then, FLOSR(|ψ⟩ , |φ⟩) ≤ max F(R, Q↓ embed) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' TrB[R] = P↓ R ∈ P↓(d2 · d′) , (7) where P and Q are the distributions defined by |ψ⟩ and |φ⟩’s Schmidt coefficients respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' More- over, this admits the following simple semidefinite program over the reals: max ∑ i∈[d2·d′] x(i) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' �diag(r) diag(x) diag(x) diag(q↓ embed) � ⪰ 0 TrB[diag(r)] = P↓ r ∈ P↓([d2 · d]) x ∈ Rd2·d′ , (8) 16 Physically, this relaxation may be seen as relaxing the isometric representation of the optimal LOSR strategy to one where one allows the ancillary en- vironment start off entangled with the local system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Mathematically, this is not too loose because we re- quire this entangled pure state has a notion of “lo- cal Schmidt coefficients” that pertain to the original target state, although this physically does not seem to have a clean interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Nonetheless, we can see that (7) will not achieve unity unless there exists a joint distribution Q = R, which would require Q↓ embed to have P↓ as it’s marginal, which seems highly restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, (7) should provide an upper bound that is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' MANY COPY PURE STATE CONVERSION WITH ZERO COMMUNICATION Having established what happens for single copies, we consider many copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We pro- vide two motivations for doing this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' First, we note that it’s not clear what the limiting be- haviour will be even in the LU setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' A reader may recall from other works that the fi- delity is multiplicative so if F(P, Q) < 1, then limn→∞ F(P⊗n, Q⊗n) = limn→∞ F(P, Q)n → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' However, we lose the multiplicativity as we are considering limn→∞ F((P⊗n)↓, (Q⊗n)↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This issue is further aggravated if we consider local opera- tions and the ancillary variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The second motivation is that what was initially considered in the literature, albeit with LOCC [23], was the conversion of many copies of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' A par- ticular focus in the referenced work and subsequent ones is the case where either the target or seed state is the maximally entangled state, known as distil- lation and dilution respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' With LOCC, we know there are ‘rates’ in the conversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' By [6] along with previous results in this work, we would not expect there to be non-negative rates without the communication assuming the error is required to be vanishing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In this section we establish convex optimiza- tion problems for dilution and distillation in the zero communication setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' These results are established in terms of the not-actually-a-norm ∥ · ∥(k,1/2), which we remind the reader is the (k, p)−norms extended to p < 1 introduced in Sec- tion III with the choice of p = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We also look at the limiting behaviour as the number of copies grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In particular, we find a closed form when trying to convert n−fold two qubit states to a dif- ferent n−fold two qubit state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Moreover, we prove the fidelity goes to zero in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We discuss the extension of this to entangled states with larger Schmidt rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Dilution Under Local Operations We begin by determining the limits of dilution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For intuition, we begin with local unitaries where there is no optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Recall that the Schmidt coefficients of the maximally entangled state are all √ d−1, so they correspond to the maximally mixed distribution under our bijection between Schmidt coefficients and probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For local unitary strategies the op- timal dilution fidelity is given by FLU � |ψ⟩ , ��Φ+ d �⊗n� = d−n ∥P∥(dn,1/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Generally, if |ψ⟩ ̸= ��Φ+ d � , 1 >F(P↓, π⊗n d )↓) =F(P↓, π⊗n d ) = � �d−n/2 ∑ i∈[dn] � P↓(i) � � 2 =d−n � � ∑ i∈[dn] � P↓(i) � � 2 =d−n∥P∥(dn,1/2) , where the first equality is because π⊗n d is invari- ant under ordering, the second is using the defi- nition of fidelity and that π⊗n d has uniform coeffi- cients, and the final equality is the definition of the (k, p)−norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In particular note we have dropped the sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We remark we could have set |φ⟩ = |φ′⟩⊗m to get a tradeoff, but this does not seem to provide any insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Just as in the one-shot setting, we know the above result isn’t as useful in general because it can’t throw out resources, so we now present the general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The optimal fidelity of converting n d−local dimensional EPR pairs to |ψ⟩ under local operations is given by FLO(|ψ⟩ , ��Φ+ d �⊗n) = d−n max P′∈P(Σ) ∥(P ⊗ P′)∥(dn,1/2) , where ∥ · ∥(k,p) is (k, p)−norm generalized to p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Moreover, for fixed n, this is a convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Starting from the result of Theorem 6, max P′∈P(Σ) F((P ⊗ P′)↓, (π⊗n d )↓) = max P′∈P(Σ) F((P ⊗ P′)↓, π⊗n d ) = � � 1 dn/2 max P′∈P(Σ) ∑ i∈[dn] � (P ⊗ P′)↓(i) � � 2 (⋆) = 1 dn max P′∈P(Σ) ∥P ⊗ P′∥(dn,1/2) , the first inequality is invariance of π⊗n d under sort- ing, the second is definition of fidelity and that each element of π⊗n d is the same, the last is the definition of (k, p)-norm extended to p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' To show this is a convex optimization problem, note that ΦP(·) := P ⊗ · is linear, −√· is opera- tor convex, and the sum of the k largest eigenvalues of a PSD P, which we will denote Σk(P) is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus, starting from (⋆), � �d−n/2 max P′∈P(Σ) ∑ i∈[dn] √ P ⊗ P′↓(i) � � 2 = � −d−n/2 min P′∈P(Σ)Σdn � − � ΦP(P′) ��2 , where we have used maxx∈C f (x) = − minx∈C − f (x) and our definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then ig- noring the −d−n/2 factor and the square, the optimization problem is over the probability simplex, which is a convex subset of the positive semidefinite matrices, and the objective function is convex over the positive semidefinite cone as − � ΦP(·) is operator convex and Σdn is a convex function over the space of Hermitian operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Unfortunately, while this gives computable bounds, it is not clear how one could determine the optimal value analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Distillation Under Local Operations We now present the same results in the distilla- tion case, where we take some state to many EPR states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For completeness, we state the local uni- taries case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The fidelity of distillation under lo- cal unitaries and zero communication is given by FLU( ��Φ+ d �⊗m , |ψ⟩⊗n) = d−m∥P⊗n∥|S|,1/2 , where S = [min{dm, rank(P)n}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The proof is effectively identical to the dilu- tion case by symmetry of the fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In contrast to the local unitary case, the symmetry is broken when one considers local operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For fixed d, m, n the optimal fidelity for dilution under local operations is given by FLO( ��Φ+ d �⊗m , |ψ⟩⊗n) =d−m � min P′∈P↓(Σ) − ∑ i∈I αi � p′(i) �2 , where P↓(Σ) is the set of decreasing distributions as defined in Section III, I ≡ [⌈rank(P)n/dm⌉], and αi := ∑j∈[(i−1)dm:min{i·dm,rank(P)n}] � p↓ n(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note the minimization is a convex optimization program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Yet again, we use the square root fidelity and then take the square at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then, using Theo- rem 6, we have FLO((Φ+ d )⊗m, ψ⊗n) = max P′∈P(Σ) F((π⊗m d ⊗ P′)↓, (P⊗n)↓) = � max P′∈P(Σ) ∑ i∈S � (π⊗m d ⊗ p′)↓(i) � p↓ n(i) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Next, note (π⊗m d ⊗ P′)↓ = d−m/2 ∑ i′∈Σ p↓(i′)1Cdm , where we have just used that π⊗m d is invariant under ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It follows that if we let I ≡ [⌈rank(P)n/dm⌉], we can rewrite, FLO((Φ+ d )⊗m, ψ⊗n) =d−m � max P′∈P(Σ) ∑ i∈I � (p′)↓(i) ∑ j∈[(i−1)dm:min{i·dm,rank(P)n}] � p↓ n(i) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Now first define αi := ∑j∈[(i−1)dm:min{i·dm,rank(P)n}] � p↓ n(i) as these co- efficients may be pre-computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Second, note that the probability simplex restricted to de- scending distributions, P↓(Σ) is itself convex as r↓ λ := λp↓ + (1 − λ)q↓ satisfies λp↓(i) + (1 − λ)q↓(i) ≥ λp↓(i + 1) + (1 − λ)q↓(i) , 18 for all i ∈ [|r|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus we have, FLO( ��Φ+ d �⊗m , |ψ⟩⊗n) = � − d−m min P′∈P↓(Σ) − ∑ i∈I αi � p′(i) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The minimization is a convex optimization problem because if we consider f (p′) := − ∑i αi � p′(i), then its Hessian is ∇2 f = ∑i[αi/4p′(i)−3/2] |i⟩⟨i|, which is positive semidefinite on the interior of the prob- ability simplex (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' when p′(i) > 0 for all i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Two Qubit Setting We have now seen that even in the basic dilu- tion and distillation setting, while we can deter- mine convex optimization programs, we can’t seem to get clean analytic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In this section we con- sider an even more tractable setting to attempt to resolve this: many copy two-qubit seed and target states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We show in this setting under certain as- sumptions the local unitary strategy is optimal and lobby this to show in particular that the optimal fi- delity of converting n copies of |φ⟩ to n copies |ψ⟩ goes to zero as n goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We note that this setting is more manageable because we effectively only have to reason about Bernoulli distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Given Bernoulli distribution P = p |0⟩⟨0| + (1 − p) |1⟩⟨1|, then P⊗n is such that the sequence xn with (n − k) zeros has probability pn−k(1 − p)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Moreover, there are (n k) sequences with probability pk(1 − p)n−k and the same for pn−k(1 − p)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The claim that xn with (n − k) zeros has prob- ability pn−k(1 − p)k is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The second point actually just follows from the fact there are (n k) sequences with k zeros, which could be proven by induction in a straightforward manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We can now use the above lemma along with Theorem 5 to get the optimal LU fidelity as a func- tion of the number of copies n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider entangled states |ψ⟩ , |φ⟩ ∈ C2 ⊗ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then, FLU(ψ⊗n, φ⊗n) = ∑ k∈[n] �n k � (pq)(n−k)/2((1 − p)(1 − q))k/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' By Theorem 5 we can reduce to the Bernoulli distributions from the Schmidt coeffi- cients, |ψ⟩⊗n �→ P⊗n, |φ⟩⊗n �→ Q⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Since these are Bernoulli distributions, if we assume without loss of generality p ≥ (1 − p), we can order the proba- bilities simply by the exponent, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' pj−k(1 − p)k ≥ pj−k−k′(1 − p)k+k′ for any 0 ≤ k′ ≤ j − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' More- over, the cardinality of each set of sequences will be the same for both P⊗n and Q⊗n because |ψ⟩ , |φ⟩ are only entangled if their Schmidt rank is two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' There- fore, F((P⊗n)↓, (Q⊗n)↓) = ∑ k∈[n] �n k � (pq)(n−k)/2((1 − p)(1 − q))k/2 where the sum is over the number of zeros in the string, the cardinality was proven in the previous lemma, and the last term is just a re-writing of � pn−k(1 − p)k � qn−k(1 − q)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We note it is straightforward to generalize the above result to the case where you have the num- ber of states differs between the seed and the target, but the form would be ugly as one would need to count how many sequences of a given probability there are and keep track of this in the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Indeed at this point the problem is elaborate enough that there is no advantage with dealing with two-qubit states as it’s a question of the type classes [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We state this as a remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider states |ψ⟩ , |φ⟩ respectively with ordered probability distributions correspond- ing to their Schmidt coefficients, P and Q respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' FLU(|ψ⟩⊗n , |φ⟩⊗m) can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This is because the probability of a given sequence drawn in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' form from a distribution has a closed form [24, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It follows that as long as one determines the type classes exactly and takes into account that the sizes of the type classes may dif- fer between P and Q, the computation is possible, albeit tedious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Rather than dealing with the computational nightmare of generalizing beyond two qubit states, we now show that the term in Corollary 3 always goes to zero as n goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider entangled states |ψ⟩ , |φ⟩ ∈ C2 ⊗ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' lim n→∞ FLU(|φ⟩⊗n , |ψ⟩⊗n) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let the probability distributions correspond- ing to their Schmidt coefficients be parameterized 19 by p and q = p + ε where ε ∈ [−1/2, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then, That way, FLU(|ψ⟩⊗n , |φ⟩⊗n) = ∑ k∈[n] �n k � (p2 + ε)(n−k)/2 [(1 − p)2 − ε(1 − p)]k/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Now note p2 + ε < 1 as otherwise |ψ⟩ is product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Define α := (p2 + ε)1/2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then we have �n k � (p2 + ε)(n−k)/2 · [(1 − p)2 − ε(1 − p)]k/2 ≤ �n · e k �k αn−k[(1 − p)2 − ε(1 − p)]k/2 = � e k α−1�k [(1 − p)2 − ε(1 − p)]k/2nk · αn =O(poly(n))O(exp(−n)) →0 , where in the inequality we have used an upper bound on the binomial coefficient, in the first equal- ity we have grouped terms by scaling, in the next equality we have used that the first portion is a polynomial in n and that α < 1, so αn scales in- verse exponentially in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The limiting factor is then because an inverse exponential times a polynomial goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We also remark that the term where k = n will also go to zero as [(1 − p)2 − ε(1 − p)]k/2 will go to zero as k goes to infinity as its magnitude will be bounded by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, each term in the sum goes to zero as n goes to infinity, so the entire sum will go to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We note our proof tells us nothing about the scal- ing as a function of the difference between p and q nor does it tell us how fast it goes to zero compared to F(P⊗n, Q⊗n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' These are shown numerically for specific cases in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It is then natural to ask if what we have seen so far is something special to local unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We show that under sufficient conditions, just like in the sin- gle copy case, when two-qubit seed states are in- volved, local unitary strategies are optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let |ψ⟩ ∈ C2 ⊗ C2 and the target state be |ψ⟩⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let the seed state |φ⟩ satisfy SR(|φ⟩) ≤ nSR(|ψ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then the optimal local operations strat- egy is the optimal local unitary strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' By Theorem 6, FLO(|ψ⟩⊗n , |φ⟩) (a) Fidelity under local unitaries as n grows for various choices of q = p + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' (b) Fidelity under local unitaries as n grows for various choices of q = p + ε compared to F(P, Q)⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 4: Degradation of fidelity of trying to convert n copies of one pure two-qubit entangle state to another for various differences in Schmidt coefficients, q = p + ε where we choose p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' (a) Shows the rate that the local unitary strategy degrades is a nonlinear function of the size of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' (b) Compares to the case where one does not re-order the Schmidt coefficients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' compares to F(P, Q)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' = max P′∈P(Σ) F((P⊗n ⊗ P′)↓, Q↓) = ∑ i∈|Q| � Q↓(i) � (P⊗n ⊗ P′)↓(i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We will show that P′ should be the delta distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' If p ̸= 1/2, p′(1) < 1, then for any 0 ≤ k ≤ n, we have the inequalities pn−k(1 − p)k >pn−k(1 − p)kp′(1) >pn−k(1 − p)kp′(2) and pn−k(1 − p)k >pn−k(1 − p)kp′(1) >pn−(k+1)(1 − p)k+1p′(1) >pn−(k+1)(1 − p)k+1p′(2) As square root is a monotone, this holds when we take the square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note that by assumption ConvergenceComparison 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='8 UnorderedE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='4 OrderedE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='6 idelity Unordered E=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='4 OrderedE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='1 UnorderedE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='2 OrderedE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='0 0 50 100 150 200 250 Numberof CopiesnConvergenceComparison 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='8 UnorderedE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='4 OrderedE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='6 idelity Unordered E=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='4 OrderedE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='1 UnorderedE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='2 OrderedE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='0 0 50 100 150 200 250 Numberof Copiesn20 P⊗n has enough entries by itself for there to be one corresponding to each q↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, given the inequalities above, it follows if p′(1) ̸= 1, each term in the sum only decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, p′(1) is optimal for every n and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus, when p ̸= 1/2, the optimal value is obtain by P′ being a delta distribution, which means it’s equivalent to the local unitary strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Finally, if p = 1/2, then pn−k(1 − p)k = 2−n for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, if p′(1) < 1, the inequalities simpli- fies for all 0 ≤ k ≤ n: pn−k(1 − p)kp′(1) =pn−(k+1)(1 − p)k+1p′(1) >pn−(k+1)(1 − p)k+1p′(2) and pn−k(1 − p)kp′(1) > pn−k(1 − p)kp′(2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Again because each q term is paired up already, this means if p′(1) ̸= 1, the value decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, we again conclude the optimal strategy is the LU strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We note that a trivial example of why we need the Schmidt rank constraint in the previous theo- rem is our original example for the advantage of LO strategies: if |φ⟩⊗n+ℓ where ℓ ≥ 1, then there is a better LO strategy than an LU strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Finally, we note it immediately follows from these previous results that Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' If |φ⟩ , |ψ⟩ ∈ C2 ⊗ C2 are both entan- gled, then lim n→∞ FLO(|ψ⟩⊗n , |φ⟩⊗n) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' ON CATALYTIC CONVERSION We now have established a rather robust theory of pure state transformations under local opera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It is natural to return to the topic of conver- sion of one state to another using an ancillary en- tanglement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' cataltyic transformations, which is a special case of the setting, and includes quantum embezzlement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Of course, it is immediate from our results so far that we know the optimization pro- gram that determines the optimal pure state cata- lyst, as we state in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For any Schmidt rank d, the opti- mal pure state catalyst for state conversion |φ⟩ to |ψ⟩ is the quantum state |ζ⟩ = vec( √ R) that is de- termined via the optimization max R∈P(d),P′∈P(Σ) F((P ⊗ P′)↓, (Q ⊗ R)↓) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This immediately follows from the input be- ing |φ⟩ ⊗ vec( √ R) and then applying Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note this means |Σ| scales as function of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' However, as we have already addressed, even without a free variable for the catalyst, the opti- mization in Theorem 6 seems unmanageable di- rectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' While in principle one could use the relax- ation in Theorem 9 to obtain efficient upper bounds, it is less obvious how often these will be non-trivial given that R is a free variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The next most natural setting would be that of catalytic state conversion under local unitaries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' we consider transformations of the form |φ⟩ |ζ⟩ LU ←→ ≈ε |ψ⟩ |ζ⟩ , where |ζ⟩ is the catalytic resource and the arrow going in both directions is because local unitaries are reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This may be seen as a generaliza- tion of embezzlement where |φ⟩ = |0⟩A |0⟩B and |ζ⟩ = |µ(n)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='2 Now as noted in the background, embezzling is known to be in effect optimal for sufficiently small ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It follows for sufficiently small error ε > 0, the strategy that embezzles out the seed state and then embezzles in the target state is roughly optimal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' |φ⟩ |µ(n)⟩ LU ←→ |0⟩ |0⟩ |µ(n)⟩ LU ←→ |ψ⟩ |µ(n)⟩ is effectively optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Nonetheless, we may explore at what point this becomes necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Using Theorem 5, we know the optimal strategy is given by3 max R∈P(d) F((P ⊗ R)↓, (Q ⊗ R)↓) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Even in the case P, Q, R ∈ P(2) this technically can’t be solved using gradient methods as one has to sort the p(1 − r) and (1 − p)r terms of p ⊗ r and likewise for q ⊗ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Nonetheless, it is hopefully clear that r ∈ [min{p, q}, max{p, q}], as it is trying to make the distributions be more similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Nonethe- less, this issue will only grow in difficulty with the dimension and it is unclear how one would prove an ansatz is optimal in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, we pro- vide two-qubit examples which characterizes the general insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 2 We refer the reader to Proposition 3 if the notation has been forgotten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 3 We stress that by the correspondence of Schmidt coefficients to probability distributions as discussed at the start of the work, even without Theorem 5, this would be a legitimate strategy, we simply wouldn’t know analytically it was optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 21 Example 4 (Resource Gap Between Embezzling and Optimal Catalyst).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider Bernoulli distri- butions P, Q, R parameterized by p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='5, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='7 and we leave r unspecified for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In other words, one of the states is the maximally entangled states and the other is, up to local unitaries, √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='7 |00⟩ + √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='3 |11⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, depending on which way one runs the transformation, we are considering entan- glement dilution or distillation with a catalytic re- source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Without the resource, FLO(|ψ⟩ , |φ⟩) = F(P↓, Q↓) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' One can verify that the optimal choice of r⋆ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='6 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For this choice FLU(|ψ⟩ |ζ⟩ , |φ⟩ |ζ⟩) =F((P ⊗ R⋆)↓, (Q ⊗ R⋆)↓) >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='979 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The first problem is that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='979 is not an accept- ably high fidelity even by contemporary standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Nonetheless, note that to get this state via embez- zling (and ignoring that embezzling out the initial state introduces error), it would require generating |µ(n)⟩ where n > m1/(1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='979) = 2 · 1014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' That is, even to embezzle a two-qubit pure state would re- quire generating an inconceivable amount of entan- glement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For this reason, specially engineered cat- alysts seems a significant improvement up to any error that can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' On the other hand, one might note that if we could generate R where r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='55, then we may as well have just used this state to begin with as F(P↓, R↓) =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='98989 >F((P ⊗ R⋆)↓, (Q ⊗ R⋆)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' From a practical perspective we agree with this cri- tique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Nonetheless, from a basic science perspec- tive, if we are interested in local unitary conversions under catalysts, then the above tells us there are bet- ter choices in general than embezzlement, although embezzling has the special property of being uni- versal and optimal for sufficiently small ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We close this consideration with two final re- marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' First, if one picks two states that are more similar to begin with, then the scaling of the embez- zling state will be even larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Second, we have not presented how the fidelity for this example scales as the local dimension of |ζ⟩ grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Both the dimen- sion scaling and two states that are more similar are considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 5 where the near-optimal fideli- ties are found via brute force numerical search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' (a) Maximum achievable fidelity of transformation under local unitaries as a function of the Schmidt rank of the catalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' (b) Order of the Schmidt Rank of embezzling state |µ(n)⟩ to achieve same fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' 5: Plots regarding dimension scaling in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' (a) The achievable fidelity of converting one two-qubit entangled state to another parameterized by p and q under local unitaries using a catalyst with a given local dimension (equivalently, Schmidt rank) using brute force search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' (b) The order (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' the power of 10) in the Schmidt rank of the embezzling state |µ(n)⟩ to obtain the same maximum fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This is calculated using 21/(1−Fmax) following Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' All optimizer catalysts provided in an appendix for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' ON EXTENSIONS OF THE THEORY As a final consideration, we discuss the applica- tion of our results beyond bipartite pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' First we remark upon extensions to multipartite pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In this case the problem is that in establish- ing all of the results, we have used that local uni- taries can take the Schmidt decomposition of the state to one of a canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' However, in the multipartite case, the Schmidt decomposition does not even exist in general [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' As such this argu- ment immediately breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Furthermore, in the proof of Theorem 5 we used Uhlmann’s theo- rem, which requires partitioning the state into two pieces, one of which is the purification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, it MaximumFidelityasaFunctionofCatalystDimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='99 L lity Fideli 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='98 p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='6,q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='65 p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='5,q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='7 Max 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='96 0 2 4 6 8 AllowedCatalvstSchmidtRankOrderofEmbezzlingStateDimforsameMaxFidelity Schmidt Rank 500 100 p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='6,q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='65 OrderofEmbezzling 50 p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='5,q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='7 10 0 2 4 6 8 AllowedCatalvstSchmidtRank22 seems no multipartite extension of this work holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Similarly, there are issues with approaching mixed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' One issue is to note that all relation- ships we have been able to establish have stemmed from the fidelity under local unitaries of pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Even in the case where local operations made a pure state no longer pure, we purified operations so that the states were pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We simply cannot do this if we start with mixed states in both arguments of the fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We also cannot purify the states as by data-processing, any optimization without tracing off the purifying space only gets us a lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Moreover, this lower bound would require estab- lishing results for tripartite systems, which returns to the issues with the multipartite pure state case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Appendix A: Randomness Embezzling Proof and Discussion on Locality In this section we provide the proof of Theorem 2 and then briefly discuss how it differs from quan- tum embezzlement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The proof is largely the same as for embezzle- ment of quantum states [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let P = ∑i p(i) |i⟩⟨i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Define Wn as Rn ⊗ P except with probabilities in de- 23 creasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note Rn ⊗ P = 1 Hn ∑ i,j p(i) j |i⟩⟨i| ⊗ |j⟩⟨j| , so there exists a relabeling on {(i, j)} that will take this to Wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In particular, letting f : [m] × [n] → [m · n] be a bijection, we have |i⟩ |j⟩ → | f (i, j)⟩ ≡ |i′⟩ |j′⟩ such that � z f (i,j) := p(i) jHn � (i,j) satisfy zk ≥ zk+1 for all k ∈ [m · n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore it suffices to approximate Wn, which means we want to bound the overlap of this with Rn ⊗ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For fixed t and i, we let Nt i := ���� � (i, j) : p(i) jHn > 1 tHn ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The inequality may be manipulated to imply 1 ≤ j < p(i)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It follows that Nt i < p(i)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' From this we obtain ∑m i=1 Nt i < ∑m i=1 p(i)t < t, where we have used ∑i p(i) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' As z1 ≥ z2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=', it follows zj ≤ 1 jHn for all 1 ≥ j ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We may restate this as for 1 ≤ j ≤ n, there are at most t′ − 1 pairs (i, j) such that p(i)/(jHn) > 1/(t′Hn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Recalling z1 ≥ z2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=', this means that z1 < 1/Hn and that there is at most one pair (i, j) pair such that p(i)/(jHn) < 1/(2Hn), which, since z1 ≥ z2, means if such a pair exists, it is z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' By applying this argument in effect recur- sively, we see that for t′, there are at most t′ − 1 (i, j) pairs such that p(i)/(jHn) > 1/(t′Hn) and since zk ≥ zk+1, if all of these pairs exist, then it must be z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=', zt′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, zj ≤ 1/(jHn) for all 1 ≤ j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We can now use this to bound the fi- delity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' F(Rn ⊗ |0⟩⟨0| , Wn) = � n ∑ j=1 � zj jHn �2 ≥ � n ∑ j=1 �zj �2 ≥ n ∑ j=1 zj, where in the equality we have used the definition of fidelity, in the second we used our established in- equality, and in the third we have used √x + √y ≥ √x + y for x, y ≥ 0 to pull the square root out around the sum and cancel with the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Now we want to lower bound this sum, which requires managing the zj terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We consider Tn = Rn ⊗ πm with probabilities t(j) where πm := 1 m ∑m i=1 |i⟩⟨i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Now note that zk ≥ tk for all k ∈ [m · n], and this is independent of what the distri- bution P is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We can then bound the relevant sum by the sum for Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It follows n ∑ j=1 tj = ⌊n/m⌋ ∑ j=1 m ∑ i=1 1 jHnm = ⌊n/m⌋ ∑ j=1 1 jHn = H⌊n/m⌋ Hm ≥ln(n/m) ln(n) = 1 − log(m) log(n) , where the second inequality is using Hn ≥ ln(n) and the final form is converting from ln to log in both the numerator and denominator so it cancels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Finally, leting 1 − log(m)/ log(n) > 1 − ε will result in n > m1/ε, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' With the proof established, we expand upon the distinction between the entangled and classical dis- tribution cases of embezzlement in terms of local- ity briefly mentioned in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In the clas- sical case, one party embezzles a distribution lo- cally by themselves, whereas in the entangled case two parties act locally on a non-local distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Mathematically, this simply follows from the fact the vec(·) map and its inverse converts between bi- partite states and a probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' How- ever, it is also physically interesting that these are the two cases that align as it is clear other varia- tions are either classically or quantumly impossible as we now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The first reasonable variation would be if there is a non-local classical case where two parties try and construct some joint distribution pXY using cat- alyst rX′Y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It is easy to see that they cannot in gen- eral satisfy the decoupling condition that is satisfied in quantum embezzlement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' they cannot satisfy pXY ⊗ rX′Y′ in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This is because without loss of generality the state will be of the form qXYX′Y′ = ∑ x,x′,y,y′ q(x|x′)q′(y|y′)r(x, y) ��x, y, x′, y′�� x, y, x′, y′�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This form means that X will be correlated to X′ and Y to Y′ unless qXY may be generated non-locally without a seed state to correlate the two which means they are (up to the allowed error) indepen- dent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' qXY ≈ε qX ⊗ qY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In this sense, there cannot be a classical non-local equivalent of quantum em- bezzlement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' On the other hand, if one does not require the decoupling, then this is a task that is possible in the classical setting and is known as distributed source simulation, where the question is the min- imal needed shared randomness as the seed state to generate the target state up to an (arbitrary) er- ror [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This was determined asymptotically in the classical case by Wyner [8], extended to sepa- rable states by Hayashi [9], and recently general- ized to the one-shot setting for separable states in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' However, as in this setting variation there is no 24 communication between the acting parties and the catalyst acts as the seed state, it follows from Propo- sition 2 that distributed source simulation cannot admit an entangled state equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' For these rea- sons, not only does the vec bijection specify the cor- respondence of embezzlement in the classical and quantum setting, but deviating from it makes either a quantum or classical version impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Appendix B: Semidefinite Program Relaxation of Max Fidelity of Pure State Transformation Under LOSR In this section we prove Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We begin by establishing (7) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Consider target state |ψ⟩ and seed state |φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Let SR(ψ) = d and SR(φ) = d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Define A = Cd, B = Cd·d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then, FLOSR(|ψ⟩ , |φ⟩) ≤ max F(R, Q↓ embed) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' TrB[R] = P↓ R ∈ P↓(d2 · d′) , where P and Q are the distributions defined by |ψ⟩ and |φ⟩’s Schmidt coefficients respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The above seems intuitively true from The- orem 6 as we have just relaxed the tensor product structure with the partial trace constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The tech- nical issue is the ordering operation ·↓ is defined in terms of a permutation of a fixed basis, so we need to make sure this works with the partial trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note the feasible set, the set we can optimizer over, in Theorem 6 is S1(P) := {(P ⊗ P′)↓ : P′ ∈ P(Σ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Now note this is the same as the set S2(P) := {(P↓ ⊗ P′↓)↓ : P′ ∈ P(Σ)} , because the ordering applied to the tensor product will result in the same thing regardless of whether or not P, P′ were ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Therefore, we can focus on P↓ ⊗ P′↓ to make the explanation clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In general, in terms of vectors, (p↓ ⊗ p′↓)↓ = � � � � � � p↓(1)p′↓ p↓(2)p′↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' p↓(d)p′↓ � � � � � � , where p(i) ≥ p(i + k) for k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Formally, we also have p↓(i)p′↓(1) ≥ p↓(i + k)p′↓(j) for all i ∈ [d], k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=', d − i}, and j ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' In partic- ular what this means is that without loss of general- ity for any i ∈ [d], p↓(i)p′↓(1) appears before any el- ement that is not of the form p↓(i − ℓ)p′↓(j) for some 0 < ℓ ≤ i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It follows that under the ordering of (p↓ ⊗ p′↓)↓, when the partial trace marginalizes to the A space, the induced ordering on the local space will be the ordering based on p↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Formally, this can be expressed as TrC|Σ|[(P↓ ⊗ P′↓)↓] = ∑ j∈Σ 1A ⊗ ⟨j| (P↓ ⊗ P′↓)↓ |j⟩ = ∑ i∈[d] p↓(i) |i⟩⟨i| , where the first equality is a representation of the partial trace and the second is using the property noted of the ordering on the joint ordered distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus, if X ∈ S2(P), TrC|Σ|(X) = P↓ and X ∈ P↓(d · |Σ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Noting that |Σ| = d · d′, this is the fea- sible set we have defined in the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The remaining point is to prove this is the semidefinite program given in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' There is much to the theory of semidefinite programs for quantum information [19], but for our purposes all we will need is the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' A semidefinite program may be ex- pressed as max Tr(AX) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Φ(X) = B XCd ⪯ 0 , where Φ ∈ T(Cd, Cd′) is a Hermitian-preserving map, A ∈ Herm(Cd), B ∈ Herm(Cd′), and Herm(·) is the space of Hermitian operators on a given Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The fidelity is known to be a semidefinite pro- gram [19], so we are really just verifying all of our constraints work and that we can write the SDP simply by making use of that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The optimization program in the pre- vious lemma, may be expressed as the following 25 semidefinite program over the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' max ∑ i∈[d2·d′] x(i) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' �diag(r) diag(x) diag(x) diag(q↓ embed) � ⪰ 0 TrB[diag(r)] = P↓ r ∈ P↓([d2 · d]) x ∈ Rd2·d′ , where d, d′ are defined in the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' We begin by expressing the objective func- tion of the previous lemma, which is in terms of fidelity, using the primal problem for the SDP for fidelity from [19, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='17]: max1 2 � Tr(X) + Tr � X†�� � R X X† Q↓ embed � ≥ 0 X ∈ L(C[d2·d′]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Now our goal is to reduce X to the diagonal of a real vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note that R, Q↓ embed are always invariant under pinching onto the computational basis of C[d2·d′], which we can denote ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Note that this pinching is a CPTP, so by the CP property, (idC2 ⊗ ∆) � R X X† Q↓ embed � = � R ∆(X) ∆(X†) Q↓ embed � ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' It also then follows as a positive semidefinite oper- ator is always Hermitian that � R ∆(X†) ∆(X) Q↓ embed � ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus by taking these two cases and averaging them, we have that � R 1 2 � ∆(X + X†) � 1 2 � ∆(X + X†) � Q↓ embed � ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Define X := 1 2 � ∆(X + X†) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Then note 1 2 � Tr(X) + Tr � X†�� =1 2 � Tr(∆(X)) + Tr � ∆(X†) �� =1 2 � Tr � X � + Tr � X†�� = Tr � X � , where the first equality is because the pinching is trace preserving, the second is by definition of X, as is the final equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus, for any X that sat- isfies the positivity constraint, we could replace it with X without loss of generality as we are con- sidering a maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Finally, note that X is a real diagonal matrix by the pinching along with the fact a + a∗ = 2 Re{a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus X = diag(x) for some x ∈ Rd2·d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Combining all these points and using Tr � X � = ∑i∈[d2·d′] x(i), we have reduced to consid- ering max ∑ i∈[d2·d′] x(i) �diag(r) diag(x) diag(x) diag(q↓ embed) � ≥ 0 x ∈ Rd2·d′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This argument works for any choice of diagonal r, so this is the major reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' What remains is to prove all the constraints are Hermitian maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' One can write the constraints for r ∈ P↓ as r(i) ≥ r(i + 1) for all i, which are semidef- inite constraints and can be written as Hermitian preserving maps on the variables r, x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' diag is a Hermitian preserving map as is the partial trace, so TrC[diag(r)] is a Hermitian preserving map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Like- wise is the block matrix mapping if one allows for the complex conjugate in the lower left block, but noting diag(x)† = diag(x), we can leave it as writ- ten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Thus all the maps are Hermitian-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The conversion to actual standard form we then omit as it provides no insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' The above two proofs establish Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content=' Appendix C: Data for Catalyst Figure For p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='5, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='7: Dimension Optimal distribution r 1 n/a 2 1 100[4, 6] 3 1 100[21, 32, 47] 4 1 100[12, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='28, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='415] 5 1 100[7, 11, 17, 26, 39] 6 1 100[5, 8, 11, 16, 24, 35] 7 1 100[3, 5, 8, 11, 16, 23, 24] 8 1 100[5, 6, 9, 9, 13, 14, 19, 25] 26 For p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='6, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} +page_content='65: Dimension Optimal distribution r 1 n/a 2 1 100[32, 63] 3 1 100[18, 31, 51] 4 1 100[10, 17, 28, 45] 5 1 100[6, 10, 16, 26, 42] 6 1 100[7, 11, 12, 18, 20, 32] 7 1 100[0, 7, 11, 12, 18, 20, 32] 8 1 100[0, 0, 7, 11, 12, 18, 20, 32]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E3T4oBgHgl3EQf0QsH/content/2301.04735v1.pdf'} diff --git a/W9FKT4oBgHgl3EQfoC5E/content/tmp_files/2301.11864v1.pdf.txt b/W9FKT4oBgHgl3EQfoC5E/content/tmp_files/2301.11864v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f426cd7fa29dae5dc5f99eb59294416c30e3c5a --- /dev/null +++ b/W9FKT4oBgHgl3EQfoC5E/content/tmp_files/2301.11864v1.pdf.txt @@ -0,0 +1,2545 @@ +arXiv:2301.11864v1 [physics.flu-dyn] 27 Jan 2023 +Under consideration for publication in J. Fluid Mech. +1 +Gravity can lead to multiple peaks in the early stages +of coffee ring formation +M. R. M O O R E1 +AND A. W. +W R A Y2 +1Department of Mathematics, School of Natural Sciences, University of Hull, Cottingham Road, Hull, HU6 7RX, +UK +2Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, +Glasgow G1 1XH, UK +(Received ?; revised ?; accepted ?. - To be entered by editorial office) +We consider the role of gravity in solute transport when a thin droplet evaporates. Under the physically- +relevant assumptions that the contact line is pinned and the solutal P´eclet number, Pe is large, we identify +two fundamental regimes that depend on the size of the Bond number, Bo. When Bo = O(1), the asymptotic +structure of solute transport follows directly from the surface tension-dominated regime, whereby advection +drives solute towards the contact line, only to be countered by local diffusive effects, leading to the for- +mation of the famous “coffee ring”. For larger Bond numbers, we identify the distinguished limit in which +Bo−1/2Pe2/3 = O(1), where the diffusive boundary layer is comparable to the surface tension boundary +layer. In each regime, we perform a systematic asymptotic analysis of the solute transport and compare +our predictions to numerical simulations of the full model. Our analysis identifies the effect of gravity on +the nascent coffee ring, providing quantitative predictions of the size, location and shape of the solute mass +profile. Furthermore, we reveal that, for certain values of Bo, Pe and the evaporation time, a secondary +peak may exist inside the classical coffee ring. We find that the onset of this secondary peak is linked to the +change in behaviour of the critical point in the droplet centre. Both the onset and the peak characteristics +are shown to be independent of Pe, but solutal diffusion may act to remove the secondary peak when the +classical coffee ring becomes so large as to subsume it. +Key words: +1. Introduction +The evaporation of sessile droplets has received significant attention in recent years, being the subject +of several major reviews (Cazabat & Guena 2010; Lohse et al. 2015; Brutin & Starov 2018; Wilson & +D’Ambrosio 2023) due to its ubiquity in theoretical, experimental and industrial settings. A particular +phenomenon of interest is the so-called “coffee ring effect”, in which a solute in such an evaporating droplet +ends up preferentially accumulated at the contact line (Deegan et al. 1997, 2000). This effect is very robust, +occurring even when the solution is initially uniformly dispersed throughout the droplet, and even when the +evaporative flux is not preferentially localised at the contact line (Boulogne et al. 2016). +Motivated by typical physical parameters, models of such systems typically assume that the P´eclet number +is sufficiently large that diffusive effects can be neglected, and so dynamics of the solute inside the droplet +are governed purely by convection (Deegan et al. 1997; Wray et al. 2021). This unphysical assumption leads +to a variety of undesirable side-effects, including singular accumulations of residue, and solute not being +conserved (Deegan et al. 2000). +A variety of attempts have been made to resolve this problem phenomenologically, including via the +incorporation of jamming effects (Popov 2005; Kaplan & Mahadevan 2015). However, jamming effects only +become significant close to the particle packing fraction, and the assumptions underpinning the model fail +long before this point. In particular, the assumption that diffusive effects can be ignored breaks down in a +diffusive boundary layer close to the contact line (Moore et al. 2021), as might be anticipated from the singular +accumulation in the na¨ıve, convection-only model. This boundary layer and its growth and dynamics have +been analysed and understood via matched asymptotics and careful numerics in situations where droplets +are small, and thus exist at quasi-static equilibrium due to surface tension (Moore et al. 2022), but little is +known for larger droplets where the effects of gravity are important. +Investigations of larger droplets have a long history, dating back to numerical integration of the appropriate +Laplace equations by Padday (1971) and Boucher & Evans (1975), with a variety of studies via asymptotics of + +2 +M. R. Moore & A. W. Wray +their shape (Rienstra 1990; O’Brien 1991; Yariv 2022) and stability (Pozrikidis 2012) in the intervening time. +The effect of gravity on droplets, and especially their internal flows, has experienced a recent resurgence of +interest due to the experiments of Edwards et al. (2018), which showed that the dynamics of binary droplets +can be sensitively dependent on droplet inclination (and hence gravity). This has since received extensive +investigation both experimentally and numerically (Li et al. 2019; Pradhan & Panigrahi 2017). +Notably, however, despite the original experiments of Deegan et al. (1997) involving large droplets, there +have been relatively few investigations of particle transport inside them, with those available being principally +experimental (Sandu & Fleaca 2011; Hampton et al. 2012; Devlin et al. 2016). This is perhaps because of the +robustness of the coffee-stain effect: asymptotic and numerical investigations (Barash et al. 2009; Kolegov +& Lobanov 2014) confirm the experimental results that the ring-stain is preserved unless additional physics +are incorporated, such as continuous particle deposition (Devlin et al. 2016). However, this neglects the bulk +of the story, including the dynamics of the residue over the course of the lifetime of the droplets: a critical +omission in situations such as continuous particle deposition. We show in the present work that the dynamics +are actually quite subtle and complex, and certainly merit detailed investigation. +The structure of this paper is therefore as follows. In §2, we describe the equations governing the fluid +flow and solute transport for the problem of a thin droplet evaporating under a diffusive flux, in particular +highlighting the effect of gravity in the model. We nondimensionalise the model and introduce the three key +dimensionless numbers in the model: the capillary, Bond and P´eclet numbers. In §3, we completely solve +for the liquid flow in the limit in which the solute is dilute, so that the flow and solute transport problems +decouple. We discuss pertinent features of the resulting fluid velocity and droplet shape, and in particular +how these features vary with the Bond number. +The bulk of the analysis in this paper concerns the influence of gravity on solute transport within the +droplet, which we analyse in the physically-relevant large-P´eclet number limit in §4. We find that there are +two distinct regimes depending on the relative sizes of the Bond and P´eclet numbers. In the first, where +the Bond number is moderate, we extend the asymptotic analysis of Moore et al. (2021) to include the +effect of gravity. However, when the Bond number is also large, a more complex asymptotic analysis is +necessary, which is presented in detail in Appendix A. In each asymptotic regime, we derive predictions for +the distribution of the solute mass within the droplet and compare the results to numerical simulations of +the full advection-diffusion problem. In particular, while we find the expected ‘nascent coffee ring’ profile in +the solute mass, for certain input parameters, we also find evidence of a novel phenomenon whereby a second +peak may also develop in the mass profile inside the classical coffee ring. +We analyse both of these peaks in detail in §5. In particular, for the classical coffee ring, we discuss +the effect of gravity in each of the two asymptotic regimes discussed in §4 and Appendix A, while for the +secondary peak, we investigate the key role gravity plays in its existence and how the secondary peak may +also be subsumed in the classical coffee ring for certain values of the Bond and P´eclet numbers. Finally, in +§6, we summarize our findings and discuss implications to various applications, as well as avenues for future +study. +2. Problem configuration +We consider the configuration depicted in figure 1 in which an axisymmetric droplet of initial volume +V ∗ +0 evaporates from a solid substrate. Here and hereafter, an asterisk denotes a dimensional variable. We +let (r∗, θ, z∗) be cylindrical polar coordinates centred along the line of symmetry of the droplet with the +substrate lying in the plane z∗ = 0: by axisymmetry, we shall assume that all the variables are independent +of θ. The droplet contact line is thus circular and we assume that it is pinned throughout the drying process, +which is observed in practice for a wide range of liquids for the majority of the drying time (Deegan et al. +1997; Hu & Larson 2002; Kajiya et al. 2008; Howard et al. 2023). We let r∗ = R∗ be the radius of the contact +line. Throughout this analysis, we shall assume that the droplet is thin, which reduces to the assumption +that +0 < δ = V ∗ +0 +R∗3 ≪ 1. +(2.1) +As we discuss presently, the thin-droplet assumption allows us to greatly simplify the flow and solute transport +models; the assumption has been extensively-validated and has shown to be reasonable even for droplets that +should realistically fall outside of this regime (Larsson & Kumar 2022). +The droplet consists of a liquid of constant density and viscosity denoted by ρ∗ and µ∗, respectively. The +droplet free surface is denoted by z∗ = h(r∗, t∗) and the air-water surface tension coefficient, σ∗ is assumed +to be constant. + +Gravity can lead to multiple peaks in the early stages of coffee ring formation +3 +z∗ +r∗ +2R∗ +z∗ = h∗(r∗, t∗) +E∗(r∗) +Figure 1: A side-on view of a solute-laden droplet evaporating under an evaporative flux E∗(r∗) from a solid +substrate that lies in the plane z∗ = 0. The droplet is axisymmetric and the contact line is assumed to be +pinned on the substrate at r∗ = R∗. The droplet free surface is denoted by h∗(r∗, t∗). The solute is assumed +to be inert and sufficiently dilute that the flow of liquid in the droplet is decoupled from the solute transport. +The liquid evaporates into the surrounding air and we assume that the evaporative process is quasi-steady, +which is a reasonable assumption for a wide range of liquid-substrate configurations (Hu & Larson 2002). +While there are a number of different viable evaporation models depending on the physical and chemical +characteristics of the problem (Murisic & Kondic 2011), for the purposes of this analysis, we assume that +the dominant process of vapour transport from the droplet surface is diffusion, so that the evaporative flux +E∗(r∗) is given by +E∗(r∗) = 2D∗(c∗ +s − c∗ +∞) +π +√ +R∗2 − r∗2 , +(2.2) +where D∗ is the diffusion coefficient and c∗ +s, c∗ +∞ are the surface and ambient vapour concentrations, respec- +tively (Deegan et al. 2000; Murisic & Kondic 2011). +The droplet contains an inert solute of initially uniform concentration φ∗ +0. The solute is assumed to be +sufficiently dilute that the flow and transport problems completely decouple. We shall discuss the validity of +the dilute assumption further in §6. +2.1. Flow model +The droplet is assumed to be sufficiently thin and the evaporation-induced flow sufficiently slow that the +flow is governed by the lubrication equations +∂h∗ +∂t∗ + 1 +r∗ +∂ +∂r∗ (r∗h∗u∗) = − E∗ +ρ∗ , +(2.3) +u∗ = − h∗2 +3µ∗ +∂p∗ +∂r∗ , +(2.4) +p∗ = p∗ +atm − ρ∗g∗(z∗ − h∗) − σ∗ 1 +r∗ +∂ +∂r∗ +� +r∗ ∂h∗ +∂r∗ +� +, +(2.5) +for 0 < r∗ < R∗, t∗ > 0, where u∗(r∗, t∗) is the depth-averaged radial fluid velocity, p∗(r∗, z∗, t∗) is the liquid +pressure and p∗ +atm denotes atmospheric pressure (Hocking 1983; Deegan et al. 2000; Oliver et al. 2015). +Equations (2.3)–(2.5) must be solved subject to the symmetry conditions +r∗h∗u∗ = ∂h∗ +∂r∗ = 0 +at +r∗ = 0, +(2.6a, b) +and the fact that the free surface touches down at, and we require no-flux of liquid through, the pinned +contact line, that is +h∗ = r∗h∗u∗ = 0 +at +r∗ = R∗. +(2.7a, b) +We close the problem by specifying the initial droplet profile, that is +h∗(r∗, 0) = h∗ +0(r∗) +for +0 < r∗ < R∗. +(2.8) +It is worth noting at this stage that, while this initial condition is needed to fully specify the mathematical +problem, in our analysis, we do not explicitly use the initial condition (2.8). In what follows, it is assumed + +4 +M. R. Moore & A. W. Wray +that the rate of evaporation is sufficiently slow that the droplet quickly relaxes under capillary action to the +quasi-steady profile found in §3 (see, for example, Lacey (1982); De Gennes (1985); Oliver et al. (2015)). +Thus, we shall for simplicity assume that h0(r) is of the same functional form of the free surface we find in +§3. While this assumption is reasonable for a wide range of applications, for extremely rapid evaporation (for +example, laser-induced evaporation, Volkov & Strizhak (2019)), a more careful consideration of the evolution +after deposition would be needed. +Assuming the contact line is pinned, the volume of the droplet V ∗(t∗) is given by +V ∗(t∗) = 2π +� R∗ +0 +r∗h∗(r∗, t∗) dr∗, +V ∗(0) = V ∗ +0 . +(2.9) +The total mass loss due to evaporation F ∗(t∗) is given by +F ∗(t∗) = 2π +� R∗ +0 +r∗E∗(r∗) dr∗ = 4D∗(c∗ +s − c∗ +∞)R∗. +(2.10) +Thus, conservation of mass in the liquid phase is +dV ∗ +dt∗ = −F ∗ +ρ∗ = −4D∗(c∗ +s − c∗ +∞)R∗ +ρ∗ +(2.11) +so that +V ∗(t∗) = V ∗ +0 − 4D∗(c∗ +s − c∗ +∞)R∗t∗ +ρ∗ +. +(2.12) +In particular, the dryout time, that is the time when the drop has fully evaporated, is +t∗ +f = +ρ∗V ∗ +0 +4D∗(c∗s − c∗∞)R∗ . +(2.13) +2.2. Solute model +The droplet is assumed to be sufficiently thin that the transport of the solute is governed by the depth- +averaged advection-diffusion equation +∂ +∂t∗ (h∗φ∗) + 1 +r∗ +∂ +∂r∗ +� +r∗ +� +h∗u∗φ∗ − D∗ +φh∗ ∂φ∗ +∂r∗ +�� += 0 +(2.14) +for 0 < r∗ < R∗, t∗ > 0, where φ∗(r∗, t∗) is the depth-averaged solute concentration and D∗ +φ is the solutal +diffusion coefficient (Wray et al. 2014; Pham & Kumar 2017; Moore et al. 2021). +While there is an acknowledged effect of the solute particles eventually being trapped at and transported +along the free surface (Kang et al. 2016; D’Ambrosio 2022), this effect is less pronounced for thin droplets, +where the capture tends to occur closer to the contact line due to the stronger outward radial flow. Thus, we +shall neglect its effects here as our study concerns the interplay between gravity, surface tension and solute +advection/diffusion. A more focused analysis on the final deposit profile would certainly need to account for +such effects. +Equation (2.14) must be solved subject to the symmetry condition +∂φ∗ +∂r∗ = 0 +at +r∗ = 0, +(2.15) +and the condition that there can be no flux of solute particles through the pinned contact line, +r∗ +� +h∗u∗φ∗ − D∗ +φh∗ ∂φ∗ +∂r∗ +� += 0 +at +r∗ = R∗. +(2.16) +Finally, we impose the initially uniform distribution of solute throughout the droplet, so that +φ∗(r∗, 0) = φ∗ +0 +for +0 < r∗ < R∗. +(2.17) +2.3. Non-dimensionalization +We assume that the fluid velocity is driven by evaporation and, for now, we retain both gravity and surface +tension, so that the pertinent scalings are +(r∗, z∗) = R∗(r, δz), +u∗ = D∗(c∗ +s − c∗ +∞) +δρ∗R∗ +u, +t∗ = t∗ +ft, +φ∗ = φ∗ +0φ, +(h∗, h∗ +0) = δR∗(h, h0), +p∗ = p∗ +atm + µ∗D∗(c∗ +s − c∗ +∞) +δ3ρ∗R∗2 +p +V ∗ = V ∗ +0 V. +(2.18) + +Gravity can lead to multiple peaks in the early stages of coffee ring formation +5 +Note, in particular, that the choice of timescale fixes the dimensionless dryout time to be t = 1. +Upon substituting the scalings (2.18) into (2.3)–(2.5), we see that +∂h +∂t + 1 +4r +∂ +∂r (rhu) = − +1 +2π +√ +1 − r2 , +(2.19) +u = h2 +3Ca +∂ +∂r +� +−Boh + 1 +r +∂ +∂r +� +r∂h +∂r +�� +, +(2.20) +for 0 < r < 1, 0 < t < 1, where the Capillary and Bond numbers are defined by +Ca = µ∗D∗(c∗ +s − c∗ +∞) +δ4ρ∗R∗σ∗ +and +Bo = ρ∗g∗R∗2 +σ∗ +, +(2.21) +respectively. +Under scalings (2.18), the symmetry conditions (2.6) become, +rhu = ∂h +∂r = 0 +at +r = 0, +(2.22a, b) +while the contact line conditions (2.7) are +h = rhu = 0 +at +r = 1. +(2.23a, b) +The initial condition (2.8) becomes +h(r, 0) = h0(r) +for +0 < r < 1. +(2.24) +Finally, the dimensionless form of conservation of liquid volume conditions (2.9) and (2.12) is +1 − t = 2π +� 1 +0 +rh(r, t) dr. +(2.25) +After scaling, the solute transport equation (2.14) becomes +∂ +∂t (hφ) + 1 +4r +∂ +∂r +� +r +� +huφ − h +Pe +∂φ +∂r +�� += 0 +(2.26) +for 0 < r <, 0 < t < 1, where the solutal P´eclet number is +Pe = D∗(c∗ +s − c∗ +∞) +δρ∗D∗ +φ +. +(2.27) +The symmetry (2.15) and boundary conditions (2.16) become +∂φ +∂r = 0 +at +r = 0 +(2.28) +and +r +� +huφ − h +Pe +∂φ +∂r +� += 0 +at +r = 1, +(2.29) +respectively. Finally, the initial condition (2.17) becomes +φ(r, 0) = 1 +for +0 < r < 1. +(2.30) +2.4. Integrated mass variable formulation +The assumption that the solute is dilute decouples the flow and solute transport problems, so that we may +solve for h and u from (2.19)–(2.25) independently of the solute concentration, φ. We shall discuss the +resulting flow solution shortly in §3. +First, however, we present a reformulation of the solute transport problem (2.26)–(2.30), which will greatly +aid us in our asymptotic and numerical investigations. In this, we follow Moore et al. (2021, 2022) by +introducing the integrated mass variable +M(r, t) = +� r +0 +sh(s, t)φ(s, t) ds. +(2.31) +By integrating the advection-diffusion equation (2.26) from 0 to r and applying the no-flux condition (2.29), + +6 +M. R. Moore & A. W. Wray +we find that +∂M +∂t + +�u +4 + +1 +4Pe +�1 +r + 1 +h +∂h +∂r +�� ∂M +∂r − +1 +4Pe +∂2M +∂r2 += 0 +for +0 < r, t < 1. +(2.32) +This must be solved subject to the boundary conditions +M(0, t) = 0, +M(1, t) = 1 +2π +for +t > 0, +(2.33a, b) +where the latter condition dictates that mass is conserved along a radial ray, which replaces the no-flux +condition (2.29). Finally, the initial condition (2.30) becomes +M(r, 0) = +� r +0 +sh(s, 0) ds +for +0 < r < 1. +(2.34) +Finally, we note that, once we have determined the integrated mass variable from (2.32)–(2.34), the solute +mass m = φh can then be retrieved from +m = 1 +r +∂M +∂r . +(2.35) +3. Flow solution in the large-Ca limit +We now suppose that surface tension dominates viscosity in the flow problem, that is Ca ≫ 1. Importantly, +this means that the problems for the free surface profile and the flow velocity decouple, an assumption that +is valid for a wide range of different liquids and evaporation models in practice (Moore et al. 2021, 2022). +Unlike these previous studies, however, we shall retain gravity in (2.20) to investigate what role it plays in +the formation of the nascent coffee ring. +To this end, we neglect the left-hand side of (2.20), so that upon integrating and applying the symmetry +condition (2.22), the contact line condition (2.23a) and the conservation of liquid volume condition (2.25), +we deduce that +h(r, t) = (1 − t) +π +I0( +√ +Bo) +I2( +√ +Bo) +� +1 − I0( +√ +Bo r) +I0( +√ +Bo) +� +, +(3.1) +where Iν(z) is the modified Bessel function of the first kind of order ν. +With the free surface found, the velocity is determined immediately from (2.19) and the no-flux condition +(2.23b) to be +u(r, t) = 1 +rh +� +2 +π +� +1 − r2 + 4I0( +√ +Bo) +πI2( +√ +Bo) +�r2 − 1 +2 ++ +1 +√ +BoI0( +√ +Bo) +(I1( +√ +Bo) − rI1( +√ +Bo r)) +�� +. +(3.2) +Notably, as in the surface tension-dominated regime where Bo → 0, time is separable in both the free +surface and fluid velocity profiles, and so merely acts to scale the functional form. In particular, this means +that the streamlines and pathlines coincide, which we shall exploit when considering the regime in which +solutal diffusion is negligible in §5.2. +We display the scaled forms of the free surface and fluid velocity for various values of the Bond number +in figure 2a,b. For the droplet free surface profile, we see the expected transition from the spherical cap for +Bo → 0 (Deegan et al. 2000) to the flat ‘pancake’ droplet for Bo → ∞ (Rienstra 1990). For each Bond +number, the velocity is singular at the contact line — as expected for a diffusive evaporative flux (see, for +example, Deegan et al. (2000)). We see that as the effect of gravity increases, the sharp increase in u occurs +closer to the contact line, corresponding to the progressively smaller region in which surface tension effects +are important. +Finally, since this will be important in our discussions of the secondary peaks seen in the solute mass profile +in §5.2, we show the divergence of the fluid velocity in figure 2c. For small Bond numbers, the divergence is +monotonically increasing with r and, as with the velocity, singular at the contact line. However, for moderate +and large Bond numbers ≳ 15, we see a clear change of behaviour, with a region of non-monotonic behaviour +in the droplet interior. This behaviour is accentuated as Bo → ∞. +For future reference, the asymptotic behaviours of the free surface and fluid velocity as r → 1− for +Bo = O(1) are given by +h = θc(t; Bo)(1 − r) + O((1 − r)2), +(3.3) +u = +2χ +θc(t; Bo)(1 − r)−1/2 + O +� +(1 − r)1/2� +, +(3.4) + +Gravity can lead to multiple peaks in the early stages of coffee ring formation +7 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.5 +1 +0 +1 +2 +3 +4 +5 +0 +0.5 +1 +0 +1 +2 +3 +4 +5 +Figure 2: (a) The quasi-steady droplet free surface, (b) the fluid velocity, and (c) the divergence of the +velocity displayed for Bo = 0.1 (black), 1 (dark purple), 10 (blue), 20 (cyan), 50 (green) and 100 (yellow). +Notably, we see the transition from the spherical cap to the ‘pancake’ droplet profile as the effect of gravity +increases. The divergence of the fluid velocity also shows a transition from a monotonic to a non-monotonic +profile as the Bond number increases. +where +θc(t; Bo) = − lim +r→1− +∂h +∂r = (1 − t)ψ(Bo), +ψ(Bo) = +√ +BoI1( +√ +Bo) +πI2( +√ +Bo) +(3.5) +is the leading order contact angle in the thin droplet limit and +χ = +√ +2 +π +(3.6) +is the dimensionless coefficient of the inverse square root singularity at the contact line in the evaporative +flux (2.2). Note that we have chosen this notation to highlight the similarities with the previous analysis of +Moore et al. (2022), who consider a surface tension-dominated droplet of arbitary contact set. +On the other hand, if we take 1 − r = O(1) and consider the large-Bo limit of (3.1), (3.2), we find that +h = h0(t) + Bo−1/2h1(t) + O(Bo−1), +(3.7) +u = u0(r, t) + Bo−1/2u1(r, t) + O(Bo−1), +(3.8) +as Bo → ∞, where +h0(t) = (1 − t) +π +, +h1(t) = 2(1 − t) +π +, +(3.9a, b) +and +u0(r, t) = 2 +√ +1 − r2 +r(1 − t) (1 − +� +1 − r2), +u1(r, t) = +4 +r(1 − t)(1 − +� +1 − r2). +(3.10a, b) +Notably, in the droplet bulk, the droplet free surface h is flat to all orders: the aforementioned characteristic +of ‘pancake’ droplets associated with large Bond numbers (Rienstra 1990). These expansions break down +close to the contact line where surface tension effects become important. We find that for 1 − r = Bo−1/2¯r, +we have +h = ¯h0(¯r, t) + Bo−1/2¯h1(¯r, t) + O(Bo−1), +(3.11) +u = Bo−1/4 � +¯u0(¯r, t) + Bo−1/4¯u1(¯r, t) + Bo−1/2¯u2(¯r, t) + O(Bo−3/4) +� +(3.12) +as Bo → ∞, where +¯h0(¯r, t) = 1 +π (1 − t)(1 − e−¯r), +(3.13) +¯h1(¯r, t) = 2(1 − t) +π +� +1 − e−¯r� +− (1 − t)¯r +2π +e−¯r, +(3.14) + +8 +M. R. Moore & A. W. Wray +and +¯u0(¯r, t) = +2 +√ +2¯r +(1 − t)(1 − e−¯r), +(3.15) +¯u1(¯r, t) = +4 +(1 − t) − +4¯r +(1 − t)(1 − e−¯r), +(3.16) +¯u2(¯r, t) = +3¯r3/2 +√ +2(1 − t)(1 − e−¯r) − +4 +√ +2¯r +(1 − t)(1 − e−¯r) + +√ +2¯r3/2e−¯r +(1 − t)(1 − e−¯r)2 . +(3.17) +We note here that as ¯r → 0, we retrieve the expect inverse square root singularity in the fluid velocity. +4. Solute transport in the large-Pe limit +Having fully determined the leading-order flow, we now seek to understand the transport of solute within +the drop and to make predictions about the early-stages of coffee ring formation. We follow the analyses of +Moore et al. (2021, 2022) by considering the physically-relevant regime in which Pe ≫ 1. In this regime, in +the bulk of the droplet, advection dominates solutal diffusion, with the latter only being relevant close to +the contact line. +Previous studies of this problem have concentrated on surface tension-dominated drops (i.e. Bo → 0) and +have shown how the competition between solutal advection and diffusion near the contact line leads to the +early stages of coffee ring formation in drying droplets. In this analysis, we wish to investigate how this +behaviour changes as we allow Bo to vary, which we pursue using a hybrid asymptotic-numerical approach. +There are naturally several different asymptotic regimes depending on the relative sizes of Bo and Pe, but +these broadly fall into two categories +i) intermediate Bond number, Bo = O(1), where the asymptotic structure of the solute transport depends +solely on the large P´eclet number; +ii) large Bond number, Bo ≫ 1, where the asymptotic structure of the solute transport now depends on +the relative sizes of Bo and Pe. +In the first regime where Bo = O(1), Pe ≫ 1, the asymptotic structure of the flow is a natural extension of +the surface tension-dominated case considered in Moore et al. (2021). In the droplet bulk where 1−r = O(1), +solute advection dominates diffusion. However, close to the contact line, a balance between solute advection +and diffusion occurs when +rhuφ ∼ rh +Pe +∂φ +∂r =⇒ 1 − r = O(Pe−2). +(4.1) +We discuss the asymptotic solution for this regime in §4.1. +In the second regime, there are several different possibilities depending on the relative sizes of the boundary +layer where surface tension enters the flow profile and the solutal diffusion boundary layer. The richest +distinguished asymptotic limit is that in which these boundary layers are comparable. As detailed in §3, for +large Bond number the free surface is flat in the bulk of the droplet, with the effect of surface tension restricted +to a boundary layer at the contact line of size 1 − r = O(Bo−1/2), where h = O(1) and u = O(Bo−1/4). +Turning to the solute transport equation (2.26), since h is order unity and u is square root bounded in this +region, advection and diffusion are comparable when +1 − r = O(Pe−2/3). +(4.2) +Hence, in the most general limit in which the size of the two boundary layers are comparable, we have +α = Bo−1/2Pe2/3 = O(1). +(4.3) +The asymptotic analysis in this regime is somewhat more involved, so for brevity, we present the details in +Appendix A. +4.1. Asymptotic solution when Bo = O(1) +In this section, we present the asymptotic solution of the solute transport problem as Pe → ∞ when +Bo = O(1). The analysis herein is a natural extension of Moore et al. (2021). For the purposes of this section, +we shall use the concentration form of the advection-diffusion equation (2.26)–(2.30) and, in particular, find +the solution in terms of the solute mass m = φh, where h is given by (3.1). + +Gravity can lead to multiple peaks in the early stages of coffee ring formation +9 +4.1.1. Outer region +In the droplet bulk where 1−r = O(1), we seek a solution of the form m = m0(r, t)+O(Pe−1) as Pe → ∞. +Substituting into (2.26), (2.30), we find that +∂m0 +∂t ++ 1 +4r +∂ +∂r (rm0u) = 0 +for +0 < r < 1, t > 0 +(4.4) +where u is given by (3.2), subject to m(r, 0) = h(r, 0). This is the usual advection equation, with solution +given by +m0(r, t) = h(R, 0) +J(R, t) , +(4.5) +where R is the initial location of the point that is at r at time t and J(R, t) is the Jacobian of the Eulerian- +Lagrangian transformation, that satisfies Euler’s identity, +D +Dt(log J) = 1 +4r +∂ +∂r(ru), +J(R, 0) = 1, +(4.6) +where D/Dt is the convective derivative. +A straightforward asymptotic analysis of (4.4) reveals that +u∂m +∂r ∼ m +r +∂ +∂r (ru) +(4.7) +as r → 1−, so that m0 = O(√1 − r) as r → 1−, and hence the concentration φ0 is square root singular. This +sharp local concentration increase necessitates the inclusion of a diffusive boundary layer. +4.1.2. Inner region +Close to the contact line, we set +r = 1 − Pe−2ˆr, +h = Pe−2ˆh, +u = Peˆu, +m = Pe2 ˆm, +(4.8) +where the last scaling on the mass comes from global conservation of solute considerations (Moore et al. +2021). We seek an asymptotic solution of the form ˆm = ˆm0(ˆr, t) + O(Pe−1) and find to leading order +∂ +∂ˆr +�� +2χ +θc(t; Bo) +√ +ˆr +− 1 +ˆr +� +ˆm0 + ∂ ˆm0 +∂ˆr +� += 0 +in +ˆr > 0, t > 0 +(4.9) +such that +� +2χ +θc(t; Bo) +√ +ˆr +− 1 +ˆr +� +ˆm0 + ∂ ˆm0 +∂ˆr += 0 +for +ˆr = 0. +(4.10) +It is straightforward to show that the solution to (4.9)–(4.10) is given by +ˆm0(ˆr, t) = C(t; Bo)ˆrexp +� +− +4χ +θc(t; Bo) +√ +ˆr +� +, +(4.11) +where, by pursuing a similar matching process to Moore et al. (2022), we find that the coefficient C(t; Bo) +is given by +C(t) = +64χ4 +3θc(t; Bo)4 N(t; Bo), +(4.12) +where N(t; Bo) is the leading-order accumulated mass advected into the contact line region up to time t, +viz. +N(t; Bo) = 1 +4 +� t +0 +m0(r, τ)u(r, τ) dτ. +(4.13) +It is worth noting that this solution follows directly from the Bo = 0 regime discussed in Moore et al. (2021, +2022), with the alterations due to gravity entering into the accumulated mass flux into the contact line and +the leading order contact angle. In particular, we note that in the limit Bo → 0, since ψ = 4/π + O(Bo), this +yields the expected form found in the surface tension-dominated problem in Moore et al. (2022) (see §3.7.2 +therein). We display the accumulated mass flux and the local contact angle for a wide range of Bond numbers +in figure 3. We see that as the influence of gravity increases, the acccumulated mass flux into the contact +line at a fixed percentage of the evaporation time is reduced from the surface tension-dominated regime. On +the other hand, the local contact angle increases, commensurate with the droplet profile transitioning from + +10 +M. R. Moore & A. W. Wray +0 +0.5 +1 +0 +1 +2 +3 +4 +Figure 3: (a) The accumulated mass flux, N(t; Bo) as defined by (4.13) and (b) the leading order local +contact angle θc(t; Bo) as defined by (3.5), for Bo = 10−2 (purple), Bo = 10−1 (purple) (dark blue), Bo = 1 +(light blue), Bo = 10 (green) and Bo = 102 (yellow). +a spherical cap to a ‘pancake’ droplet. We note that this combined behaviour leads to C(t; Bo) decreasing +as Bo increases. We discuss how these findings impact coffee ring formation in more detail in §5.1.1. +4.1.3. Composite solution +We may use van Dyke’s rule (Van Dyke 1964) to formulate a leading-order composite solution for the +solute mass that is valid throughout the drop by combining the leading-order-outer solution (4.5) and the +leading-order-inner solution (4.11), finding +mcomp(r, t) = mouter(r, t) + Pe2m +� +Pe2(1 − r), t +� +. +(4.14) +4.2. Comparisons between the numerical and asymptotic results +Our asymptotic predictions are compared to numerical simulations of the full advection-diffusion problem for +the integrated mass variable given by (2.32)–(2.34). The integrated mass variable is chosen over the solute +mass m or the concentration φ since it is better behaved close to the contact line. The numerical procedure +requires careful consideration of the thin diffusive boundary layer and we follow a similar approach to that +described for the surface tension-dominated problem by Moore et al. (2021). We give a summary of the +methodologies in Appendix B. +We begin by comparing the asymptotic predictions of the solute mass profiles to numerical solutions in +the regime where Bo = O(1). In figure 4, we display asymptotic (dashed, red) and numerical (solid, blue) +curves at 10% intervals of the total drying time for Pe = 102, Bo = 1 (a,b) and Pe = 102, Bo = 30 (c,d). +In each figure, we see excellent agreement between the simulations of the full system and the leading-order +composite solution (4.14). There is a clear formation of the expected coffee ring in the region near the contact +line, where solutal diffusion and advection interact. We see that increasing the Bond number in this regime +leads to a slight reduction of the size of the coffee ring. +This behaviour is reminiscent of the Bo = 0 regime considered previously by Moore et al. (2021). However, +in the later stages of the Pe = 102, Bo = 30 example, we see evidence of a qualitative difference in behaviour, +with the formation of another peak in the mass profile in the droplet interior (see inset in figure 4(c)). +Henceforth, we shall refer to the classical coffee ring as the primary peak and this new feature as the +secondary peak. The presence of the secondary peak depends on the Bond number, as there is no secondary +peak in any of the profiles when Bo = 1, but it also depends on the drying time, as the peak only develops +in the later stages of evaporation when Bo = 30 (between 60 − 70% of the drying time). Noticeably, the +secondary peak is significantly smaller in magnitude than the primary peak. +For larger Bond numbers, we compare the numerical results to the asymptotic predictions in Appendix +A. In figure 5, we display results for Pe = 102, Bo = 105 (α ≈ 0.07) (a,b) and Pe = 103 and Bo = 104 +(α = 1) (c,d). In each case, we display the composite profile for the solute mass given by (A 34). In each +figure, we see that after an initial transient the asymptotic predictions and numerical results are again in +excellent agreement. Moreover, we see further evidence of the existence of a secondary peak in the case + +Gravity can lead to multiple peaks in the early stages of coffee ring formation +11 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +10 -6 +10 -4 +10 -2 +10 0 +10 -3 +10 -1 +10 1 +10 3 +10 5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +10 -6 +10 -4 +10 -2 +10 0 +10 -3 +10 -1 +10 1 +10 3 +10 5 +0.4 +0.6 +0.1 +0.105 +Figure 4: Profiles of the solute mass when an axisymmetric droplet evaporates under a diffusive evaporative +flux for (a,b) Pe = 102 and Bo = 1 (c,d) Pe = 102 and Bo = 30. In each figure, the bold, black curve +represents the initial mass profile, which corresponds to the droplet free surface profile (3.1). We also display +plots at time intervals of 0.1 up to t = 0.9 in which solid, blue curves represent the results from the numerical +solution of (2.32)–(2.34) and the dashed, red curves show the leading-order composite mass profile, given by +(4.14). The right-hand figures display a close-up of the profiles near the contact line. In (c), the inset shows +a close up of the mass profile in the droplet interior at t = 0.9 where we see a clear formation of a secondary +peak. +Pe = 103, Bo = 104 regimes, where the peak appears much earlier and is noticeably larger than that in +the previous example (cf. figure 4c, where Pe = 102, Bo = 30). However, we also note again the strong +dependence of the secondary peak on Bo and, possibly, Pe, as there is no evidence of such an interior peak +when Pe = 102, Bo = 105. +These findings prompt us to investigate this new feature more closely, alongside a discussion of how the +characteristics of the primary peak — and hence the classical coffee ring — depend on the Bond number. +5. Properties of the two peaks +Given the excellent comparisons displayed in the previous section, we seek to use our asymptotic results to +investigate properties of the nascent coffee ring and, in particular, the new feature of these moderate-to-large +Bond number regimes: the secondary peak. +5.1. Primary peak +We shall begin by discussing the effect of the Bond number on the primary peak. As in previous studies +of the surface tension-dominated regime, the formation of the primary peak is driven by the competing +diffusive and advective solute fluxes (Moore et al. 2021, 2022) and is always present in the large-Pe regime. +Furthermore, since all of the features of interest are well within the solutal diffusion boundary layer, we will + +12 +M. R. Moore & A. W. Wray +Mass +Mass +Figure 5: Profiles of the solute mass when an axisymmetric droplet evaporates under a diffusive evaporative +flux for (a,b) Pe = 102 and Bo = 105 (α ≈ 0.07) (c,d) Pe = 103 and Bo = 104 (α = 1). In each figure, +the bold, black curve represents the initial mass profile (2.34). We also display plots at time intervals of 0.1 +up to t = 0.9 in which solid, blue curves represent the results from the numerical solution of (2.32)–(2.34) +and the dashed, red curves show the composite mass profiles, given by (A 33) for the integrate mass variable +and (A 34) for the solute mass, respectively. Note that in (c,d), we can clearly see the development of the +secondary peak behind the primary peak. +use the inner solution — as discussed in §4.1.2 in the Bo = O(1) regime and §A.2 in the large-Bo regime — +to do this. +5.1.1. Bo = O(1) regime +When the Bond number is order unity, the analysis is a natural extension of that in Moore et al. (2021, +2022). The local solute profile is dominated by the leading-order inner solution (4.11). Introducing the time- +dependent P´eclet number +Pet = +Pe +1 − t, +(5.1) +the nascent coffee ring profile may be seen to have the similarity form +ˆm0(R, t) +Pe2 +t N(t; Bo) = +2χ +3ψ(Bo)f +�√ +R, 3, +4χ +ψ(Bo) +� +, +R = Pe2 +t(1 − r) +(5.2) +where ψ and χ retain their definitions from (3.5) and (3.6) as the initial local contact angle and the coefficient +of the evaporative flux singularity, respectively, and f(x, k, l) = lkxk−1e−lx/Γ(k) is the probability density +function of a gamma distribution. It is this functional form which describes the characteristic narrow, sharp +peak of the coffee ring. +Since the definition of R only depends on the time-dependent P´eclet number, we can clearly illustrate the + +Gravity can lead to multiple peaks in the early stages of coffee ring formation +13 +0 +20 +40 +60 +80 +100 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +Figure 6: The similarity profile (5.2) of the leading-order-inner solute mass profile for Bo = 10−2 (purple), +Bo = 10−1 (purple) (dark blue), Bo = 1 (light blue), Bo = 10 (green) and Bo = 102 (yellow). +effect of gravity by plotting the similarity profile (5.2) for a range of Bond numbers in figure 6. We see that, +as the effect of gravity increases, the height of the primary peak decreases, and the peak moves further from +the pinned contact line. Moreover, the shape of the primary peak tends towards a shallower, wider profile. +Notably, this behaviour is driven purely by changes in ψ(Bo); as we saw in figure 3a, the accumulated mass +flux into the contact line decreases with the Bond number, clearly this acts to accentuate this behaviour. +We can expand upon these results by finding the leading order asymptotic prediction of the primary peak +height and location, which are given by +rpeak,I(t; Bo) = 1 − ψ(Bo)2 +4Pe2 +t χ2 , +mpeak,I(t; Bo) = 16Pe2 +tN(t; Bo)χ2 +3e2ψ(Bo)2 +, +(5.3) +respectively. Notably, while gravity only influences the location of the primary peak through the initial local +contact angle, ψ(Bo), the height depends on gravity through both the contact angle and the accumulated +mass flux, N(t; Bo). In particular, referring back to figure 3, this means that gravity has a stronger effect on +the peak height than its location. +We illustrate the veracity of these asymptotic predictions by comparing them to the corresponding numer- +ical results for Pe = 102 and a range of Bond numbers in figure 7. As anticipated from the comparisons of +the solute mass profiles, we see excellent agreement between the asymptotic predictions and the numerical +results. In particular, in figure 7a, we note that as the influence of gravity increases (i.e. Bo increases), the +coffee ring effect is inhibited: although a peak clearly still forms, it is lower for large Bond number at a similar +stage of the drying process. This effect varies nonlinearly with time (cf. figure 3a). For example, considering +the cases Bo = 1/2 and Bo = 30, after 50% of the drying time, the peak height is reduced by a factor of +≈ 3.97, while at 60% of the drying time, the reduction is a factor of ≈ 3.85 and at 90% of the drying time, +it is ≈ 3.63. +Similarly, in figure 7b, we see that as the Bond number increases, the location of primary peak moves +further from the contact line and that this significantly increases as the Bond number gets larger. For +Bo = 1/10, 1/2, 1 the location is almost indistinguishable from the zero-Bond number solution — where +Pe2 +t (1 − r) = 2 (Moore et al. (2022)) — but for Bo = 30, this has increased to ≈ 6.79. +It is worth noting that in all this analysis, the P´eclet number simply acts to scale the above findings. For +a larger P´eclet number, the height of the primary peak increases, while it is located closer to the contact +line. This is precisely what is seen for the Bo = 0 regime (Moore et al. 2021). +5.1.2. Large-Bo regime +In the large-Bo regime, given the size of the primary peak, we anticipate that the leading-order-inner +solution +˜ +M0(˜r, t) as given by (A 16) should reasonably capture the features of the primary peak. However, + +14 +M. R. Moore & A. W. Wray +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.5 +1 +1.5 +2 +2.5 +3 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +2 +4 +6 +8 +10 +Figure 7: Numerical (circles) and asymptotic predictions (solid lines) of (a)) the height of the primary peak, +mpeak,I(t)/Pe2/3 +t +and (b)) its location Pe2/3 +t +(1 − rpeak,I(t)) in the Bo = O(1) regime as given by (5.3). For +each curve, Pe = 102, while the Bond number varies according to Bo = 1/10 (dark purple), Bo = 1/2 (blue), +Bo = 1 (green) and Bo = 30 (yellow). +0 +0.2 +0.4 +0.6 +0.8 +1 +10-6 +10-4 +10-2 +100 +102 +104 +106 +0 +0.2 +0.4 +0.6 +0.8 +1 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +Figure 8: Numerical (circles) and asymptotic predictions (solid lines) of (a)) the height of the primary peak, +MI(t) = mpeak,I(t)/Pe2/3 and (b)) its location ηI(t) = Pe2/3(1−rpeak,I(t)) as given by (5.8)–(5.9). Results are +presented for Pe = 102, Bo = 103 (α ≈ 0.68, yellow), Pe = 103, Bo = 104 (α = 1, green), Pe = 104, Bo = 105 +(α ≈ 1.47, blue) and Pe = 105, Bo = 105 (α ≈ 6.81, dark purple). +unlike its moderate-Bo counterpart, there is no simple similarity form for the solution in this regime, so that +we proceed more carefully. +We denote the height and location of the primary peak by +mpeak,I(t) = Pe2/3MI(t), +rpeak,I(t) = 1 − Pe−2/3ηI(t), +(5.4a, b) +respectively. By (2.35), the location of the maximum ηI(t) satisfies +0 = ∂2 ˜ +M0 +∂˜r2 (ηI(t), t). +(5.5) + +Gravity can lead to multiple peaks in the early stages of coffee ring formation +15 +Utilizing (A 13), we find that +∂2 ˜ +M0 +∂˜r2 (ηI(t), t) = − +� +˜u0 − 1 +˜h0 +∂˜h0 +∂˜r +� +∂ ˜ +M0 +∂˜r +����� +(ηI(t),t) += 0. +(5.6) +Since ∂ ˜ +M0/∂˜r > 0 for ˜r > 0, we conclude +˜u0(ηI(t), t) − +1 +˜h0(ηI(t), t) +∂˜h0 +∂˜r (ηI(t), t) = 0 +(5.7) +so that +ηI(t) = α +2 W0 +�(1 − t)2 +4α3 +� +, +(5.8) +where W0(x) is the Lambert-W function (i.e. the solution to wew = x). +With ηI(t) in hand, the corresponding height of the ring at the peak is then given by +MI(t) = +� −B0(t) +I(ηI(t), t) +� += +� +N(t) +I(ηI(t), t) +�� ∞ +0 +1 +I(s, t) ds +�−1� +, +(5.9) +where I(r, t) is given by (A 15) and N(t) is the leading-order accumulated mass flux into the boundary layer +(A 18). Note that, in this regime, N(t) is independent of α and, hence, the Bond number, but the function +I(r, t) does change with α. +In figure 8, we plot the asymptotic predictions of the location and height of the primary deposit peak +against the simulation results for a range of different P´eclet and Bond numbers (and, correspondingly, α). +There are several discernible features. After an initial transient, the location of the peak is captured extremely +well by the asymptotic prediction (5.8) for each case presented. This initial transient is primarily due to the +lack of a distinct peak at early stages of the drying process; a period of time is necessary for sufficient solute +to be advected to the contact line. This process takes longer for smaller P´eclet numbers, i.e. when diffusion +is relatively stronger. The height of the primary peak is captured quite well by the asymptotic prediction +(5.9), particularly for larger P´eclet numbers and as time increases. It is worth noting that the error in the +approximation of the height is O(Pe1/3), so for an improved estimation of the primary peak height, it would +be necessary to consider the first two inner solutions ˜ +M0(˜r, t) and ˜ +M1(˜r, t). While this is possible, the results +do not have a simple analytic form, so are not practical to work with. We also note that, as the droplet +evaporates, the primary peak both increases in size and moves closer to the contact line, i.e. MI(t) increases +and ηI(t) decreases as t increases. +5.2. Secondary peak +As evidenced by the solute mass profiles, the behaviour of the secondary peak — and indeed, even its presence +— is more complex than that of the primary peak, which always forms in the large-Pe regime. We have seen, +for example in figure 4 in the Bo = O(1) regime, that the presence of the peak varies with both Bo and +drying time, while when Bo ≫ 1, we have also seen variation with Pe (and hence α), see for example figure +5. This gives a clear indication that we need to treat this feature more carefully. +To begin, we will consider whether or not the secondary peak is present. We shall first fix the P´eclet +number and use the numerical results to produce a regime diagram in (Bo, t)-parameter space indicating +whether one or two peaks are present in the solute mass profile. We note here that these are the only options +that we have been able to find — we have found no instances of more than two peaks appearing. +We show the results for Pe = 102 in figure 9a. In the figure, solute profiles with one peak — i.e. only the +classical coffee ring — are denoted by blue circles, while solute profiles exhibiting two peaks are denoted +by red circles. We see a strong nonlinear dependence on both Bond number and dryout time. In particular, +there is a band of Bond numbers between around Bo ≈ 10 and Bo ≈ 30000 that may lead to secondary peak +formation, although the existence of a peak also depends strongly on t for a fixed Bond number. We note +that for Bo ≲ 10, there is only one peak for any t, in agreement with the classical Bo = 0 regime. Moreover, +for very large Bond number Bo ≳ 30000, again we see that there is only one peak. +We illustrate the effect of the P´eclet number by plotting the equivalent regime diagram for Pe = 103 +in figure 9b. Remarkably, the onset of the secondary peak appears to be unaffected by the increase of the +P´eclet number, although the band of Bond numbers for which we see two peaks is significantly widened into +larger Bo. Notably, however, the shape of the curve delineating between two peaks / one peak for large Bond +number appears to be independent of Pe, only its location has shifted. + +16 +M. R. Moore & A. W. Wray +10-3 +10-2 +10-1 +100 +101 +102 +103 +104 +105 +0 +0.2 +0.4 +0.6 +0.8 +1 +One peak +Two peaks +10-3 +10-2 +10-1 +100 +101 +102 +103 +104 +105 +0 +0.2 +0.4 +0.6 +0.8 +1 +One peak +Two peaks +Figure 9: (Bo, t)-regime diagram illustrating the presence of either one (blue circles) or two (red circles) +peaks in the solute mass profile for (a) Pe = 102 and (b) Pe = 103. The data is extracted from the numerical +simulations and demonstrates a clear band of Bond numbers for which two peaks may exist in the profile. +In each figure, the black curve denotes the asymptotic prediction of when the centre of the droplet changes +from a maximum to a minimum as given by (5.21). + +Gravity can lead to multiple peaks in the early stages of coffee ring formation +17 +10-6 +10-3 +100 +10-2 +10-1 +100 +101 +102 +103 +10-6 +10-3 +100 +10-2 +10-1 +100 +101 +102 +103 +10-6 +10-3 +100 +10-2 +10-1 +100 +101 +102 +103 +Figure 10: Solute profiles for an evaporating droplet with Pe = 102 and Bo = 20. The deposit profile is +displayed on a doubly-logarithmic plot at 25% (a)), 35% (b)) and 75% (c)) of the drying time in order to +catch the emergence of the secondary peak. In each of a) − c), the primary peak is indicated by a red circle, +while the secondary peak is indicated by a black circle (when it exists). +5.2.1. Onset of the secondary peak +In this section, we seek to investigate some of the phenomena around the onset of the secondary peak in +more detail. We saw that for a fixed P´eclet number, there was a distinct switch from one to two peaks for +Bond number Bo ≈ 10 and that this switch appears to be independent of Pe. This suggests that secondary +peak formation is not a result of the interplay between solutal advection and diffusion that drives the classical +coffee ring. +In order to investigate the reasons behind the presence of or lack of a secondary peak, in figure 10, we +plot numerical results for the solute profiles in a droplet with Pe = 102, Bo = 20 at 25%, 35% and 75% of +the drying time. In the figure, the primary and secondary peaks are indicated by the red and black circles, +respectively. We clearly see in figure 10a that at 25% of the drying time there is only one peak, but by 35% +of the drying time, the secondary peak has emerged close to the droplet centre. As the droplet evaporates +further to 75% of the drying time the secondary peak has moved further towards the droplet contact line. +This particular example gives us a strong indication that the secondary peak initially arises from the centre +of the drop and, in particular, appears to be linked with a transition from the centre being a maximum in +solute mass profile — as it is for the classical coffee ring of Deegan et al. (1997, 2000) — to a minimum. +To investigate this postulate, we consider the behvaiour close to the droplet centre. To simplify things, +since the initial emergence of the secondary peak appears to be independent of the P´eclet number, we neglect +solutal diffusion completely, taking Pe = ∞, so that the solute mass m satisfies the first-order semi-linear +equation +∂m +∂t + 1 +4r +∂ +∂r (rmu) = 0, +m(r, 0) = h(r, 0), +(5.10) +where, since the emergence appears to be rooted in the region where Bo ≈ 10, we consider the moderate +Bond number regime and retain the full expressions for the droplet free surface h and fluid velocity u given +by (3.1)–(3.2). +We seek an asymptotic solution of (5.10) as r → 0. First, we note that for small arguments, the free surface +and velocity have the following asymptotic expansions: +h(r, t) ∼ (1 − t) +� +H0(Bo) + H1(Bo)r2 + o(r2) +� +, +(5.11) +u(r, t) ∼ +1 +(1 − t) +� +U0(Bo)r + U1(Bo)r3 + o(r3) +� +(5.12) + +18 +M. R. Moore & A. W. Wray +as r → 0, where +H0(Bo) = (I0( +√ +Bo) − 1) +πI2( +√ +Bo) +, +(5.13) +H1(Bo) = − +Bo +4πI2( +√ +Bo) +, +(5.14) +U0(Bo) = 2 +√ +Bo − +√ +BoI0( +√ +Bo) − 2I1( +√ +Bo) +√ +Bo(1 − I0( +√ +Bo)) +, +(5.15) +U1(Bo) = −(Bo3/2 − +√ +BoI0( +√ +Bo) + +√ +BoI0( +√ +Bo)2 + 2I1( +√ +Bo) − 2BoI1( +√ +Bo) − 2I0( +√ +Bo)I1( +√ +Bo) +4 +√ +Bo(1 − I0( +√ +Bo))2 +. +(5.16) +Now, by the symmetry of the problem, the droplet centre must be a critical point, so we seek a solution +of the form m = m0(t) + m1(t)r2 + o(r2) as r → 0. Upon substituting this ansatz and the above forms for h +and u into (5.10), straightforward calculation yields +m0(t) = H0(1 − t)U0/2, +(5.17) +m1(t) = +�2U1H0 +U0 ++ H1 +� +(1 − t)U0 − 2U1H0 +U0 +(1 − t)U0/2. +(5.18) +Hence, given that initially the droplet has a maximum at its centre for any Bo, we deduce that the +maximum becomes a minimum at the critical time tc such that +m1(tc) = 0. +(5.19) +Since 2U1H0/U0 + H1 < 0, H0 > 0, U0 > 0 for all Bo, (5.19) only has solutions for Bo > Boc where +U1(Boc) = 0 +=⇒ +Boc ≈ 15.21. +(5.20) +When Bo > Boc, we may solve (5.19) explicitly to find +tc(Bo) = 1 − +� +2U1(Bo)H0(Bo) +2U1(Bo)H0(Bo) + H1(Bo)U0(Bo) +�2/U0(Bo) +. +(5.21) +This critical curve in figure 9 is displayed as the solid black curve and we see that there is excellent +agreement between this prediction and the transition from one to two peaks. But, what is causing the +transition? Since the phenomenon is independent of the P´eclet number, it is purely a result of the droplet +geometry and the evaporation-driven flow. In particular, we note that the critical Bond number Boc given by +(5.20) is linked to the change in sign of U1, which is equivalent to requiring that (1 − t)∇ · (uer) is decreasing +near r = 0. This correlates with the profiles of the divergence of u displayed in figure 2c, where we see this +change in sign clearly as the Bond number increases. +Notably, considering the curve displayed in figure 9, we see that for Bo close to Boc, the secondary peak only +emerges very late in the dryout process, but as the Bond number increases, it appears almost instantaneously. +Hence, from this analysis alone, we might expect there to always be two peaks for Bo > Boc, but clearly this +is not the case. We now investigate why in more detail. +5.2.2. Loss of the secondary peak +Given its clear variation with each of t, Bo and Pe, it is perhaps unsurprising that it is more challenging to +determine an analytical expression for the location of the right-hand boundary between two peaks and one +peak in figure 9), and unfortunately we have been unable to do so. However, it is relatively straightforward +to illustrate why the transition occurs by considering a specific example. +In figure 11, we plot solute mass profiles for Pe = 102 and Bo = 103 at 5%, 20%, 50% and 90% of the +drying time indicating the primary and secondary peaks by red and black circles where appropriate. Note +that, for such a large Bond number, the critical time at which we would expect a secondary peak to be +present may be found from (5.21) to be tc ≈ 2.8 × 10−10. We see in figure 11a that, indeed, after 5% of the +drying time, the secondary peak has emerged and is visible close to the droplet centre — moreover, at this +stage, the primary peak associated with the coffee ring has yet to fully develop (so that the ‘one peak’ at +this stage in figure 9a is in fact the secondary peak!). However, by the time we reach 20% of the drying time, +both peaks are clearly visible, with the primary peak now approximately 50% larger than the secondary +peak. + +Gravity can lead to multiple peaks in the early stages of coffee ring formation +19 +10-6 +10-4 +10-2 +100 +10-2 +10-1 +100 +10-6 +10-4 +10-2 +100 +10-2 +10-1 +100 +10-6 +10-4 +10-2 +100 +10-2 +10-1 +100 +101 +10-6 +10-4 +10-2 +100 +10-2 +100 +102 +Figure 11: Solute profiles for an evaporating droplet with Pe = 102 and Bo = 103 displayed on a doubly- +logarithmic plot at 5% (a)), 20% (b)), 50% (c)) and 90% (d)) of the drying time. In each figure, the primary +peak is indicated by a red circle, while the secondary peak is indicated by a black circle when either exists. +Increasing time further, we see that the primary peak continues to grow rapidly so that, by 50% of the +drying time, it is so large, that it has subsumed the secondary peak into its upstream tail. That is, the +secondary peak is still present according to the Pe = ∞ theory, but due to the fact that Pe is actually finite +and the corresponding presence of the classical coffee ring, we do not see the secondary peak. +If we then increase t even further, we see that by 90% of the drying time, the secondary peak has reemerged +from the lee of the primary peak. By this stage of the evaporation process, the primary peak has moved +significantly closer to the contact line — here 1 − rpeak,I ≈ 1.4 × 10−4, while the secondary peak is located +at 1 − r ≈ 4.8 × 10−2, so that it is sufficiently far behind the primary peak to be visible. +Thus, the loss of the secondary peak appears to be intrinsically tied to both the location, size and shape of +the primary peak. Given that this behaviour largely occurs in the regime in which Bo ≫ 1, these properties +of the primary peak are given by (5.8), (5.9) and the derivative of (A 16), respectively. Clearly, therefore, the +behaviour is strongly dependent on t, Bo and Pe (cf. figure 8, for example). +5.2.3. Properties of the secondary peak +Given its dependence on the various parameters of the model, discerning the properties of the secondary +peak analytically is challenging, particularly in the Bo = O(1)-regime since, in this case, the peak tends to +be situated in the droplet bulk, so that we are unable to use the simpler forms of the inner solution described +in §4.1.2. +Hence, we utilize the numerical results to track the height mpeak,II(t) and location rpeak,II(t) of the +secondary peak when it exists and we display the results for several different values of Pe, Bo in figure +12. In the figure, results for Pe = 102 and Pe = 103 are denoted by circles and squares, respectively. The +Bond number is represented by the colour, with results for Bo = 20 (purple), 50 (dark blue), 100 (light +blue) and 1000 (green). It is evident that for each of the Bond numbers represented, increasing the P´eclet +number appears to have negligible effect on both the size and location of the secondary peak. However, both + +20 +M. R. Moore & A. W. Wray +0 +0.5 +1 +0 +0.1 +0.2 +0.3 +0.4 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +Figure 12: Numerical predictions of (a)) the height of the secondary peak, mpeak,II(t) and (b)) its location +rpeak,II(t) for different values of Pe, Bo. The symbols denote different P´eclet numbers: Pe = 100 (circles), +Pe = 1000 (squares); while the colours denote different Bond numbers: Bo = 20 (purple), Bo = 50 (dark +blue), Bo = 100 (light blue), Bo = 1000 (green). +properties do vary with the Bond number. In particular, as the Bond number increases, the secondary peak +is situated closer to the contact line at the same stage of the drying process, and similarly, for a fixed Bond +number, the peak gets closer to the contact line as the droplet evaporates. On the other hand, variations +of the secondary peak height with Bo are much less trivial, although for all of the displayed results, we see +that the height of the secondary peak decreases as the droplet evaporates. This is in stark contrast to the +primary peak, which always grows as more solute is transported to the contact line. +Thus, we conclude that the secondary peak is predominantly driven by the Bond number. Indeed, it is +only for sufficiently large Bond numbers that we find a second peak at all, and the properties of that peak +then depend strongly on the size of Bo. The only role played by the P´eclet number appears to be in the +disappearance of the secondary peak when it is subsumed by the primary peak, which is typically orders of +magnitude larger and always closer to the contact line. +6. Summary and discussion +In this paper, we have performed a detailed asymptotic and numerical analysis into the effect of gravity on +the famous coffee ring phenomenon observed in solute-laden droplets. In the physically-relevant limit of small +droplet capillary number, Ca ≪ 1, and large solutal P´eclet number, Pe ≫ 1, we identified two asymptotic +regimes based on the size of the Bond number, Bo: +i) a moderate Bond number regime, where Bo = O(1); +ii) a large Bond number regime, Bo ≫ 1. +In the first of these regimes, gravity acts to flatten the droplet profile from the spherical cap of the zero- +gravity problem, while reducing the liquid velocity. Moreover, the asymptotic structure of the solute transport +follows exactly that presented by Moore et al. (2021) for surface tension-dominated droplets, with advection +dominating in the droplet bulk, while the competition between advection and diffusion in a boundary layer +of width of O(Pe−2) near the pinned contact line drives the nascent coffee ring. Gravity acts to modify the +surface tension-dominated solution both through the accumulated mass flux of solute into the contact line +and a parameter dependent on the local contact angle. In particular, as the Bond number increases, the +height of the nascent coffee ring is reduced — which is consistent with the reduced flow velocity as Bo is +increased. Moreover, the peak is situated further from the contact line. +To categorize the role of gravity more explicitly, we derived an approximate similarity profile, ˆm0, for the + +Gravity can lead to multiple peaks in the early stages of coffee ring formation +21 +nascent coffee ring profile, given by +ˆm0(R, t) +Pe2 +t N(t; Bo) = +2χ +3ψ(Bo)f +�√ +R, 3, +4χ +ψ(Bo) +� +, +R = Pe2 +t(1 − r) +(6.1) +where Pet = Pe/(1−t) is the time-dependent P´eclet number, N(t; Bo) is the accumulated mass flux of solute +at the contact line from the droplet bulk, χ is the coefficient of the inverse square root singularity in the +evaporative flux at the contact line; ψ(Bo) is the leading order initial local contact angle; and f(x, k, l) = +lkxk−1e−lx/Γ(k)! is the probability density function of a gamma distribution. Clearly, the Bond number acts +to scale the coffee ring profile through the accumulated mass flux, while it acts to change the shape of the +profile through the initial contact angle ψ(Bo). +In the second regime, the Bond number is large, so that the droplet is approximately flat, with surface +tension confined to a narrow region near the pinned contact line — a ‘pancake’ or ‘puddle’ droplet. Thus, +the asymptotic analysis discussed above is no longer valid, since there are two competing boundary layers +near the edge of the droplet — the diffusion boundary layer in the solute transport and the surface tension +boundary layer in the droplet free surface profile (and, hence, the liquid velocity). We derived the resulting +solute distribution in the most general regime in which the two boundary layers are comparable, which +reduces to the assumption that α = Bo−1/2Pe2/3 = O(1). Under this assumption, diffusion and advection +balance in a region of size Pe−2/3 near the contact line, noticeably larger than in the moderate gravity +regime. This is a further indication of gravity acting to shift the coffee ring further from the contact line +and, moreover, tends to cause shallower solute profiles in the boundary layer region. +The asymptotic analysis in the large-Bond number regime is more challenging than that in the moderate +Bond number regime and, in particular, the nascent coffee ring no longer collapses onto an approximate +similarity profile. However, we were able to derive expressions for the location (5.8) and height (5.9) of the +peak, demonstrating that it still strongly depends on the accumulated mass flux of solute into the contact +line alongside the parameter α. In particular, increasing α leads to higher coffee rings that are located closer +to the contact line. +In each regime, we demonstrated that our asymptotic predictions were in excellent agreement with nu- +merical simulations of the full advection-diffusion problem for the solute mass distribution. +Alongside the anticipated nascent coffee ring driven by the competition between advection and diffusion +of the solute, our asymptotic and numerical analysis also revealed a novel phenomenon: that the solute +profile may have a secondary peak. The secondary peak was characterized by being situated upstream of and +significantly smaller than the primary coffee ring. Moreover, the presence of this peak strongly depended on +the Bond number, P´eclet number and evaporation time. +Further investigation revealed that, for a fixed P´eclet number, there exists a band in (Bo, t)-space at which +two peaks are present in the profile. We demonstrated that the onset of this band is independent of the P´eclet +number and is caused by the critical point at the centre of the droplet changing in type from a maximum +(as in the spherical cap droplet in the Bo = 0 regime) to a minimum. When the critical point at the droplet +centre changes type, an internal maximum forms downstream of the centre and it is this that corresponds to +the secondary peak. This behaviour only occurs above a critical Bond number, Boc ≈ 15.21, and then only +after a given drying time, given by +tc(Bo) = 1 − +� +2U1(Bo)H0(Bo) +2U1(Bo)H0(Bo) + H1(Bo)U0(Bo) +�2/U0(Bo) +. +(6.2) +In particular, as Bo increases, tc decreases, so the secondary peak emerges earlier in the evaporative process. +These predictions were shown to be in excellent agreement to the numerical results and, remarkably, are +independent of the P´eclet number. +However, the above analysis suggests that for all Bo > Boc and t > tc a secondary peak exists — something +that we did not find in our analysis. The reason for this discrepancy was shown to be due to the presence +of the primary peak. In particular, as time increases, the secondary peak is located further from the droplet +centre so that it may get subsumed in the tail of the primary peak. For a fixed Bond number, this possibility +was shown to depend strongly on both the P´eclet number and the evaporation time; this is due to the fact +that the size of the primary peak increases with both t and Pe, while the size of the secondary peak only +varies with t. +Beyond this subsuming effect, however, we were able to demonstrate that the P´eclet number plays negligible +role in the size and location of the secondary peak for a range of Bond numbers, suggesting that this feature +may be reliably controlled simply by altering Bo. +In previous studies of coffee ring formation (e.g. Deegan et al. (2000); Popov (2005); Moore et al. (2021), + +22 +M. R. Moore & A. W. Wray +gravity has frequently been neglected under the assumption of small Bond number, which is a reasonable +assumption for sufficiently small droplets. However, given that the Bond number may be increased in an +experimental or industrial setting by steadily increasing the droplet radius, the influence of gravity may be +of fundamental interest in applications that utilize droplet drying to adaptively control the shape of the +residual deposit, such as colloidal patterning (Harris et al. 2007; Choi et al. 2010) and fabrication techniques +using inkjet printing (Layani et al. 2009). Our analysis thus plays a dual role in the field. First, we have +presented the first formal categorization of the role of gravity in the early-stages of coffee ring formation and +given a quantitative prediction of the resulting features of the solute profile. Second, we have found a novel +phenomenon — the secondary peak — which may also be exploited in such processes, particularly when the +size of the primary peak can be carefully controlled. This is particularly relevant given that the secondary +peak emerges at a relatively moderate critical Bond number. +There are, naturally, limitations to our analysis. Throughout, we have assumed that the contact line +remains pinned as the droplet evaporates. This has been shown to be a reasonable assumption for many +configurations (see, for example, the experiments in Deegan et al. (2000); Kajiya et al. (2008); Howard et al. +(2023)) and may further be enhanced by solute aggregation near the edge of the droplet (Orejon et al. +(2011); Weon & Je (2013); Larson (2014)). However, at late stages of the evaporation, the contact line may +depin and become mobile, moving inwards towards the droplet centre. The contact line may then become +pinned at a new location and the process may repeat. This behaviour is known as ‘stick-slip’ evaporation +and represents an important class within the field that is beyond the scope of the present study, but may +represent an interesting future direction in terms of the effect of gravity, particularly with the presence of +the secondary peak and its associated increased solute mass, which may promote re-pinning. +Another effect that we have neglected in the present analysis is the possibility of solute becoming trapped +at the free surface of the droplet. If this occurs, the solute is then transported to the contact line along +the free surface, and has been suggested as an alternative mechanism for coffee ring formation (Kang et al. +(2016)). This behaviour has been demonstrated to occur for a wide variety of droplets but is more pronounced +for droplets with large contact angles Kang et al. (2016) or when vertical diffusion happens over a longer +timescale than evaporation D’Ambrosio (2022). Since we deal with the opposite case of a thin drop with +fast vertical diffusion (i.e. so that the solute concentration may be assumed to be uniform across the droplet +to leading order), we have not considered this phenomenon here. It would be interesting to see how such +behaviour impacted the solute profile in the current case, although it should be noted that the aforementioned +studies neglect gravity entirely. +A further aspect that would form the basis of an exciting future study surrounds the assumption that +the solute is dilute in the droplet. Naturally, the build up of the solute in the coffee ring means that the +concentration rapidly approaches levels where finite particle size effects can no longer be ignored. This has +been analysed in detail for surface tension-dominated droplets in Moore et al. (2021, 2022) and a similar +analysis would follow here with the inclusion of gravity. One possible aspect that would differentiate droplets +where gravity is included is in the vicinity of the secondary peak. It is an interesting open question as to +whether the dilute assumption may also break down in the vicinity of the secondary peak as well as the +primary. Once finite particle size effects become important, there are a number of different approaches that +can be followed to continue the analysis, such as a sharp transition between a dilute and jammed region +(Popov (2005)), using a viscosity and solute diffusivity that vary with concentration (Kaplan & Mahadevan +(2015)) or through more complicated two-phase suspension models (see, for example, Guazzelli & Pouliquen +(2018)). +Our analysis has concentrated on a diffusive evaporative model, while there are many situations where +other evaporative models may be appropriate. Examples include water evaporating on glass, which may more +appropriately be modelled using a kinetic evaporative model (Murisic & Kondic (2011)), droplets evaporating +above a bath of the same liquid, where the evaporation is effectively constant (Boulogne et al. (2016)) and +situations where a mask is used to control the evaporative flux so that it is stronger towards the centre +(Vodolazskaya et al. (2017)). The analysis herein could readily be extended to such situations, although we +note that for evaporative fluxes with different — including non-singular — behaviour close to the contact +line, the size of the boundary layer regions near the contact line in which solutal diffusion and surface tension +are relevant will change accordingly, as for surface tension-dominated droplets (Moore et al. 2021). +A future direction of interest would be to extend the analysis herein to non-axisymmetric droplets. Such +droplets occur widely in applications, particularly in printing OLED/AMOLED screens (see, for example, +Mai & Richerzhagen (2007); Huo et al. (2020)). It is well-known that droplet geometry plays a strong role in +the behaviour of the evaporative flux (S´aenz et al. (2017); Wray & Moore (2023)) and the transient and final + +Gravity can lead to multiple peaks in the early stages of coffee ring formation +23 +deposit profiles (Freed-Brown (2015); S´aenz et al. (2017); Moore et al. (2022)). It would be of significant +theoretical and practical interest to explore the behaviour of the secondary peak in such problems as well. +Finally, we note that another context in which gravity may play an important role is that of binary/multi- +component droplets, particularly in situations where the different fluids have different densities. Multi- +component droplets occur widely, from commercial alcohols such as whiskey and ouzo (Tan et al. (2019); +Carrithers et al. (2020)) to various inks (Shargaieva et al. (2020)). While it would be certainly of interest to +extend the analysis presente here to such droplets, a careful treatment of the internal flow would be needed, +as the multi-component nature of the droplet significantly complicates the dynamics (Li et al. (2019)). +Acknowledgments MRM would like to acknowledge the support of EPSRC grant EP/X035646/1. +Declaration of Interests. The authors report no conflict of interests. +Appendix A. Matyched asymptotic analysis in the limit of large Bo, α = O(1) +In this appendix, we present the asymptotic solution of the solute transport problem in the limit in which +Bo, Pe ≫ 1 and +α = Bo−1/2Pe2/3 = O(1). +(A 1) +For convenience, we choose to use Pe−2/3 as our small parameter in the asymptotic expansions. Moreover, it +transpires that it is easier to analyse the integrated mass variable formulation of the problem (2.32)–(2.35). +A.1. Outer region +In the droplet bulk, 1 − r is O(1), and we recall from (3.9) that the droplet free surface h is flat to all orders +and that the velocity is given by (3.10). Upon substituting these expressions into (2.32) and (2.34), and then +expanding M(r, t) = M0(r, t) + Pe−2/3M1(r, t) + O(Pe−4/3) as Pe → ∞, we find to leading-order +∂M0 +∂t ++ u0 +4 +∂M0 +∂r += 0 +for +0 < r, t < 1, +M0(r, 0) = r2 +2π +for +0 < r < 1. +(A 2a, b) +This may be solved using the method of characteristics, yielding +M0(r, t) = (1 − t)r2 +2π ++ +√1 − t(1 − √1 − t) +π +(1 − +� +1 − r2). +(A 3) +We see that this solution automatically satisfies the boundary condition (2.33a). +At O(Pe−2/3), the problem for M1(r, t) is given by +∂M1 +∂t ++ u0 +4 +∂M1 +∂r += −αu1 +4 +∂M0 +∂r +for +0 < r, t < 1, +M1(r, 0) = 0 +for +0 < r < 1. +(A 4a, b) +for 0 < r < 1, 0 < t < 1, while the initial condition is given by M1(r, 0) = αr2/π for 0 < r < 1. This may +be solved in a similar manner using the method of characteristics, yielding +M1(r, t) = 2ακ(r, t) +π +(1 − κ(r, t)) log +�√1 − t − κ(r, t) +1 − κ(r, t) +� ++ α +π (1 − (1 − κ(r, t))2), +(A 5) +where κ(r, t) = √1 − t(1 − +√ +1 − r2). +Expanding the leading-order solution (A 3) as we approach the contact line, we have +M0(r, t) ∼ +√1 − t +π +− (1 − t) +2π +− +� +2(1 − t) +π +(1 − +√ +1 − t) +√ +1 − r − (1 − t) +π +(1 − r) + O((1 − r)3/2) +(A 6) +as r → 1−. Notably, this means that the leading-order outer solute mass m0 is singular at the contact line, +which gives a strong indication of the importance of diffusive effects local to the edge of the droplet. This is +in stark contrast to the Bo = O(1) solution, where the outer solute mass was square root bounded as r → 1−. +A similar expansion of (A 5), yields +M1(r, t) ∼ α√1 − t(1 − √1 − t) +π +log(1 − r) + α√1 − t(1 − √1 − t) +π +log +� +2(1 − t) +(1 − √1 − t)2 +� ++ +α +π (1 − (1 − +√ +1 − t)2) + O( +√ +1 − r log(1 − r)) +(A 7) +as r → 1−. We can clearly see this will necessitate an inner expansion that contains logarithmic terms; +similar behaviour is displayed for surface tension-dominated drops under different evaporative fluxes (Moore +et al. 2021). + +24 +M. R. Moore & A. W. Wray +Finally, if we expand the solute mass m ∼ m0 as Pe → 0 in (2.35), we find +m0(r, t) = +√1 − t +π +√ +1 − r2 +� +1 − +√ +1 − t(1 − +� +1 − r2) +� +. +(A 8) +Whilst we could proceed to O(Pe−2/3) in the solute mass expansion in the outer region, we shall not require +this when constructing a composite profile that is valid to O(1) throughout the droplet, so we do not present +this here. +A.2. Inner region +Recalling (3.11)–(3.12), (4.2) and (4.3), in order to retain a balance between the advective and diffusive +effects in (2.32) close to the contact line, we set +r = 1 − Pe−2/3˜r, +u = Pe−1/3˜u, +h = ˜h, +M = ˜ +M, +m = Pe2/3 ˜m +(A 9) +in (2.32)–(2.35). Note that we therefore have +˜h = ˜h0 + Pe−2/3˜h1 + O(Pe−4/3), +˜u = ˜u0 + Pe−1/3˜u1 + Pe−2/3˜u2 + O(Pe−1) +(A 10) +as Pe → ∞ where +˜h0(˜r, t) = ¯h0(˜r/α, t), +˜h1(˜r, t) = α¯h1(˜r/α, t), +(A 11) +and ¯h0, ¯h1 are given by (3.13)–(3.14), and +˜u0(˜r, t) = √α¯u0(˜r/α, t), +˜u1(˜r, t) = α¯u1(˜r/α, t), +˜u2(˜r, t) = α3/2 ¯u2(˜r/α, t) +(A 12) +and ¯u0, ¯u1, ¯u2 are given by (3.15))–(3.17). +Seeking an asymptotic expansion of the integrated mass of the form ˜ +M = ˜ +M0+Pe−1/3 ˜ +M1+Pe−2/3 log Pe−2/3 ˜ +M2+ +Pe−2/3 ˜ +M3 + o(Pe−2/3) as Pe → ∞, we find that the leading-order inner problem is given by +∂2 ˜ +M0 +∂˜r2 ++ +� +˜u0 − 1 +˜h0 +∂˜h0 +∂˜r +� +∂ ˜ +M0 +∂r += 0, +for +˜r > 0, 0 < t < 1, +(A 13) +subject to the boundary condition +˜ +M0(0, t) = 1/2π for 0 < t < 1 and, in order to match with the local +expansion of leading-order-outer solution at the contact line (A 6), we must have +˜ +M0 → +√1 − t +π +− (1 − t) +2π +as +˜r → ∞, +(A 14) +Defining the integrating factor +I(˜r, t) = +� +1 +1 − e−˜r/α +� +exp +� +2 +√ +2 +(1 − t) +� ˜r +0 +√ξ +1 − e−ξ/α dξ +� +, +(A 15) +we find that the solution is given by +˜ +M0(˜r, t) = 1 +2π + B0(t) +� ˜r +0 +1 +I(s, t) ds, +(A 16) +where +B0(t) = − 1 +π +� +1 − +√ +1 − t − t +2 +� �� ∞ +0 +1 +I(s, t) ds +�−1 +. +(A 17) +We note here that the first term on the right-hand side of B0(t) is simply the leading-order accumulated +mass at the contact line as a function of time, N(t), that is +N(t) = 1 +4 +� t +0 +(m0u0)(1−, τ) dτ = 1 +π +� +1 − +√ +1 − t − t +2 +� +. +(A 18) +It is worth noting the similarities between (A 18) and the equivalent expression for a surface-tension domi- +nated drop evaporating under a constant evaporative flux (Freed-Brown 2015; Moore et al. 2021). +At O(Pe−1/3), we have +∂2 ˜ +M1 +∂˜r2 ++ +� +˜u0 − 1 +˜h0 +∂˜h0 +∂˜r +� +∂ ˜ +M1 +∂r += 4∂ ˜ +M0 +∂t +− ˜u1 +∂ ˜ +M0 +∂˜r +for +˜r > 0, 0 < t < 1, +(A 19) + +Gravity can lead to multiple peaks in the early stages of coffee ring formation +25 +subject to +˜ +M1(0, t) = 0 for 0 < t < 1 and the far-field matching condition +˜ +M1 → − +� +2(1 − t) +π +(1 − +√ +1 − t) +√ +˜r +as +˜r → ∞. +(A 20) +While in practice it may be easier to find +˜ +M1(˜r, t) from (A 19)–(A 20) numerically, for posterity, we state +that this boundary value problem has solution +˜ +M1(˜r, t) = +� ˜r +0 +1 +I(s, t) +�� s +0 +� +4∂ ˜ +M0 +∂t +− ˜u1 +∂ ˜ +M0 +∂˜r +� +I(σ, t) dσ +� +ds + B1(t) +� ˜r +0 +1 +I(s, t) ds, +(A 21) +where +B1(t) = − +�� ∞ +0 +� +1 +I(s, t) +�� s +0 +� +4∂ ˜ +M0 +∂t +− ˜u1 +∂ ˜ +M0 +∂˜r +� +I(σ, t) dσ +� +− +√ +2(1 − t)∂ ˜ +M0 +∂t +1 +√s +� +ds +� �� ∞ +0 +1 +I(s, t) ds +�−1 +, +(A 22) +is chosen to kill the O(1)-term in the far-field expansion of +˜ +M1(˜r, t). +The O(Pe−2/3 log Pe−2/3)-problem is given by +∂2 ˜ +M2 +∂˜r2 ++ +� +˜u0 − 1 +˜h0 +∂˜h0 +∂˜r +� +∂ ˜ +M2 +∂r += 0 +for +˜r > 0, 0 < t < 1, +(A 23) +subject to +˜ +M2(0, t) = 0 for 0 < t < 1 and the far-field matching condition +˜ +M2 → α√1 − t(1 − √1 − t) +π +as +˜r → ∞. +(A 24) +The solution may be found in a similar manner to the leading-order problem, yielding +˜ +M2(˜r, t) = B2(t) +� ˜r +0 +1 +I(s, t) ds, +(A 25) +where +B2(t) = α√1 − t(1 − √1 − t) +π +�� ∞ +0 +1 +I(s, t) ds +�−1 +. +(A 26) +Lastly, at O(Pe−2/3), we have +∂2 ˜ +M3 +∂˜r2 ++ +� +˜u0 − 1 +˜h0 +∂˜h0 +∂˜r +� +∂ ˜ +M3 +∂r += 4∂ ˜ +M1 +∂t +−˜u1 +∂ ˜ +M1 +∂˜r −˜u2 +∂ ˜ +M0 +∂˜r − 1 +˜h0 +�˜h1 +˜h0 +∂˜h0 +∂˜r − ∂˜h1 +∂˜r +� +∂ ˜ +M0 +∂˜r − ∂ ˜ +M0 +∂˜r +=: V(˜r, t) +(A 27) +for ˜r > 0, 0 < t < 1, subject to +˜ +M3(0, t) = 0 for 0 < t < 1 and the far-field condition +˜ +M3 → −(1 − t) +π +˜r+ +�α√1 − t(1 − √1 − t) +π +� � +log ˜r + log +� +2(1 − t) +(1 − √1 − t)2 +�� ++α +π (1−(1− +√ +1 − t)2) +as +˜r → ∞. +(A 28) +The solution is given by +˜ +M3(˜r, t) = +� ˜r +0 +1 +I(s, t) +�� s +0 +V(σ, t)I(σ, t) dσ +� +ds + B3(t) +� ˜r +0 +1 +I(s, t) ds, +(A 29) +where +B3(t) = +� +− +� ∞ +1 +� +1 +I(s, t) +�� s +0 +V(σ, t)I(σ, t) dσ +� ++ (1 − t) +π +− α√1 − t(1 − √1 − t) +πs +� +ds +− +� 1 +0 +1 +I(s, t) +�� s +0 +V(σ, t)I(σ, t) dσ +� +ds − (1 − t) +π ++ α +π (1 − (1 − +√ +1 − t)2) ++α√1 − t(1 − √1 − t) +π +log +� +2(1 − t) +(1 − √1 − t)2 +�� �� ∞ +0 +1 +I(s, t) ds +�−1 +, +(A 30) +has been chosen to satisfy the correct far-field behaviour. +We are now in a position to find the inner solution for the solute mass. By substituting the scalings (A 9) + +26 +M. R. Moore & A. W. Wray +into (2.35), we see that +˜m = − +1 +1 − Pe−2/3˜r +∂ ˜ +M +∂˜r , +(A 31) +so that expanding ˜m = ˜m0 + Pe−1/3 ˜m1 + Pe−2/3 log Pe−2/3 ˜m1 + Pe−2/3 ˜m2 as Pe → ∞, we have +˜m0 = −∂ ˜ +M0 +∂˜r , +˜m1 = −∂ ˜ +M1 +∂˜r , +˜m2 = −∂ ˜ +M2 +∂˜r , +˜m3 = −∂ ˜ +M3 +∂˜r +− ˜r∂ ˜ +M0 +∂˜r . +(A 32) +A.3. Composite solutions +We now have all of the necessary components needed to construct (additive) composite solutions for com- +parison to the numerical results. +To construct a composite solution for the integrated mass variable, we combine the first two outer solutions +(A 3) and (A 5), the first four inner solutions (A 16), (A 21), (A 25) and (A 29), the overlap terms given by +(A 6)–(A 7) using Van Dyke’s matching rule Van Dyke (1964), which yields +Mcomp(r, t) = M0(r, t) + Pe−2/3M1(r, t) + ˜ +M0 +� +Pe2/3(1 − r), t +� ++ Pe−1/3 ˜ +M1 +� +Pe2/3(1 − r), t +� ++ +Pe−2/3 log Pe−2/3 ˜ +M2 +� +Pe2/3(1 − r), t +� ++ Pe−2/3 ˜ +M3 +� +Pe2/3(1 − r), t +� +− +�√1 − t +π +− (1 − t) +2π +− +� +2(1 − t) +π +(1 − +√ +1 − t) +√ +1 − r − (1 − t) +π +(1 − r)+ +Pe−2/3 +�α +π (1 − (1 − +√ +1 − t)2) + α√1 − t(1 − √1 − t) +π +� +log(1 − r) + log +� +2(1 − t) +(1 − √1 − t)2 +���� +. +(A 33) +This composite solution is valid up to and including O(Pe−2/3) throughout the whole of the droplet. +Similarly, for the solute mass, the equivalent composite profile is compiled by taking the first outer solution +(A 8) as well as the first four inner solutions given by (A 32), so that, accounting for the overlap contributions, +mcomp(r, t) = m0(r, t) + Pe2/3 ˜m0 +� +Pe2/3(1 − r), t +� ++ Pe1/3 ˜m1 +� +Pe2/3(1 − r), t +� ++ +log Pe−2/3 ˜m2 +� +Pe2/3(1 − r), t +� ++ ˜m3 +� +Pe2/3(1 − r), t +� +− +� +(1 − t)(1 − √1 − t) +√ +2π√1 − r +− (1 − t) +π +. +(A 34) +We note that this composite solution is valid up to and including O(1) throughout the droplet. +Appendix B. Numerical solution of the solute transport problem +In this section, we outline the numerical scheme for solving the advection-diffusion problem (2.32)–(2.34) +for the integrated mass variable M(r, t). As discussed previously, the integrated mass variable formulation +is advantageous when solving numerically, since it is mass-preserving and has simple-to-implement Dirichlet +boundary conditions. +Our numerical method is an adaptation of that discussed in Moore et al. (2021) for the Bo = 0 regime. We +utilize central differences with gridpoints clustered close to the contact line, where there are rapid changes +in behaviour associated with the coffee ring. We choose a uniform grid in the variable ζ ∈ [0, 1], where +r = 1 − ℓζ +1 − ℓ , +(B 1) +and ℓ is taken to coincide with the smallest of the two boundary layers; that is, ℓ = κ(1 − tc) where +κ = min +� +Bo−1/2, Pe−2/3� +and tc is the final computation time. 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H. 2013 Self-pinning by colloids confined at a contact line. Phys. Rev. Lett. 110 (2), 028303. +Wilson, Stephen K & D’Ambrosio, Hannah-May 2023 Evaporation of sessile droplets. Annual Review of Fluid +Mechanics 55. +Wray, A. W. & Moore, M. R. 2023 Evaporation of non-circular droplets. J. Fluid Mech. p. (Under review). +Wray, A. W., Papageorgiou, D. T., Craster, R. V., Sefiane, K. & Matar, O. K. 2014 Electrostatic suppres- +sion of the “coffee stain effect”. Langmuir 30 (20), 5849–5858. +Wray, A. W., Wray, P. S., Duffy, B. R. & Wilson, S. K. 2021 Contact-line deposits from multiple evaporating +droplets. arXiv preprint arXiv:2103.07221 . +Yariv, Ehud 2022 Shape of sessile drops at small contact angles. Journal of Fluid Mechanics 950, R4. + diff --git a/W9FKT4oBgHgl3EQfoC5E/content/tmp_files/load_file.txt b/W9FKT4oBgHgl3EQfoC5E/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6e70a2306ba162847c14c9053a2aeea8f8d9262f --- /dev/null +++ b/W9FKT4oBgHgl3EQfoC5E/content/tmp_files/load_file.txt @@ -0,0 +1,1359 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf,len=1358 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='11864v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='flu-dyn] 27 Jan 2023 Under consideration for publication in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 1 Gravity can lead to multiple peaks in the early stages of coffee ring formation M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' M O O R E1 AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W R A Y2 1Department of Mathematics, School of Natural Sciences, University of Hull, Cottingham Road, Hull, HU6 7RX, UK 2Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK (Received ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' revised ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' accepted ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='. - To be entered by editorial office) We consider the role of gravity in solute transport when a thin droplet evaporates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Under the physically- relevant assumptions that the contact line is pinned and the solutal P´eclet number, Pe is large, we identify two fundamental regimes that depend on the size of the Bond number, Bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' When Bo = O(1), the asymptotic structure of solute transport follows directly from the surface tension-dominated regime, whereby advection drives solute towards the contact line, only to be countered by local diffusive effects, leading to the for- mation of the famous “coffee ring”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' For larger Bond numbers, we identify the distinguished limit in which Bo−1/2Pe2/3 = O(1), where the diffusive boundary layer is comparable to the surface tension boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In each regime, we perform a systematic asymptotic analysis of the solute transport and compare our predictions to numerical simulations of the full model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Our analysis identifies the effect of gravity on the nascent coffee ring, providing quantitative predictions of the size, location and shape of the solute mass profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Furthermore, we reveal that, for certain values of Bo, Pe and the evaporation time, a secondary peak may exist inside the classical coffee ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We find that the onset of this secondary peak is linked to the change in behaviour of the critical point in the droplet centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Both the onset and the peak characteristics are shown to be independent of Pe, but solutal diffusion may act to remove the secondary peak when the classical coffee ring becomes so large as to subsume it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Key words: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Introduction The evaporation of sessile droplets has received significant attention in recent years, being the subject of several major reviews (Cazabat & Guena 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Lohse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Brutin & Starov 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wilson & D’Ambrosio 2023) due to its ubiquity in theoretical, experimental and industrial settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' A particular phenomenon of interest is the so-called “coffee ring effect”, in which a solute in such an evaporating droplet ends up preferentially accumulated at the contact line (Deegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 1997, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This effect is very robust, occurring even when the solution is initially uniformly dispersed throughout the droplet, and even when the evaporative flux is not preferentially localised at the contact line (Boulogne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Motivated by typical physical parameters, models of such systems typically assume that the P´eclet number is sufficiently large that diffusive effects can be neglected, and so dynamics of the solute inside the droplet are governed purely by convection (Deegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This unphysical assumption leads to a variety of undesirable side-effects, including singular accumulations of residue, and solute not being conserved (Deegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' A variety of attempts have been made to resolve this problem phenomenologically, including via the incorporation of jamming effects (Popov 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Kaplan & Mahadevan 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, jamming effects only become significant close to the particle packing fraction, and the assumptions underpinning the model fail long before this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In particular, the assumption that diffusive effects can be ignored breaks down in a diffusive boundary layer close to the contact line (Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2021), as might be anticipated from the singular accumulation in the na¨ıve, convection-only model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This boundary layer and its growth and dynamics have been analysed and understood via matched asymptotics and careful numerics in situations where droplets are small, and thus exist at quasi-static equilibrium due to surface tension (Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2022), but little is known for larger droplets where the effects of gravity are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Investigations of larger droplets have a long history, dating back to numerical integration of the appropriate Laplace equations by Padday (1971) and Boucher & Evans (1975), with a variety of studies via asymptotics of 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray their shape (Rienstra 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' O’Brien 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Yariv 2022) and stability (Pozrikidis 2012) in the intervening time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The effect of gravity on droplets, and especially their internal flows, has experienced a recent resurgence of interest due to the experiments of Edwards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2018), which showed that the dynamics of binary droplets can be sensitively dependent on droplet inclination (and hence gravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This has since received extensive investigation both experimentally and numerically (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Pradhan & Panigrahi 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Notably, however, despite the original experiments of Deegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (1997) involving large droplets, there have been relatively few investigations of particle transport inside them, with those available being principally experimental (Sandu & Fleaca 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Hampton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This is perhaps because of the robustness of the coffee-stain effect: asymptotic and numerical investigations (Barash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Kolegov & Lobanov 2014) confirm the experimental results that the ring-stain is preserved unless additional physics are incorporated, such as continuous particle deposition (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, this neglects the bulk of the story, including the dynamics of the residue over the course of the lifetime of the droplets: a critical omission in situations such as continuous particle deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We show in the present work that the dynamics are actually quite subtle and complex, and certainly merit detailed investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The structure of this paper is therefore as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In §2, we describe the equations governing the fluid flow and solute transport for the problem of a thin droplet evaporating under a diffusive flux, in particular highlighting the effect of gravity in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We nondimensionalise the model and introduce the three key dimensionless numbers in the model: the capillary, Bond and P´eclet numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In §3, we completely solve for the liquid flow in the limit in which the solute is dilute, so that the flow and solute transport problems decouple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We discuss pertinent features of the resulting fluid velocity and droplet shape, and in particular how these features vary with the Bond number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The bulk of the analysis in this paper concerns the influence of gravity on solute transport within the droplet, which we analyse in the physically-relevant large-P´eclet number limit in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We find that there are two distinct regimes depending on the relative sizes of the Bond and P´eclet numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In the first, where the Bond number is moderate, we extend the asymptotic analysis of Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2021) to include the effect of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, when the Bond number is also large, a more complex asymptotic analysis is necessary, which is presented in detail in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In each asymptotic regime, we derive predictions for the distribution of the solute mass within the droplet and compare the results to numerical simulations of the full advection-diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In particular, while we find the expected ‘nascent coffee ring’ profile in the solute mass, for certain input parameters, we also find evidence of a novel phenomenon whereby a second peak may also develop in the mass profile inside the classical coffee ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We analyse both of these peaks in detail in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In particular, for the classical coffee ring, we discuss the effect of gravity in each of the two asymptotic regimes discussed in §4 and Appendix A, while for the secondary peak, we investigate the key role gravity plays in its existence and how the secondary peak may also be subsumed in the classical coffee ring for certain values of the Bond and P´eclet numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Finally, in §6, we summarize our findings and discuss implications to various applications, as well as avenues for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Problem configuration We consider the configuration depicted in figure 1 in which an axisymmetric droplet of initial volume V ∗ 0 evaporates from a solid substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Here and hereafter, an asterisk denotes a dimensional variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We let (r∗, θ, z∗) be cylindrical polar coordinates centred along the line of symmetry of the droplet with the substrate lying in the plane z∗ = 0: by axisymmetry, we shall assume that all the variables are independent of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The droplet contact line is thus circular and we assume that it is pinned throughout the drying process, which is observed in practice for a wide range of liquids for the majority of the drying time (Deegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Hu & Larson 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Kajiya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We let r∗ = R∗ be the radius of the contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Throughout this analysis, we shall assume that the droplet is thin, which reduces to the assumption that 0 < δ = V ∗ 0 R∗3 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1) As we discuss presently, the thin-droplet assumption allows us to greatly simplify the flow and solute transport models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' the assumption has been extensively-validated and has shown to be reasonable even for droplets that should realistically fall outside of this regime (Larsson & Kumar 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The droplet consists of a liquid of constant density and viscosity denoted by ρ∗ and µ∗, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The droplet free surface is denoted by z∗ = h(r∗, t∗) and the air-water surface tension coefficient, σ∗ is assumed to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Gravity can lead to multiple peaks in the early stages of coffee ring formation 3 z∗ r∗ 2R∗ z∗ = h∗(r∗, t∗) E∗(r∗) Figure 1: A side-on view of a solute-laden droplet evaporating under an evaporative flux E∗(r∗) from a solid substrate that lies in the plane z∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The droplet is axisymmetric and the contact line is assumed to be pinned on the substrate at r∗ = R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The droplet free surface is denoted by h∗(r∗, t∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The solute is assumed to be inert and sufficiently dilute that the flow of liquid in the droplet is decoupled from the solute transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The liquid evaporates into the surrounding air and we assume that the evaporative process is quasi-steady, which is a reasonable assumption for a wide range of liquid-substrate configurations (Hu & Larson 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' While there are a number of different viable evaporation models depending on the physical and chemical characteristics of the problem (Murisic & Kondic 2011), for the purposes of this analysis, we assume that the dominant process of vapour transport from the droplet surface is diffusion, so that the evaporative flux E∗(r∗) is given by E∗(r∗) = 2D∗(c∗ s − c∗ ∞) π √ R∗2 − r∗2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2) where D∗ is the diffusion coefficient and c∗ s, c∗ ∞ are the surface and ambient vapour concentrations, respec- tively (Deegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Murisic & Kondic 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The droplet contains an inert solute of initially uniform concentration φ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The solute is assumed to be sufficiently dilute that the flow and transport problems completely decouple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We shall discuss the validity of the dilute assumption further in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Flow model The droplet is assumed to be sufficiently thin and the evaporation-induced flow sufficiently slow that the flow is governed by the lubrication equations ∂h∗ ∂t∗ + 1 r∗ ∂ ∂r∗ (r∗h∗u∗) = − E∗ ρ∗ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='3) u∗ = − h∗2 3µ∗ ∂p∗ ∂r∗ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4) p∗ = p∗ atm − ρ∗g∗(z∗ − h∗) − σ∗ 1 r∗ ∂ ∂r∗ � r∗ ∂h∗ ∂r∗ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5) for 0 < r∗ < R∗, t∗ > 0, where u∗(r∗, t∗) is the depth-averaged radial fluid velocity, p∗(r∗, z∗, t∗) is the liquid pressure and p∗ atm denotes atmospheric pressure (Hocking 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Deegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Oliver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='3)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5) must be solved subject to the symmetry conditions r∗h∗u∗ = ∂h∗ ∂r∗ = 0 at r∗ = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6a, b) and the fact that the free surface touches down at, and we require no-flux of liquid through, the pinned contact line, that is h∗ = r∗h∗u∗ = 0 at r∗ = R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='7a, b) We close the problem by specifying the initial droplet profile, that is h∗(r∗, 0) = h∗ 0(r∗) for 0 < r∗ < R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8) It is worth noting at this stage that, while this initial condition is needed to fully specify the mathematical problem, in our analysis, we do not explicitly use the initial condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In what follows, it is assumed 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray that the rate of evaporation is sufficiently slow that the droplet quickly relaxes under capillary action to the quasi-steady profile found in §3 (see, for example, Lacey (1982);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' De Gennes (1985);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Oliver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Thus, we shall for simplicity assume that h0(r) is of the same functional form of the free surface we find in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' While this assumption is reasonable for a wide range of applications, for extremely rapid evaporation (for example, laser-induced evaporation, Volkov & Strizhak (2019)), a more careful consideration of the evolution after deposition would be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Assuming the contact line is pinned, the volume of the droplet V ∗(t∗) is given by V ∗(t∗) = 2π � R∗ 0 r∗h∗(r∗, t∗) dr∗, V ∗(0) = V ∗ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='9) The total mass loss due to evaporation F ∗(t∗) is given by F ∗(t∗) = 2π � R∗ 0 r∗E∗(r∗) dr∗ = 4D∗(c∗ s − c∗ ∞)R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='10) Thus, conservation of mass in the liquid phase is dV ∗ dt∗ = −F ∗ ρ∗ = −4D∗(c∗ s − c∗ ∞)R∗ ρ∗ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='11) so that V ∗(t∗) = V ∗ 0 − 4D∗(c∗ s − c∗ ∞)R∗t∗ ρ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='12) In particular, the dryout time, that is the time when the drop has fully evaporated, is t∗ f = ρ∗V ∗ 0 4D∗(c∗s − c∗∞)R∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='13) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Solute model The droplet is assumed to be sufficiently thin that the transport of the solute is governed by the depth- averaged advection-diffusion equation ∂ ∂t∗ (h∗φ∗) + 1 r∗ ∂ ∂r∗ � r∗ � h∗u∗φ∗ − D∗ φh∗ ∂φ∗ ∂r∗ �� = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='14) for 0 < r∗ < R∗, t∗ > 0, where φ∗(r∗, t∗) is the depth-averaged solute concentration and D∗ φ is the solutal diffusion coefficient (Wray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Pham & Kumar 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' While there is an acknowledged effect of the solute particles eventually being trapped at and transported along the free surface (Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' D’Ambrosio 2022), this effect is less pronounced for thin droplets, where the capture tends to occur closer to the contact line due to the stronger outward radial flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Thus, we shall neglect its effects here as our study concerns the interplay between gravity, surface tension and solute advection/diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' A more focused analysis on the final deposit profile would certainly need to account for such effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='14) must be solved subject to the symmetry condition ∂φ∗ ∂r∗ = 0 at r∗ = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='15) and the condition that there can be no flux of solute particles through the pinned contact line, r∗ � h∗u∗φ∗ − D∗ φh∗ ∂φ∗ ∂r∗ � = 0 at r∗ = R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='16) Finally, we impose the initially uniform distribution of solute throughout the droplet, so that φ∗(r∗, 0) = φ∗ 0 for 0 < r∗ < R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='17) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Non-dimensionalization We assume that the fluid velocity is driven by evaporation and, for now, we retain both gravity and surface tension, so that the pertinent scalings are (r∗, z∗) = R∗(r, δz), u∗ = D∗(c∗ s − c∗ ∞) δρ∗R∗ u, t∗ = t∗ ft, φ∗ = φ∗ 0φ, (h∗, h∗ 0) = δR∗(h, h0), p∗ = p∗ atm + µ∗D∗(c∗ s − c∗ ∞) δ3ρ∗R∗2 p V ∗ = V ∗ 0 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='18) Gravity can lead to multiple peaks in the early stages of coffee ring formation 5 Note, in particular, that the choice of timescale fixes the dimensionless dryout time to be t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Upon substituting the scalings (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='18) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='3)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5), we see that ∂h ∂t + 1 4r ∂ ∂r (rhu) = − 1 2π √ 1 − r2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='19) u = h2 3Ca ∂ ∂r � −Boh + 1 r ∂ ∂r � r∂h ∂r �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='20) for 0 < r < 1, 0 < t < 1, where the Capillary and Bond numbers are defined by Ca = µ∗D∗(c∗ s − c∗ ∞) δ4ρ∗R∗σ∗ and Bo = ρ∗g∗R∗2 σ∗ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='21) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Under scalings (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='18), the symmetry conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6) become, rhu = ∂h ∂r = 0 at r = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='22a, b) while the contact line conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='7) are h = rhu = 0 at r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='23a, b) The initial condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8) becomes h(r, 0) = h0(r) for 0 < r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='24) Finally, the dimensionless form of conservation of liquid volume conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='12) is 1 − t = 2π � 1 0 rh(r, t) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='25) After scaling, the solute transport equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='14) becomes ∂ ∂t (hφ) + 1 4r ∂ ∂r � r � huφ − h Pe ∂φ ∂r �� = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='26) for 0 < r <, 0 < t < 1, where the solutal P´eclet number is Pe = D∗(c∗ s − c∗ ∞) δρ∗D∗ φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='27) The symmetry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='15) and boundary conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='16) become ∂φ ∂r = 0 at r = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='28) and r � huφ − h Pe ∂φ ∂r � = 0 at r = 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='29) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Finally, the initial condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='17) becomes φ(r, 0) = 1 for 0 < r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='30) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Integrated mass variable formulation The assumption that the solute is dilute decouples the flow and solute transport problems, so that we may solve for h and u from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='19)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='25) independently of the solute concentration, φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We shall discuss the resulting flow solution shortly in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' First, however, we present a reformulation of the solute transport problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='26)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='30), which will greatly aid us in our asymptotic and numerical investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In this, we follow Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2021, 2022) by introducing the integrated mass variable M(r, t) = � r 0 sh(s, t)φ(s, t) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='31) By integrating the advection-diffusion equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='26) from 0 to r and applying the no-flux condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='29), 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray we find that ∂M ∂t + �u 4 + 1 4Pe �1 r + 1 h ∂h ∂r �� ∂M ∂r − 1 4Pe ∂2M ∂r2 = 0 for 0 < r, t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='32) This must be solved subject to the boundary conditions M(0, t) = 0, M(1, t) = 1 2π for t > 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='33a, b) where the latter condition dictates that mass is conserved along a radial ray, which replaces the no-flux condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Finally, the initial condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='30) becomes M(r, 0) = � r 0 sh(s, 0) ds for 0 < r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='34) Finally, we note that, once we have determined the integrated mass variable from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='32)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='34), the solute mass m = φh can then be retrieved from m = 1 r ∂M ∂r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='35) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Flow solution in the large-Ca limit We now suppose that surface tension dominates viscosity in the flow problem, that is Ca ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Importantly, this means that the problems for the free surface profile and the flow velocity decouple, an assumption that is valid for a wide range of different liquids and evaporation models in practice (Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Unlike these previous studies, however, we shall retain gravity in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='20) to investigate what role it plays in the formation of the nascent coffee ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' To this end, we neglect the left-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='20), so that upon integrating and applying the symmetry condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='22), the contact line condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='23a) and the conservation of liquid volume condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='25), we deduce that h(r, t) = (1 − t) π I0( √ Bo) I2( √ Bo) � 1 − I0( √ Bo r) I0( √ Bo) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1) where Iν(z) is the modified Bessel function of the first kind of order ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' With the free surface found, the velocity is determined immediately from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='19) and the no-flux condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='23b) to be u(r, t) = 1 rh � 2 π � 1 − r2 + 4I0( √ Bo) πI2( √ Bo) �r2 − 1 2 + 1 √ BoI0( √ Bo) (I1( √ Bo) − rI1( √ Bo r)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2) Notably, as in the surface tension-dominated regime where Bo → 0, time is separable in both the free surface and fluid velocity profiles, and so merely acts to scale the functional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In particular, this means that the streamlines and pathlines coincide, which we shall exploit when considering the regime in which solutal diffusion is negligible in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We display the scaled forms of the free surface and fluid velocity for various values of the Bond number in figure 2a,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' For the droplet free surface profile, we see the expected transition from the spherical cap for Bo → 0 (Deegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2000) to the flat ‘pancake’ droplet for Bo → ∞ (Rienstra 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' For each Bond number, the velocity is singular at the contact line — as expected for a diffusive evaporative flux (see, for example, Deegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2000)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We see that as the effect of gravity increases, the sharp increase in u occurs closer to the contact line, corresponding to the progressively smaller region in which surface tension effects are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Finally, since this will be important in our discussions of the secondary peaks seen in the solute mass profile in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2, we show the divergence of the fluid velocity in figure 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' For small Bond numbers, the divergence is monotonically increasing with r and, as with the velocity, singular at the contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, for moderate and large Bond numbers ≳ 15, we see a clear change of behaviour, with a region of non-monotonic behaviour in the droplet interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This behaviour is accentuated as Bo → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' For future reference, the asymptotic behaviours of the free surface and fluid velocity as r → 1− for Bo = O(1) are given by h = θc(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo)(1 − r) + O((1 − r)2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='3) u = 2χ θc(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo)(1 − r)−1/2 + O � (1 − r)1/2� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4) Gravity can lead to multiple peaks in the early stages of coffee ring formation 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5 1 0 1 2 3 4 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5 1 0 1 2 3 4 5 Figure 2: (a) The quasi-steady droplet free surface, (b) the fluid velocity, and (c) the divergence of the velocity displayed for Bo = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1 (black), 1 (dark purple), 10 (blue), 20 (cyan), 50 (green) and 100 (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Notably, we see the transition from the spherical cap to the ‘pancake’ droplet profile as the effect of gravity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The divergence of the fluid velocity also shows a transition from a monotonic to a non-monotonic profile as the Bond number increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' where θc(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) = − lim r→1− ∂h ∂r = (1 − t)ψ(Bo), ψ(Bo) = √ BoI1( √ Bo) πI2( √ Bo) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5) is the leading order contact angle in the thin droplet limit and χ = √ 2 π (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6) is the dimensionless coefficient of the inverse square root singularity at the contact line in the evaporative flux (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Note that we have chosen this notation to highlight the similarities with the previous analysis of Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2022), who consider a surface tension-dominated droplet of arbitary contact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' On the other hand, if we take 1 − r = O(1) and consider the large-Bo limit of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2), we find that h = h0(t) + Bo−1/2h1(t) + O(Bo−1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='7) u = u0(r, t) + Bo−1/2u1(r, t) + O(Bo−1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8) as Bo → ∞, where h0(t) = (1 − t) π , h1(t) = 2(1 − t) π , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='9a, b) and u0(r, t) = 2 √ 1 − r2 r(1 − t) (1 − � 1 − r2), u1(r, t) = 4 r(1 − t)(1 − � 1 − r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='10a, b) Notably, in the droplet bulk, the droplet free surface h is flat to all orders: the aforementioned characteristic of ‘pancake’ droplets associated with large Bond numbers (Rienstra 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' These expansions break down close to the contact line where surface tension effects become important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We find that for 1 − r = Bo−1/2¯r, we have h = ¯h0(¯r, t) + Bo−1/2¯h1(¯r, t) + O(Bo−1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='11) u = Bo−1/4 � ¯u0(¯r, t) + Bo−1/4¯u1(¯r, t) + Bo−1/2¯u2(¯r, t) + O(Bo−3/4) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='12) as Bo → ∞, where ¯h0(¯r, t) = 1 π (1 − t)(1 − e−¯r), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='13) ¯h1(¯r, t) = 2(1 − t) π � 1 − e−¯r� − (1 − t)¯r 2π e−¯r, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='14) 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray and ¯u0(¯r, t) = 2 √ 2¯r (1 − t)(1 − e−¯r), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='15) ¯u1(¯r, t) = 4 (1 − t) − 4¯r (1 − t)(1 − e−¯r), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='16) ¯u2(¯r, t) = 3¯r3/2 √ 2(1 − t)(1 − e−¯r) − 4 √ 2¯r (1 − t)(1 − e−¯r) + √ 2¯r3/2e−¯r (1 − t)(1 − e−¯r)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='17) We note here that as ¯r → 0, we retrieve the expect inverse square root singularity in the fluid velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Solute transport in the large-Pe limit Having fully determined the leading-order flow, we now seek to understand the transport of solute within the drop and to make predictions about the early-stages of coffee ring formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We follow the analyses of Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2021, 2022) by considering the physically-relevant regime in which Pe ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In this regime, in the bulk of the droplet, advection dominates solutal diffusion, with the latter only being relevant close to the contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Previous studies of this problem have concentrated on surface tension-dominated drops (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo → 0) and have shown how the competition between solutal advection and diffusion near the contact line leads to the early stages of coffee ring formation in drying droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In this analysis, we wish to investigate how this behaviour changes as we allow Bo to vary, which we pursue using a hybrid asymptotic-numerical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' There are naturally several different asymptotic regimes depending on the relative sizes of Bo and Pe, but these broadly fall into two categories i) intermediate Bond number, Bo = O(1), where the asymptotic structure of the solute transport depends solely on the large P´eclet number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' ii) large Bond number, Bo ≫ 1, where the asymptotic structure of the solute transport now depends on the relative sizes of Bo and Pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In the first regime where Bo = O(1), Pe ≫ 1, the asymptotic structure of the flow is a natural extension of the surface tension-dominated case considered in Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In the droplet bulk where 1−r = O(1), solute advection dominates diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, close to the contact line, a balance between solute advection and diffusion occurs when rhuφ ∼ rh Pe ∂φ ∂r =⇒ 1 − r = O(Pe−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1) We discuss the asymptotic solution for this regime in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In the second regime, there are several different possibilities depending on the relative sizes of the boundary layer where surface tension enters the flow profile and the solutal diffusion boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The richest distinguished asymptotic limit is that in which these boundary layers are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' As detailed in §3, for large Bond number the free surface is flat in the bulk of the droplet, with the effect of surface tension restricted to a boundary layer at the contact line of size 1 − r = O(Bo−1/2), where h = O(1) and u = O(Bo−1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Turning to the solute transport equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='26), since h is order unity and u is square root bounded in this region, advection and diffusion are comparable when 1 − r = O(Pe−2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2) Hence, in the most general limit in which the size of the two boundary layers are comparable, we have α = Bo−1/2Pe2/3 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='3) The asymptotic analysis in this regime is somewhat more involved, so for brevity, we present the details in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Asymptotic solution when Bo = O(1) In this section, we present the asymptotic solution of the solute transport problem as Pe → ∞ when Bo = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The analysis herein is a natural extension of Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' For the purposes of this section, we shall use the concentration form of the advection-diffusion equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='26)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='30) and, in particular, find the solution in terms of the solute mass m = φh, where h is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Gravity can lead to multiple peaks in the early stages of coffee ring formation 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Outer region In the droplet bulk where 1−r = O(1), we seek a solution of the form m = m0(r, t)+O(Pe−1) as Pe → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Substituting into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='26), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='30), we find that ∂m0 ∂t + 1 4r ∂ ∂r (rm0u) = 0 for 0 < r < 1, t > 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4) where u is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2), subject to m(r, 0) = h(r, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This is the usual advection equation, with solution given by m0(r, t) = h(R, 0) J(R, t) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5) where R is the initial location of the point that is at r at time t and J(R, t) is the Jacobian of the Eulerian- Lagrangian transformation, that satisfies Euler’s identity, D Dt(log J) = 1 4r ∂ ∂r(ru), J(R, 0) = 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6) where D/Dt is the convective derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' A straightforward asymptotic analysis of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4) reveals that u∂m ∂r ∼ m r ∂ ∂r (ru) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='7) as r → 1−, so that m0 = O(√1 − r) as r → 1−, and hence the concentration φ0 is square root singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This sharp local concentration increase necessitates the inclusion of a diffusive boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Inner region Close to the contact line, we set r = 1 − Pe−2ˆr, h = Pe−2ˆh, u = Peˆu, m = Pe2 ˆm, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8) where the last scaling on the mass comes from global conservation of solute considerations (Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We seek an asymptotic solution of the form ˆm = ˆm0(ˆr, t) + O(Pe−1) and find to leading order ∂ ∂ˆr �� 2χ θc(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) √ ˆr − 1 ˆr � ˆm0 + ∂ ˆm0 ∂ˆr � = 0 in ˆr > 0, t > 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='9) such that � 2χ θc(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) √ ˆr − 1 ˆr � ˆm0 + ∂ ˆm0 ∂ˆr = 0 for ˆr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='10) It is straightforward to show that the solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='9)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='10) is given by ˆm0(ˆr, t) = C(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo)ˆrexp � − 4χ θc(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) √ ˆr � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='11) where, by pursuing a similar matching process to Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2022), we find that the coefficient C(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) is given by C(t) = 64χ4 3θc(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo)4 N(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='12) where N(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) is the leading-order accumulated mass advected into the contact line region up to time t, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' N(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) = 1 4 � t 0 m0(r, τ)u(r, τ) dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='13) It is worth noting that this solution follows directly from the Bo = 0 regime discussed in Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2021, 2022), with the alterations due to gravity entering into the accumulated mass flux into the contact line and the leading order contact angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In particular, we note that in the limit Bo → 0, since ψ = 4/π + O(Bo), this yields the expected form found in the surface tension-dominated problem in Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2022) (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We display the accumulated mass flux and the local contact angle for a wide range of Bond numbers in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We see that as the influence of gravity increases, the acccumulated mass flux into the contact line at a fixed percentage of the evaporation time is reduced from the surface tension-dominated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' On the other hand, the local contact angle increases, commensurate with the droplet profile transitioning from 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5 1 0 1 2 3 4 Figure 3: (a) The accumulated mass flux, N(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) as defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='13) and (b) the leading order local contact angle θc(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) as defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5), for Bo = 10−2 (purple), Bo = 10−1 (purple) (dark blue), Bo = 1 (light blue), Bo = 10 (green) and Bo = 102 (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' a spherical cap to a ‘pancake’ droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We note that this combined behaviour leads to C(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) decreasing as Bo increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We discuss how these findings impact coffee ring formation in more detail in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Composite solution We may use van Dyke’s rule (Van Dyke 1964) to formulate a leading-order composite solution for the solute mass that is valid throughout the drop by combining the leading-order-outer solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5) and the leading-order-inner solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='11), finding mcomp(r, t) = mouter(r, t) + Pe2m � Pe2(1 − r), t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='14) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Comparisons between the numerical and asymptotic results Our asymptotic predictions are compared to numerical simulations of the full advection-diffusion problem for the integrated mass variable given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='32)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The integrated mass variable is chosen over the solute mass m or the concentration φ since it is better behaved close to the contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The numerical procedure requires careful consideration of the thin diffusive boundary layer and we follow a similar approach to that described for the surface tension-dominated problem by Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We give a summary of the methodologies in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We begin by comparing the asymptotic predictions of the solute mass profiles to numerical solutions in the regime where Bo = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In figure 4, we display asymptotic (dashed, red) and numerical (solid, blue) curves at 10% intervals of the total drying time for Pe = 102, Bo = 1 (a,b) and Pe = 102, Bo = 30 (c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In each figure, we see excellent agreement between the simulations of the full system and the leading-order composite solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' There is a clear formation of the expected coffee ring in the region near the contact line, where solutal diffusion and advection interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We see that increasing the Bond number in this regime leads to a slight reduction of the size of the coffee ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This behaviour is reminiscent of the Bo = 0 regime considered previously by Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, in the later stages of the Pe = 102, Bo = 30 example, we see evidence of a qualitative difference in behaviour, with the formation of another peak in the mass profile in the droplet interior (see inset in figure 4(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Henceforth, we shall refer to the classical coffee ring as the primary peak and this new feature as the secondary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The presence of the secondary peak depends on the Bond number, as there is no secondary peak in any of the profiles when Bo = 1, but it also depends on the drying time, as the peak only develops in the later stages of evaporation when Bo = 30 (between 60 − 70% of the drying time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Noticeably, the secondary peak is significantly smaller in magnitude than the primary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' For larger Bond numbers, we compare the numerical results to the asymptotic predictions in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In figure 5, we display results for Pe = 102, Bo = 105 (α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='07) (a,b) and Pe = 103 and Bo = 104 (α = 1) (c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In each case, we display the composite profile for the solute mass given by (A 34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In each figure, we see that after an initial transient the asymptotic predictions and numerical results are again in excellent agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moreover, we see further evidence of the existence of a secondary peak in the case Gravity can lead to multiple peaks in the early stages of coffee ring formation 11 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8 1 10 -6 10 -4 10 -2 10 0 10 -3 10 -1 10 1 10 3 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8 1 10 -6 10 -4 10 -2 10 0 10 -3 10 -1 10 1 10 3 10 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='105 Figure 4: Profiles of the solute mass when an axisymmetric droplet evaporates under a diffusive evaporative flux for (a,b) Pe = 102 and Bo = 1 (c,d) Pe = 102 and Bo = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In each figure, the bold, black curve represents the initial mass profile, which corresponds to the droplet free surface profile (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We also display plots at time intervals of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1 up to t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='9 in which solid, blue curves represent the results from the numerical solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='32)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='34) and the dashed, red curves show the leading-order composite mass profile, given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The right-hand figures display a close-up of the profiles near the contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In (c), the inset shows a close up of the mass profile in the droplet interior at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='9 where we see a clear formation of a secondary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Pe = 103, Bo = 104 regimes, where the peak appears much earlier and is noticeably larger than that in the previous example (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' figure 4c, where Pe = 102, Bo = 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, we also note again the strong dependence of the secondary peak on Bo and, possibly, Pe, as there is no evidence of such an interior peak when Pe = 102, Bo = 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' These findings prompt us to investigate this new feature more closely, alongside a discussion of how the characteristics of the primary peak — and hence the classical coffee ring — depend on the Bond number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Properties of the two peaks Given the excellent comparisons displayed in the previous section, we seek to use our asymptotic results to investigate properties of the nascent coffee ring and, in particular, the new feature of these moderate-to-large Bond number regimes: the secondary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Primary peak We shall begin by discussing the effect of the Bond number on the primary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' As in previous studies of the surface tension-dominated regime, the formation of the primary peak is driven by the competing diffusive and advective solute fluxes (Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2021, 2022) and is always present in the large-Pe regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Furthermore, since all of the features of interest are well within the solutal diffusion boundary layer, we will 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray Mass Mass Figure 5: Profiles of the solute mass when an axisymmetric droplet evaporates under a diffusive evaporative flux for (a,b) Pe = 102 and Bo = 105 (α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='07) (c,d) Pe = 103 and Bo = 104 (α = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In each figure, the bold, black curve represents the initial mass profile (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We also display plots at time intervals of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1 up to t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='9 in which solid, blue curves represent the results from the numerical solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='32)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='34) and the dashed, red curves show the composite mass profiles, given by (A 33) for the integrate mass variable and (A 34) for the solute mass, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Note that in (c,d), we can clearly see the development of the secondary peak behind the primary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' use the inner solution — as discussed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2 in the Bo = O(1) regime and §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2 in the large-Bo regime — to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo = O(1) regime When the Bond number is order unity, the analysis is a natural extension of that in Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The local solute profile is dominated by the leading-order inner solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Introducing the time- dependent P´eclet number Pet = Pe 1 − t, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1) the nascent coffee ring profile may be seen to have the similarity form ˆm0(R, t) Pe2 t N(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) = 2χ 3ψ(Bo)f �√ R, 3, 4χ ψ(Bo) � , R = Pe2 t(1 − r) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2) where ψ and χ retain their definitions from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6) as the initial local contact angle and the coefficient of the evaporative flux singularity, respectively, and f(x, k, l) = lkxk−1e−lx/Γ(k) is the probability density function of a gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' It is this functional form which describes the characteristic narrow, sharp peak of the coffee ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Since the definition of R only depends on the time-dependent P´eclet number, we can clearly illustrate the Gravity can lead to multiple peaks in the early stages of coffee ring formation 13 0 20 40 60 80 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1 Figure 6: The similarity profile (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2) of the leading-order-inner solute mass profile for Bo = 10−2 (purple), Bo = 10−1 (purple) (dark blue), Bo = 1 (light blue), Bo = 10 (green) and Bo = 102 (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' effect of gravity by plotting the similarity profile (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2) for a range of Bond numbers in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We see that, as the effect of gravity increases, the height of the primary peak decreases, and the peak moves further from the pinned contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moreover, the shape of the primary peak tends towards a shallower, wider profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Notably, this behaviour is driven purely by changes in ψ(Bo);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' as we saw in figure 3a, the accumulated mass flux into the contact line decreases with the Bond number, clearly this acts to accentuate this behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We can expand upon these results by finding the leading order asymptotic prediction of the primary peak height and location, which are given by rpeak,I(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) = 1 − ψ(Bo)2 4Pe2 t χ2 , mpeak,I(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) = 16Pe2 tN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo)χ2 3e2ψ(Bo)2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='3) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Notably, while gravity only influences the location of the primary peak through the initial local contact angle, ψ(Bo), the height depends on gravity through both the contact angle and the accumulated mass flux, N(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In particular, referring back to figure 3, this means that gravity has a stronger effect on the peak height than its location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We illustrate the veracity of these asymptotic predictions by comparing them to the corresponding numer- ical results for Pe = 102 and a range of Bond numbers in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' As anticipated from the comparisons of the solute mass profiles, we see excellent agreement between the asymptotic predictions and the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In particular, in figure 7a, we note that as the influence of gravity increases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo increases), the coffee ring effect is inhibited: although a peak clearly still forms, it is lower for large Bond number at a similar stage of the drying process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This effect varies nonlinearly with time (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' For example, considering the cases Bo = 1/2 and Bo = 30, after 50% of the drying time, the peak height is reduced by a factor of ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='97, while at 60% of the drying time, the reduction is a factor of ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='85 and at 90% of the drying time, it is ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Similarly, in figure 7b, we see that as the Bond number increases, the location of primary peak moves further from the contact line and that this significantly increases as the Bond number gets larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' For Bo = 1/10, 1/2, 1 the location is almost indistinguishable from the zero-Bond number solution — where Pe2 t (1 − r) = 2 (Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2022)) — but for Bo = 30, this has increased to ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' It is worth noting that in all this analysis, the P´eclet number simply acts to scale the above findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' For a larger P´eclet number, the height of the primary peak increases, while it is located closer to the contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This is precisely what is seen for the Bo = 0 regime (Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Large-Bo regime In the large-Bo regime, given the size of the primary peak, we anticipate that the leading-order-inner solution ˜ M0(˜r, t) as given by (A 16) should reasonably capture the features of the primary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8 1 0 2 4 6 8 10 Figure 7: Numerical (circles) and asymptotic predictions (solid lines) of (a)) the height of the primary peak, mpeak,I(t)/Pe2/3 t and (b)) its location Pe2/3 t (1 − rpeak,I(t)) in the Bo = O(1) regime as given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' For each curve, Pe = 102, while the Bond number varies according to Bo = 1/10 (dark purple), Bo = 1/2 (blue), Bo = 1 (green) and Bo = 30 (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8 1 10-6 10-4 10-2 100 102 104 106 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8 1 10-6 10-5 10-4 10-3 10-2 10-1 100 Figure 8: Numerical (circles) and asymptotic predictions (solid lines) of (a)) the height of the primary peak, MI(t) = mpeak,I(t)/Pe2/3 and (b)) its location ηI(t) = Pe2/3(1−rpeak,I(t)) as given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Results are presented for Pe = 102, Bo = 103 (α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='68, yellow), Pe = 103, Bo = 104 (α = 1, green), Pe = 104, Bo = 105 (α ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='47, blue) and Pe = 105, Bo = 105 (α ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='81, dark purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' unlike its moderate-Bo counterpart, there is no simple similarity form for the solution in this regime, so that we proceed more carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We denote the height and location of the primary peak by mpeak,I(t) = Pe2/3MI(t), rpeak,I(t) = 1 − Pe−2/3ηI(t), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4a, b) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='35), the location of the maximum ηI(t) satisfies 0 = ∂2 ˜ M0 ∂˜r2 (ηI(t), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5) Gravity can lead to multiple peaks in the early stages of coffee ring formation 15 Utilizing (A 13), we find that ∂2 ˜ M0 ∂˜r2 (ηI(t), t) = − � ˜u0 − 1 ˜h0 ∂˜h0 ∂˜r � ∂ ˜ M0 ∂˜r ����� (ηI(t),t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6) Since ∂ ˜ M0/∂˜r > 0 for ˜r > 0, we conclude ˜u0(ηI(t), t) − 1 ˜h0(ηI(t), t) ∂˜h0 ∂˜r (ηI(t), t) = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='7) so that ηI(t) = α 2 W0 �(1 − t)2 4α3 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8) where W0(x) is the Lambert-W function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' the solution to wew = x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' With ηI(t) in hand, the corresponding height of the ring at the peak is then given by MI(t) = � −B0(t) I(ηI(t), t) � = � N(t) I(ηI(t), t) �� ∞ 0 1 I(s, t) ds �−1� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='9) where I(r, t) is given by (A 15) and N(t) is the leading-order accumulated mass flux into the boundary layer (A 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Note that, in this regime, N(t) is independent of α and, hence, the Bond number, but the function I(r, t) does change with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In figure 8, we plot the asymptotic predictions of the location and height of the primary deposit peak against the simulation results for a range of different P´eclet and Bond numbers (and, correspondingly, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' There are several discernible features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' After an initial transient, the location of the peak is captured extremely well by the asymptotic prediction (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8) for each case presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This initial transient is primarily due to the lack of a distinct peak at early stages of the drying process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' a period of time is necessary for sufficient solute to be advected to the contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This process takes longer for smaller P´eclet numbers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' when diffusion is relatively stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The height of the primary peak is captured quite well by the asymptotic prediction (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='9), particularly for larger P´eclet numbers and as time increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' It is worth noting that the error in the approximation of the height is O(Pe1/3), so for an improved estimation of the primary peak height, it would be necessary to consider the first two inner solutions ˜ M0(˜r, t) and ˜ M1(˜r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' While this is possible, the results do not have a simple analytic form, so are not practical to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We also note that, as the droplet evaporates, the primary peak both increases in size and moves closer to the contact line, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' MI(t) increases and ηI(t) decreases as t increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Secondary peak As evidenced by the solute mass profiles, the behaviour of the secondary peak — and indeed, even its presence — is more complex than that of the primary peak, which always forms in the large-Pe regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We have seen, for example in figure 4 in the Bo = O(1) regime, that the presence of the peak varies with both Bo and drying time, while when Bo ≫ 1, we have also seen variation with Pe (and hence α), see for example figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This gives a clear indication that we need to treat this feature more carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' To begin, we will consider whether or not the secondary peak is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We shall first fix the P´eclet number and use the numerical results to produce a regime diagram in (Bo, t)-parameter space indicating whether one or two peaks are present in the solute mass profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We note here that these are the only options that we have been able to find — we have found no instances of more than two peaks appearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We show the results for Pe = 102 in figure 9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In the figure, solute profiles with one peak — i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' only the classical coffee ring — are denoted by blue circles, while solute profiles exhibiting two peaks are denoted by red circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We see a strong nonlinear dependence on both Bond number and dryout time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In particular, there is a band of Bond numbers between around Bo ≈ 10 and Bo ≈ 30000 that may lead to secondary peak formation, although the existence of a peak also depends strongly on t for a fixed Bond number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We note that for Bo ≲ 10, there is only one peak for any t, in agreement with the classical Bo = 0 regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moreover, for very large Bond number Bo ≳ 30000, again we see that there is only one peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We illustrate the effect of the P´eclet number by plotting the equivalent regime diagram for Pe = 103 in figure 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Remarkably, the onset of the secondary peak appears to be unaffected by the increase of the P´eclet number, although the band of Bond numbers for which we see two peaks is significantly widened into larger Bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Notably, however, the shape of the curve delineating between two peaks / one peak for large Bond number appears to be independent of Pe, only its location has shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray 10-3 10-2 10-1 100 101 102 103 104 105 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8 1 One peak Two peaks 10-3 10-2 10-1 100 101 102 103 104 105 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8 1 One peak Two peaks Figure 9: (Bo, t)-regime diagram illustrating the presence of either one (blue circles) or two (red circles) peaks in the solute mass profile for (a) Pe = 102 and (b) Pe = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The data is extracted from the numerical simulations and demonstrates a clear band of Bond numbers for which two peaks may exist in the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In each figure, the black curve denotes the asymptotic prediction of when the centre of the droplet changes from a maximum to a minimum as given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Gravity can lead to multiple peaks in the early stages of coffee ring formation 17 10-6 10-3 100 10-2 10-1 100 101 102 103 10-6 10-3 100 10-2 10-1 100 101 102 103 10-6 10-3 100 10-2 10-1 100 101 102 103 Figure 10: Solute profiles for an evaporating droplet with Pe = 102 and Bo = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The deposit profile is displayed on a doubly-logarithmic plot at 25% (a)), 35% (b)) and 75% (c)) of the drying time in order to catch the emergence of the secondary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In each of a) − c), the primary peak is indicated by a red circle, while the secondary peak is indicated by a black circle (when it exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Onset of the secondary peak In this section, we seek to investigate some of the phenomena around the onset of the secondary peak in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We saw that for a fixed P´eclet number, there was a distinct switch from one to two peaks for Bond number Bo ≈ 10 and that this switch appears to be independent of Pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This suggests that secondary peak formation is not a result of the interplay between solutal advection and diffusion that drives the classical coffee ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In order to investigate the reasons behind the presence of or lack of a secondary peak, in figure 10, we plot numerical results for the solute profiles in a droplet with Pe = 102, Bo = 20 at 25%, 35% and 75% of the drying time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In the figure, the primary and secondary peaks are indicated by the red and black circles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We clearly see in figure 10a that at 25% of the drying time there is only one peak, but by 35% of the drying time, the secondary peak has emerged close to the droplet centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' As the droplet evaporates further to 75% of the drying time the secondary peak has moved further towards the droplet contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This particular example gives us a strong indication that the secondary peak initially arises from the centre of the drop and, in particular, appears to be linked with a transition from the centre being a maximum in solute mass profile — as it is for the classical coffee ring of Deegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (1997, 2000) — to a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' To investigate this postulate, we consider the behvaiour close to the droplet centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' To simplify things, since the initial emergence of the secondary peak appears to be independent of the P´eclet number, we neglect solutal diffusion completely, taking Pe = ∞, so that the solute mass m satisfies the first-order semi-linear equation ∂m ∂t + 1 4r ∂ ∂r (rmu) = 0, m(r, 0) = h(r, 0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='10) where, since the emergence appears to be rooted in the region where Bo ≈ 10, we consider the moderate Bond number regime and retain the full expressions for the droplet free surface h and fluid velocity u given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We seek an asymptotic solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='10) as r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' First, we note that for small arguments, the free surface and velocity have the following asymptotic expansions: h(r, t) ∼ (1 − t) � H0(Bo) + H1(Bo)r2 + o(r2) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='11) u(r, t) ∼ 1 (1 − t) � U0(Bo)r + U1(Bo)r3 + o(r3) � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='12) 18 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray as r → 0, where H0(Bo) = (I0( √ Bo) − 1) πI2( √ Bo) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='13) H1(Bo) = − Bo 4πI2( √ Bo) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='14) U0(Bo) = 2 √ Bo − √ BoI0( √ Bo) − 2I1( √ Bo) √ Bo(1 − I0( √ Bo)) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='15) U1(Bo) = −(Bo3/2 − √ BoI0( √ Bo) + √ BoI0( √ Bo)2 + 2I1( √ Bo) − 2BoI1( √ Bo) − 2I0( √ Bo)I1( √ Bo) 4 √ Bo(1 − I0( √ Bo))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='16) Now, by the symmetry of the problem, the droplet centre must be a critical point, so we seek a solution of the form m = m0(t) + m1(t)r2 + o(r2) as r → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Upon substituting this ansatz and the above forms for h and u into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='10), straightforward calculation yields m0(t) = H0(1 − t)U0/2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='17) m1(t) = �2U1H0 U0 + H1 � (1 − t)U0 − 2U1H0 U0 (1 − t)U0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='18) Hence, given that initially the droplet has a maximum at its centre for any Bo, we deduce that the maximum becomes a minimum at the critical time tc such that m1(tc) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='19) Since 2U1H0/U0 + H1 < 0, H0 > 0, U0 > 0 for all Bo, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='19) only has solutions for Bo > Boc where U1(Boc) = 0 =⇒ Boc ≈ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='20) When Bo > Boc, we may solve (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='19) explicitly to find tc(Bo) = 1 − � 2U1(Bo)H0(Bo) 2U1(Bo)H0(Bo) + H1(Bo)U0(Bo) �2/U0(Bo) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='21) This critical curve in figure 9 is displayed as the solid black curve and we see that there is excellent agreement between this prediction and the transition from one to two peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' But, what is causing the transition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Since the phenomenon is independent of the P´eclet number, it is purely a result of the droplet geometry and the evaporation-driven flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In particular, we note that the critical Bond number Boc given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='20) is linked to the change in sign of U1, which is equivalent to requiring that (1 − t)∇ · (uer) is decreasing near r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This correlates with the profiles of the divergence of u displayed in figure 2c, where we see this change in sign clearly as the Bond number increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Notably, considering the curve displayed in figure 9, we see that for Bo close to Boc, the secondary peak only emerges very late in the dryout process, but as the Bond number increases, it appears almost instantaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Hence, from this analysis alone, we might expect there to always be two peaks for Bo > Boc, but clearly this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We now investigate why in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Loss of the secondary peak Given its clear variation with each of t, Bo and Pe, it is perhaps unsurprising that it is more challenging to determine an analytical expression for the location of the right-hand boundary between two peaks and one peak in figure 9), and unfortunately we have been unable to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, it is relatively straightforward to illustrate why the transition occurs by considering a specific example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In figure 11, we plot solute mass profiles for Pe = 102 and Bo = 103 at 5%, 20%, 50% and 90% of the drying time indicating the primary and secondary peaks by red and black circles where appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Note that, for such a large Bond number, the critical time at which we would expect a secondary peak to be present may be found from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='21) to be tc ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8 × 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We see in figure 11a that, indeed, after 5% of the drying time, the secondary peak has emerged and is visible close to the droplet centre — moreover, at this stage, the primary peak associated with the coffee ring has yet to fully develop (so that the ‘one peak’ at this stage in figure 9a is in fact the secondary peak!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, by the time we reach 20% of the drying time, both peaks are clearly visible, with the primary peak now approximately 50% larger than the secondary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Gravity can lead to multiple peaks in the early stages of coffee ring formation 19 10-6 10-4 10-2 100 10-2 10-1 100 10-6 10-4 10-2 100 10-2 10-1 100 10-6 10-4 10-2 100 10-2 10-1 100 101 10-6 10-4 10-2 100 10-2 100 102 Figure 11: Solute profiles for an evaporating droplet with Pe = 102 and Bo = 103 displayed on a doubly- logarithmic plot at 5% (a)), 20% (b)), 50% (c)) and 90% (d)) of the drying time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In each figure, the primary peak is indicated by a red circle, while the secondary peak is indicated by a black circle when either exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Increasing time further, we see that the primary peak continues to grow rapidly so that, by 50% of the drying time, it is so large, that it has subsumed the secondary peak into its upstream tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' That is, the secondary peak is still present according to the Pe = ∞ theory, but due to the fact that Pe is actually finite and the corresponding presence of the classical coffee ring, we do not see the secondary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' If we then increase t even further, we see that by 90% of the drying time, the secondary peak has reemerged from the lee of the primary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' By this stage of the evaporation process, the primary peak has moved significantly closer to the contact line — here 1 − rpeak,I ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4 × 10−4, while the secondary peak is located at 1 − r ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8 × 10−2, so that it is sufficiently far behind the primary peak to be visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Thus, the loss of the secondary peak appears to be intrinsically tied to both the location, size and shape of the primary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Given that this behaviour largely occurs in the regime in which Bo ≫ 1, these properties of the primary peak are given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='9) and the derivative of (A 16), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Clearly, therefore, the behaviour is strongly dependent on t, Bo and Pe (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' figure 8, for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Properties of the secondary peak Given its dependence on the various parameters of the model, discerning the properties of the secondary peak analytically is challenging, particularly in the Bo = O(1)-regime since, in this case, the peak tends to be situated in the droplet bulk, so that we are unable to use the simpler forms of the inner solution described in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Hence, we utilize the numerical results to track the height mpeak,II(t) and location rpeak,II(t) of the secondary peak when it exists and we display the results for several different values of Pe, Bo in figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In the figure, results for Pe = 102 and Pe = 103 are denoted by circles and squares, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The Bond number is represented by the colour, with results for Bo = 20 (purple), 50 (dark blue), 100 (light blue) and 1000 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' It is evident that for each of the Bond numbers represented, increasing the P´eclet number appears to have negligible effect on both the size and location of the secondary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, both 20 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8 1 Figure 12: Numerical predictions of (a)) the height of the secondary peak, mpeak,II(t) and (b)) its location rpeak,II(t) for different values of Pe, Bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The symbols denote different P´eclet numbers: Pe = 100 (circles), Pe = 1000 (squares);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' while the colours denote different Bond numbers: Bo = 20 (purple), Bo = 50 (dark blue), Bo = 100 (light blue), Bo = 1000 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' properties do vary with the Bond number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In particular, as the Bond number increases, the secondary peak is situated closer to the contact line at the same stage of the drying process, and similarly, for a fixed Bond number, the peak gets closer to the contact line as the droplet evaporates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' On the other hand, variations of the secondary peak height with Bo are much less trivial, although for all of the displayed results, we see that the height of the secondary peak decreases as the droplet evaporates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This is in stark contrast to the primary peak, which always grows as more solute is transported to the contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Thus, we conclude that the secondary peak is predominantly driven by the Bond number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Indeed, it is only for sufficiently large Bond numbers that we find a second peak at all, and the properties of that peak then depend strongly on the size of Bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The only role played by the P´eclet number appears to be in the disappearance of the secondary peak when it is subsumed by the primary peak, which is typically orders of magnitude larger and always closer to the contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Summary and discussion In this paper, we have performed a detailed asymptotic and numerical analysis into the effect of gravity on the famous coffee ring phenomenon observed in solute-laden droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In the physically-relevant limit of small droplet capillary number, Ca ≪ 1, and large solutal P´eclet number, Pe ≫ 1, we identified two asymptotic regimes based on the size of the Bond number, Bo: i) a moderate Bond number regime, where Bo = O(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' ii) a large Bond number regime, Bo ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In the first of these regimes, gravity acts to flatten the droplet profile from the spherical cap of the zero- gravity problem, while reducing the liquid velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moreover, the asymptotic structure of the solute transport follows exactly that presented by Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2021) for surface tension-dominated droplets, with advection dominating in the droplet bulk, while the competition between advection and diffusion in a boundary layer of width of O(Pe−2) near the pinned contact line drives the nascent coffee ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Gravity acts to modify the surface tension-dominated solution both through the accumulated mass flux of solute into the contact line and a parameter dependent on the local contact angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In particular, as the Bond number increases, the height of the nascent coffee ring is reduced — which is consistent with the reduced flow velocity as Bo is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moreover, the peak is situated further from the contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' To categorize the role of gravity more explicitly, we derived an approximate similarity profile, ˆm0, for the Gravity can lead to multiple peaks in the early stages of coffee ring formation 21 nascent coffee ring profile, given by ˆm0(R, t) Pe2 t N(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) = 2χ 3ψ(Bo)f �√ R, 3, 4χ ψ(Bo) � , R = Pe2 t(1 − r) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1) where Pet = Pe/(1−t) is the time-dependent P´eclet number, N(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Bo) is the accumulated mass flux of solute at the contact line from the droplet bulk, χ is the coefficient of the inverse square root singularity in the evaporative flux at the contact line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' ψ(Bo) is the leading order initial local contact angle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' and f(x, k, l) = lkxk−1e−lx/Γ(k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' is the probability density function of a gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Clearly, the Bond number acts to scale the coffee ring profile through the accumulated mass flux, while it acts to change the shape of the profile through the initial contact angle ψ(Bo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In the second regime, the Bond number is large, so that the droplet is approximately flat, with surface tension confined to a narrow region near the pinned contact line — a ‘pancake’ or ‘puddle’ droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Thus, the asymptotic analysis discussed above is no longer valid, since there are two competing boundary layers near the edge of the droplet — the diffusion boundary layer in the solute transport and the surface tension boundary layer in the droplet free surface profile (and, hence, the liquid velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We derived the resulting solute distribution in the most general regime in which the two boundary layers are comparable, which reduces to the assumption that α = Bo−1/2Pe2/3 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Under this assumption, diffusion and advection balance in a region of size Pe−2/3 near the contact line, noticeably larger than in the moderate gravity regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This is a further indication of gravity acting to shift the coffee ring further from the contact line and, moreover, tends to cause shallower solute profiles in the boundary layer region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The asymptotic analysis in the large-Bond number regime is more challenging than that in the moderate Bond number regime and, in particular, the nascent coffee ring no longer collapses onto an approximate similarity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, we were able to derive expressions for the location (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='8) and height (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='9) of the peak, demonstrating that it still strongly depends on the accumulated mass flux of solute into the contact line alongside the parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In particular, increasing α leads to higher coffee rings that are located closer to the contact line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In each regime, we demonstrated that our asymptotic predictions were in excellent agreement with nu- merical simulations of the full advection-diffusion problem for the solute mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Alongside the anticipated nascent coffee ring driven by the competition between advection and diffusion of the solute, our asymptotic and numerical analysis also revealed a novel phenomenon: that the solute profile may have a secondary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The secondary peak was characterized by being situated upstream of and significantly smaller than the primary coffee ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moreover, the presence of this peak strongly depended on the Bond number, P´eclet number and evaporation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Further investigation revealed that, for a fixed P´eclet number, there exists a band in (Bo, t)-space at which two peaks are present in the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We demonstrated that the onset of this band is independent of the P´eclet number and is caused by the critical point at the centre of the droplet changing in type from a maximum (as in the spherical cap droplet in the Bo = 0 regime) to a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' When the critical point at the droplet centre changes type, an internal maximum forms downstream of the centre and it is this that corresponds to the secondary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This behaviour only occurs above a critical Bond number, Boc ≈ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='21, and then only after a given drying time, given by tc(Bo) = 1 − � 2U1(Bo)H0(Bo) 2U1(Bo)H0(Bo) + H1(Bo)U0(Bo) �2/U0(Bo) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2) In particular, as Bo increases, tc decreases, so the secondary peak emerges earlier in the evaporative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' These predictions were shown to be in excellent agreement to the numerical results and, remarkably, are independent of the P´eclet number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, the above analysis suggests that for all Bo > Boc and t > tc a secondary peak exists — something that we did not find in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The reason for this discrepancy was shown to be due to the presence of the primary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In particular, as time increases, the secondary peak is located further from the droplet centre so that it may get subsumed in the tail of the primary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' For a fixed Bond number, this possibility was shown to depend strongly on both the P´eclet number and the evaporation time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' this is due to the fact that the size of the primary peak increases with both t and Pe, while the size of the secondary peak only varies with t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Beyond this subsuming effect, however, we were able to demonstrate that the P´eclet number plays negligible role in the size and location of the secondary peak for a range of Bond numbers, suggesting that this feature may be reliably controlled simply by altering Bo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' In previous studies of coffee ring formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Deegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Popov (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2021), 22 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray gravity has frequently been neglected under the assumption of small Bond number, which is a reasonable assumption for sufficiently small droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, given that the Bond number may be increased in an experimental or industrial setting by steadily increasing the droplet radius, the influence of gravity may be of fundamental interest in applications that utilize droplet drying to adaptively control the shape of the residual deposit, such as colloidal patterning (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2010) and fabrication techniques using inkjet printing (Layani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Our analysis thus plays a dual role in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' First, we have presented the first formal categorization of the role of gravity in the early-stages of coffee ring formation and given a quantitative prediction of the resulting features of the solute profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Second, we have found a novel phenomenon — the secondary peak — which may also be exploited in such processes, particularly when the size of the primary peak can be carefully controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This is particularly relevant given that the secondary peak emerges at a relatively moderate critical Bond number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' There are, naturally, limitations to our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Throughout, we have assumed that the contact line remains pinned as the droplet evaporates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This has been shown to be a reasonable assumption for many configurations (see, for example, the experiments in Deegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Kajiya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2023)) and may further be enhanced by solute aggregation near the edge of the droplet (Orejon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Weon & Je (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Larson (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' However, at late stages of the evaporation, the contact line may depin and become mobile, moving inwards towards the droplet centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The contact line may then become pinned at a new location and the process may repeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This behaviour is known as ‘stick-slip’ evaporation and represents an important class within the field that is beyond the scope of the present study, but may represent an interesting future direction in terms of the effect of gravity, particularly with the presence of the secondary peak and its associated increased solute mass, which may promote re-pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Another effect that we have neglected in the present analysis is the possibility of solute becoming trapped at the free surface of the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' If this occurs, the solute is then transported to the contact line along the free surface, and has been suggested as an alternative mechanism for coffee ring formation (Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This behaviour has been demonstrated to occur for a wide variety of droplets but is more pronounced for droplets with large contact angles Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2016) or when vertical diffusion happens over a longer timescale than evaporation D’Ambrosio (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Since we deal with the opposite case of a thin drop with fast vertical diffusion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' so that the solute concentration may be assumed to be uniform across the droplet to leading order), we have not considered this phenomenon here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' It would be interesting to see how such behaviour impacted the solute profile in the current case, although it should be noted that the aforementioned studies neglect gravity entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' A further aspect that would form the basis of an exciting future study surrounds the assumption that the solute is dilute in the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Naturally, the build up of the solute in the coffee ring means that the concentration rapidly approaches levels where finite particle size effects can no longer be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This has been analysed in detail for surface tension-dominated droplets in Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2021, 2022) and a similar analysis would follow here with the inclusion of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' One possible aspect that would differentiate droplets where gravity is included is in the vicinity of the secondary peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' It is an interesting open question as to whether the dilute assumption may also break down in the vicinity of the secondary peak as well as the primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Once finite particle size effects become important, there are a number of different approaches that can be followed to continue the analysis, such as a sharp transition between a dilute and jammed region (Popov (2005)), using a viscosity and solute diffusivity that vary with concentration (Kaplan & Mahadevan (2015)) or through more complicated two-phase suspension models (see, for example, Guazzelli & Pouliquen (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Our analysis has concentrated on a diffusive evaporative model, while there are many situations where other evaporative models may be appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Examples include water evaporating on glass, which may more appropriately be modelled using a kinetic evaporative model (Murisic & Kondic (2011)), droplets evaporating above a bath of the same liquid, where the evaporation is effectively constant (Boulogne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2016)) and situations where a mask is used to control the evaporative flux so that it is stronger towards the centre (Vodolazskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The analysis herein could readily be extended to such situations, although we note that for evaporative fluxes with different — including non-singular — behaviour close to the contact line, the size of the boundary layer regions near the contact line in which solutal diffusion and surface tension are relevant will change accordingly, as for surface tension-dominated droplets (Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' A future direction of interest would be to extend the analysis herein to non-axisymmetric droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Such droplets occur widely in applications, particularly in printing OLED/AMOLED screens (see, for example, Mai & Richerzhagen (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Huo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' It is well-known that droplet geometry plays a strong role in the behaviour of the evaporative flux (S´aenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray & Moore (2023)) and the transient and final Gravity can lead to multiple peaks in the early stages of coffee ring formation 23 deposit profiles (Freed-Brown (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' S´aenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' It would be of significant theoretical and practical interest to explore the behaviour of the secondary peak in such problems as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Finally, we note that another context in which gravity may play an important role is that of binary/multi- component droplets, particularly in situations where the different fluids have different densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Multi- component droplets occur widely, from commercial alcohols such as whiskey and ouzo (Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Carrithers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2020)) to various inks (Shargaieva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' While it would be certainly of interest to extend the analysis presente here to such droplets, a careful treatment of the internal flow would be needed, as the multi-component nature of the droplet significantly complicates the dynamics (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Acknowledgments MRM would like to acknowledge the support of EPSRC grant EP/X035646/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Declaration of Interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The authors report no conflict of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Matyched asymptotic analysis in the limit of large Bo, α = O(1) In this appendix, we present the asymptotic solution of the solute transport problem in the limit in which Bo, Pe ≫ 1 and α = Bo−1/2Pe2/3 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 1) For convenience, we choose to use Pe−2/3 as our small parameter in the asymptotic expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moreover, it transpires that it is easier to analyse the integrated mass variable formulation of the problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='32)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Outer region In the droplet bulk, 1 − r is O(1), and we recall from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='9) that the droplet free surface h is flat to all orders and that the velocity is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Upon substituting these expressions into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='32) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='34), and then expanding M(r, t) = M0(r, t) + Pe−2/3M1(r, t) + O(Pe−4/3) as Pe → ∞, we find to leading-order ∂M0 ∂t + u0 4 ∂M0 ∂r = 0 for 0 < r, t < 1, M0(r, 0) = r2 2π for 0 < r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 2a, b) This may be solved using the method of characteristics, yielding M0(r, t) = (1 − t)r2 2π + √1 − t(1 − √1 − t) π (1 − � 1 − r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 3) We see that this solution automatically satisfies the boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='33a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' At O(Pe−2/3), the problem for M1(r, t) is given by ∂M1 ∂t + u0 4 ∂M1 ∂r = −αu1 4 ∂M0 ∂r for 0 < r, t < 1, M1(r, 0) = 0 for 0 < r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 4a, b) for 0 < r < 1, 0 < t < 1, while the initial condition is given by M1(r, 0) = αr2/π for 0 < r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This may be solved in a similar manner using the method of characteristics, yielding M1(r, t) = 2ακ(r, t) π (1 − κ(r, t)) log �√1 − t − κ(r, t) 1 − κ(r, t) � + α π (1 − (1 − κ(r, t))2), (A 5) where κ(r, t) = √1 − t(1 − √ 1 − r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Expanding the leading-order solution (A 3) as we approach the contact line, we have M0(r, t) ∼ √1 − t π − (1 − t) 2π − � 2(1 − t) π (1 − √ 1 − t) √ 1 − r − (1 − t) π (1 − r) + O((1 − r)3/2) (A 6) as r → 1−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Notably, this means that the leading-order outer solute mass m0 is singular at the contact line, which gives a strong indication of the importance of diffusive effects local to the edge of the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' This is in stark contrast to the Bo = O(1) solution, where the outer solute mass was square root bounded as r → 1−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' A similar expansion of (A 5), yields M1(r, t) ∼ α√1 − t(1 − √1 − t) π log(1 − r) + α√1 − t(1 − √1 − t) π log � 2(1 − t) (1 − √1 − t)2 � + α π (1 − (1 − √ 1 − t)2) + O( √ 1 − r log(1 − r)) (A 7) as r → 1−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We can clearly see this will necessitate an inner expansion that contains logarithmic terms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' similar behaviour is displayed for surface tension-dominated drops under different evaporative fluxes (Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 24 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray Finally, if we expand the solute mass m ∼ m0 as Pe → 0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='35), we find m0(r, t) = √1 − t π √ 1 − r2 � 1 − √ 1 − t(1 − � 1 − r2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 8) Whilst we could proceed to O(Pe−2/3) in the solute mass expansion in the outer region, we shall not require this when constructing a composite profile that is valid to O(1) throughout the droplet, so we do not present this here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Inner region Recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='11)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='12), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='3), in order to retain a balance between the advective and diffusive effects in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='32) close to the contact line, we set r = 1 − Pe−2/3˜r, u = Pe−1/3˜u, h = ˜h, M = ˜ M, m = Pe2/3 ˜m (A 9) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='32)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Note that we therefore have ˜h = ˜h0 + Pe−2/3˜h1 + O(Pe−4/3), ˜u = ˜u0 + Pe−1/3˜u1 + Pe−2/3˜u2 + O(Pe−1) (A 10) as Pe → ∞ where ˜h0(˜r, t) = ¯h0(˜r/α, t), ˜h1(˜r, t) = α¯h1(˜r/α, t), (A 11) and ¯h0, ¯h1 are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='13)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='14), and ˜u0(˜r, t) = √α¯u0(˜r/α, t), ˜u1(˜r, t) = α¯u1(˜r/α, t), ˜u2(˜r, t) = α3/2 ¯u2(˜r/α, t) (A 12) and ¯u0, ¯u1, ¯u2 are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='15))–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Seeking an asymptotic expansion of the integrated mass of the form ˜ M = ˜ M0+Pe−1/3 ˜ M1+Pe−2/3 log Pe−2/3 ˜ M2+ Pe−2/3 ˜ M3 + o(Pe−2/3) as Pe → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' we find that the leading-order inner problem is given by ∂2 ˜ M0 ∂˜r2 + � ˜u0 − 1 ˜h0 ∂˜h0 ∂˜r � ∂ ˜ M0 ∂r = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' for ˜r > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 0 < t < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 13) subject to the boundary condition ˜ M0(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) = 1/2π for 0 < t < 1 and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' in order to match with the local expansion of leading-order-outer solution at the contact line (A 6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' we must have ˜ M0 → √1 − t π − (1 − t) 2π as ˜r → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 14) Defining the integrating factor I(˜r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) = � 1 1 − e−˜r/α � exp � 2 √ 2 (1 − t) � ˜r 0 √ξ 1 − e−ξ/α dξ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 15) we find that the solution is given by ˜ M0(˜r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) = 1 2π + B0(t) � ˜r 0 1 I(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 16) where B0(t) = − 1 π � 1 − √ 1 − t − t 2 � �� ∞ 0 1 I(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) ds �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 17) We note here that the first term on the right-hand side of B0(t) is simply the leading-order accumulated mass at the contact line as a function of time, N(t), that is N(t) = 1 4 � t 0 (m0u0)(1−, τ) dτ = 1 π � 1 − √ 1 − t − t 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 18) It is worth noting the similarities between (A 18) and the equivalent expression for a surface-tension domi- nated drop evaporating under a constant evaporative flux (Freed-Brown 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' At O(Pe−1/3), we have ∂2 ˜ M1 ∂˜r2 + � ˜u0 − 1 ˜h0 ∂˜h0 ∂˜r � ∂ ˜ M1 ∂r = 4∂ ˜ M0 ∂t − ˜u1 ∂ ˜ M0 ∂˜r for ˜r > 0, 0 < t < 1, (A 19) Gravity can lead to multiple peaks in the early stages of coffee ring formation 25 subject to ˜ M1(0, t) = 0 for 0 < t < 1 and the far-field matching condition ˜ M1 → − � 2(1 − t) π (1 − √ 1 − t) √ ˜r as ˜r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 20) While in practice it may be easier to find ˜ M1(˜r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) from (A 19)–(A 20) numerically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' for posterity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' we state that this boundary value problem has solution ˜ M1(˜r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) = � ˜r 0 1 I(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) �� s 0 � 4∂ ˜ M0 ∂t − ˜u1 ∂ ˜ M0 ∂˜r � I(σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) dσ � ds + B1(t) � ˜r 0 1 I(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 21) where B1(t) = − �� ∞ 0 � 1 I(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) �� s 0 � 4∂ ˜ M0 ∂t − ˜u1 ∂ ˜ M0 ∂˜r � I(σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) dσ � − √ 2(1 − t)∂ ˜ M0 ∂t 1 √s � ds � �� ∞ 0 1 I(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) ds �−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 22) is chosen to kill the O(1)-term in the far-field expansion of ˜ M1(˜r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The O(Pe−2/3 log Pe−2/3)-problem is given by ∂2 ˜ M2 ∂˜r2 + � ˜u0 − 1 ˜h0 ∂˜h0 ∂˜r � ∂ ˜ M2 ∂r = 0 for ˜r > 0, 0 < t < 1, (A 23) subject to ˜ M2(0, t) = 0 for 0 < t < 1 and the far-field matching condition ˜ M2 → α√1 − t(1 − √1 − t) π as ˜r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 24) The solution may be found in a similar manner to the leading-order problem, yielding ˜ M2(˜r, t) = B2(t) � ˜r 0 1 I(s, t) ds, (A 25) where B2(t) = α√1 − t(1 − √1 − t) π �� ∞ 0 1 I(s, t) ds �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 26) Lastly, at O(Pe−2/3), we have ∂2 ˜ M3 ∂˜r2 + � ˜u0 − 1 ˜h0 ∂˜h0 ∂˜r � ∂ ˜ M3 ∂r = 4∂ ˜ M1 ∂t −˜u1 ∂ ˜ M1 ∂˜r −˜u2 ∂ ˜ M0 ∂˜r − 1 ˜h0 �˜h1 ˜h0 ∂˜h0 ∂˜r − ∂˜h1 ∂˜r � ∂ ˜ M0 ∂˜r − ∂ ˜ M0 ∂˜r =: V(˜r, t) (A 27) for ˜r > 0, 0 < t < 1, subject to ˜ M3(0, t) = 0 for 0 < t < 1 and the far-field condition ˜ M3 → −(1 − t) π ˜r+ �α√1 − t(1 − √1 − t) π � � log ˜r + log � 2(1 − t) (1 − √1 − t)2 �� +α π (1−(1− √ 1 − t)2) as ˜r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 28) The solution is given by ˜ M3(˜r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) = � ˜r 0 1 I(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) �� s 0 V(σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t)I(σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) dσ � ds + B3(t) � ˜r 0 1 I(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 29) where B3(t) = � − � ∞ 1 � 1 I(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) �� s 0 V(σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t)I(σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) dσ � + (1 − t) π − α√1 − t(1 − √1 − t) πs � ds − � 1 0 1 I(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) �� s 0 V(σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t)I(σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) dσ � ds − (1 − t) π + α π (1 − (1 − √ 1 − t)2) +α√1 − t(1 − √1 − t) π log � 2(1 − t) (1 − √1 − t)2 �� �� ∞ 0 1 I(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) ds �−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 30) has been chosen to satisfy the correct far-field behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We are now in a position to find the inner solution for the solute mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' By substituting the scalings (A 9) 26 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Moore & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Wray into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='35), we see that ˜m = − 1 1 − Pe−2/3˜r ∂ ˜ M ∂˜r , (A 31) so that expanding ˜m = ˜m0 + Pe−1/3 ˜m1 + Pe−2/3 log Pe−2/3 ˜m1 + Pe−2/3 ˜m2 as Pe → ∞, we have ˜m0 = −∂ ˜ M0 ∂˜r , ˜m1 = −∂ ˜ M1 ∂˜r , ˜m2 = −∂ ˜ M2 ∂˜r , ˜m3 = −∂ ˜ M3 ∂˜r − ˜r∂ ˜ M0 ∂˜r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 32) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Composite solutions We now have all of the necessary components needed to construct (additive) composite solutions for com- parison to the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' To construct a composite solution for the integrated mass variable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' we combine the first two outer solutions (A 3) and (A 5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' the first four inner solutions (A 16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 21),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 25) and (A 29),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' the overlap terms given by (A 6)–(A 7) using Van Dyke’s matching rule Van Dyke (1964),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' which yields Mcomp(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) = M0(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) + Pe−2/3M1(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t) + ˜ M0 � Pe2/3(1 − r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t � + Pe−1/3 ˜ M1 � Pe2/3(1 − r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t � + Pe−2/3 log Pe−2/3 ˜ M2 � Pe2/3(1 − r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t � + Pe−2/3 ˜ M3 � Pe2/3(1 − r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' t � − �√1 − t π − (1 − t) 2π − � 2(1 − t) π (1 − √ 1 − t) √ 1 − r − (1 − t) π (1 − r)+ Pe−2/3 �α π (1 − (1 − √ 1 − t)2) + α√1 − t(1 − √1 − t) π � log(1 − r) + log � 2(1 − t) (1 − √1 − t)2 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 33) This composite solution is valid up to and including O(Pe−2/3) throughout the whole of the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Similarly, for the solute mass, the equivalent composite profile is compiled by taking the first outer solution (A 8) as well as the first four inner solutions given by (A 32), so that, accounting for the overlap contributions, mcomp(r, t) = m0(r, t) + Pe2/3 ˜m0 � Pe2/3(1 − r), t � + Pe1/3 ˜m1 � Pe2/3(1 − r), t � + log Pe−2/3 ˜m2 � Pe2/3(1 − r), t � + ˜m3 � Pe2/3(1 − r), t � − � (1 − t)(1 − √1 − t) √ 2π√1 − r − (1 − t) π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (A 34) We note that this composite solution is valid up to and including O(1) throughout the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Numerical solution of the solute transport problem In this section, we outline the numerical scheme for solving the advection-diffusion problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='32)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content='34) for the integrated mass variable M(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' As discussed previously, the integrated mass variable formulation is advantageous when solving numerically, since it is mass-preserving and has simple-to-implement Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Our numerical method is an adaptation of that discussed in Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' (2021) for the Bo = 0 regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We utilize central differences with gridpoints clustered close to the contact line, where there are rapid changes in behaviour associated with the coffee ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' We choose a uniform grid in the variable ζ ∈ [0, 1], where r = 1 − ℓζ 1 − ℓ , (B 1) and ℓ is taken to coincide with the smallest of the two boundary layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' that is, ℓ = κ(1 − tc) where κ = min � Bo−1/2, Pe−2/3� and tc is the final computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Note that these boundary layers are in the context of large Bond number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' when Bo = O(1), we have both increased the number of nodes in the discretization and chosen ℓ = Pe−2 to ensure we capture the diffusive boundary layer in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Even when it is present, the secondary peak does not exhibit such extreme behaviour, with a much shallower profile than the primary peak, so provided that the discretization is chosen suitably small, the secondary peak is captured well without special considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' The resulting system is solved using ode15s in MATLAB and incorporates complex step differentiation to compute the Jacobian (Shampine (2007)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' Gravity can lead to multiple peaks in the early stages of coffee ring formation 27 The veracity of the simulations has been confirmed with stringent convergent checks alongside the excellent comparisons to the asymptotic results in both the order unity Bond number regime and the large Bond number regime (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' figures 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FKT4oBgHgl3EQfoC5E/content/2301.11864v1.pdf'} +page_content=' REFERENCES Barash, L Yu, Bigioni, TP, Vinokur, VM & Shchur, LN 2009 Evaporation and fluid dynamics of a sessile drop of capillary size.' metadata={'source': 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a/YNE2T4oBgHgl3EQfvAgF/content/tmp_files/2301.04085v1.pdf.txt b/YNE2T4oBgHgl3EQfvAgF/content/tmp_files/2301.04085v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b511351ef376ff6c149bb226cd0fb8bd4444755e --- /dev/null +++ b/YNE2T4oBgHgl3EQfvAgF/content/tmp_files/2301.04085v1.pdf.txt @@ -0,0 +1,1609 @@ +MNRAS 000, 1–9 (2022) +Preprint 11 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Propagating photo-z uncertainties: a functional derivative approach +Robert Reischke⋆1 +1 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), +German Centre for Cosmological Lensing, 44780 Bochum, Germany +11 January 2023 +ABSTRACT +Photometric redshifts are a key ingredient in the analysis and interpretation of large-scale +structure (LSS) surveys. The accuracy and precision of these redshifts estimates is directly +linked to the constraining power of photometric surveys. It is hence necessary to define preci- +sion and accuracy requirements for the redshift calibration to not to infer biased results in the +final analysis. For weak gravitational lensing of the LSS the photometry culminates in the es- +timation of the source redshift distribution (SRD) in each of the tomographic bins used in the +analysis. The focus has been on shifts of the mean of the SRDs and how well the calibration +must be able to recover those. Since the estimated SRDs are usually given as a normalized +histogram with corresponding errors, it would be advantageous to propagate these uncertain- +ties accordingly to see whether the requirements of the given survey are indeed fulfilled. Here +we propose the use of functional derivatives to calculate the sensitivity of the final observ- +ables, e.g. the lensing angular power spectrum, with respect to the SRD at a specific redshift. +This allows the propagation of arbitrarily shaped small perturbations to the SRD, without hav- +ing to run the whole analysis pipeline for each realization again. We apply our method to a +EUCLID survey and demonstrate it with SRDs of the KV450 data set, recovering previous +results. Lastly, we note that for cosmic shear moments of order larger than two will probably +be not relevant when propagating redshift uncertainties. +Key words: cosmology: theory, large-scale structure of Universe, surveys, galaxies: photom- +etry +1 +INTRODUCTION +Cosmic shear, the weak gravitational lensing effect imprinted on +distant galaxies by the large-scale structure (LSS), is one of the pri- +mal science goals for EUCLID and Rubin-LSST. The blueprint for +these missions has been set by current stage-3 surveys, including +the Kilo-Degree Survey (Kuijken et al. 2019; Asgari et al. 2021, +KiDS), the Dark Energy Survey (Abbott et al. 2018; Gatti et al. +2022, DES) or the Subaru Hyper Suprime-Cam (Hamana et al. +2020, HST), yielding tight constraints ob the matter distribution +in the late Universe. +The cosmic shear signal is estimated by measuring the coher- +ent distortion of background galaxies. Since the intrinsic elliptic- +ity of galaxies is much larger than the lensing effect, millions of +galaxies are required to measure a significant signal. This casts a +complete spectroscopic survey unfeasible. Hence, one has to rely +on redshift estimates from photometry. In order to interpret the ob- +served ellipticity correlations, the potometric redshifts have to be +calibrated. There are different approaches for the calibration pro- +cedure on the market. These include the calibration with a spec- +troscopic reference sample (possibly with re-weighting) (e.g. Lima +et al. 2008; Newman 2008; Matthews & Newman 2010; Masters +⋆ E-mail: reischke@astro.ruhr-uni-bochum.de +et al. 2015; Bonnett et al. 2016; McLeod et al. 2017; Hildebrandt +et al. 2020; Wright et al. 2020; Myles et al. 2021), using photome- +try measurements in conjunction with clustering measurements of +tracer populations (e.g. Sánchez & Bernstein 2019; van den Busch +et al. 2020; Alarcon et al. 2020) and self-organising maps (Wright +et al. 2020). It is also possible to partially self-calibrate the pho- +tometric redshifts in weak lensing data (e.g. Schaan et al. 2020). +In order to account for general shapes of the source-redshift dis- +tributions (SRDs) different mixture models have been employed +(see for example Rau et al. 2020). These Gaussian processes are +non-parametric, but they are by definition non-linear, which makes +their implementation in cosmology pipelines in general very diffi- +cult. Stölzner et al. (2021) used linear fit parameters to circumvent +this problem to self-calibrate the data, as it can be implemented in +existing pipelines very easily. +Currently it is best practice to propagate the redshift uncer- +tainty in the SRDs by introducing shift parameters in the mean of +the distribution (Hildebrandt et al. 2021; Hikage et al. 2019; Abbott +et al. 2022). As the sensitivity of surveys rises, however, the re- +quirements on the SRD uncertainties become larger as well. There- +fore, the contributions from higher order cumulants of the SRD be- +come important. As discussed above, previous works have focused +on Gaussian mixture models to self-calibrate the cosmic shear mea- +surement. In this paper we investigate the general sensitivity of the +© 2022 The Authors +arXiv:2301.04085v1 [astro-ph.CO] 10 Jan 2023 + +2 +Reischke +lensing power spectrum to perturbations in the SRD. In particu- +lar we are calculating the functional derivative of the cosmic shear +angular power spectrum with respect to the SRD at a particular +co-moving distance. This can then be mapped to a total error in +the cosmic shear power spectrum if a perturbation in the SRD in +a co-moving interval is applied. We take the constraint of the nor- +malisation of the SRD into account when calculating the functional +derivative. Therefore we can propagate arbitrary perturbations to +the SRDs (subject to some underlying covariance) and propagate +them into the Cℓ of cosmic shear. This allows us to estimate the dif- +ference in χ2 induced by the uncertainty in the SRD, without hav- +ing to run thousands of realizations of the analysis pipeline used. +By using a Fisher matrix for the cosmological parameters, this ∆χ2 +can then be mapped to potential biases in cosmological parameters. +Here we studied a rather idealised scenario by working in Fourier +space, assuming a Gaussian likelihood and ignoring intrinsic align- +ments. The method, however, easily generalises and including these +effects is straightforward. +We structure the paper as follows: In Section 2 we briefly +review cosmic shear basics and introduce the methodology used +by calculating the functional derivative of the weak lensing angu- +lar power spectrum. The results are presented in Section 3, where +we apply the procedure to a survey with EUCLIDs specifications +and to KiDS-VIKING-450 (KV450). We conclude in Section 4. +In the appendices we also investigate the possibility of an Edge- +worth expansion of the SRD (Appendix A), discuss photometric +galaxy clustering (Appendix B), the distribution of the mean and +standard deviation of the SRD in Appendix C, the general relation- +ship to observables (Appendix D), the functional derivative of the +non-Limber projection in Appendix E and inrinsic alignments (Ap- +pendix F). +2 +METHODOLOGY +In this section we present the basic methodology of our analysis. +In particular we describe the basics of cosmic shear and derive the +function derivative of the lensing angular power spectrum with re- +spect to the SRDs. +2.1 +Cosmic shear basics +The equation for the cosmic shear power spectrum in tomographic +bins i and j in the Limber proejction is (Limber 1954; Loverde & +Afshordi 2008) +C +κiκj +ℓ += +� χH +0 +dχ +χ2 W(i) +κ (χ)W(j) +κ (χ)Pδ +�ℓ + 0.5 +χ +, χ +� +, +(1) +where Pδ is the matter power spectrum, for which we use the em- +ulated spectrum from Mead et al. (2015). W(i) +κ (χ) is the lensing +weight of the i-th tomographic bin as given by: +W(i) +κ (χ) = 3Ωm0 +2χ2 +H +χ +a(χ) +� χH +χ +dχ′n(i) +s (χ′)χ′ − χ +χ′ +. +(2) +Here χ is the co-moving distance, a the scale factor, Ωm0 the matter +density parameter today, χH the Hubble radius and n(i) +s is the SRD +in the i-th tomographic bin which builds on photo-z measurements +and its calibration. It is normalized in each tomographic bin such +that +� +dz n(i) +s (z) = 1 = +� +dχ n(i) +s (z(χ)) dz +dχ ≡ +� +dχ n(i) +s (χ) . +(3) +Since photo-z is just an estimate of the true redshift, the estimated +source-redshift distribution, n(i) +s , is not exactly known. Here we in- +vestigate two approaches: +i) Use functional derivatives to evaluate the change of the lens- +ing power spectrum when perturbing the n(i) +s at different redshifts. +Given specific survey settings and precision goals, limits on the al- +lowed change of the n(i) +s can be determined, which in turn can be +mapped to changes in the cumulants or moments of the underlying +distribution (see Section 2.2). +ii) We expand the underlying source-redshift distribution in an +asymptotic Edgeworth series and investigate the requirements on +the cumulants directly in a Fisher analysis. The second approach is +not feasible for realistic SRDs (see Appendix A). +2.2 +Functional derivative of the lensing power spectrum +Here we wish to investigate the sensitivity of the weak lensing +power spectrum to the full shape of the source-redshift distribution +using functional derivatives. In particular we start by perturbing +n(i) +s (χ(z)) at a certain redshift z0, such that χ0 = χ(z0). The corre- +sponding perturbed lensing weight is thus +∆W(i) +κ (χ, χ0) = δW(i) +κ (χ) +δn(i) +s (χ0) +∆n(i) +s (χ0) . +(4) +This expression evaluates, how the lensing weight changes if the +source-redshift distribution is perturbed by an amount ∆n(i) +s at the +co-moving distance χ0 corresponding to the redshift z0. +Ultimately, we are interested in the change of the lensing +power spectrum, Equation (B1). First, by applying the Leibniz rule +δC(ij) +ℓ +δn(a)(χ0) = +� +dx δC(ij) +ℓ +δW(a)(x) +δW(a)(x) +δn(a)(χ0) += +� +dx δW(a)(x) +δn(a)(χ0) +Pδ +� +ℓ+0.5 +x , x +� +x2 +� +W(j)(x)δD +ia + W(i)(x)δD +ja +� +, +(5) +The missing ingredient is the functional derivative of the lensing +kernel, for which we find +δW(i)(x) +δn(j) +s (χ0) += 3Ωm0 +2χ2 +H +x +a(x) +χ0 − x +χ0 +δD +ijΘ(χ0 − x) . +(6) +Θ(x) is the Heaviside function to ensure that the functional deriva- +tive vanishes if the SRD is perturbed outside the integration bounds. +Using Equation (4) and Equation (5) we can write the change in +angular power spectrum ∆C(ij) +ℓ (χ′) due to a change in the source- +redshift distribution at co-moving distance χ0 as +∆C(ij) +ℓ,a (χ0) ≡ +δC(ij) +ℓ +δn(a)(χ0)∆n(a)(χ0) += 3Ωm0 +2χ2 +H +∆n(χ0) +� +dx +a(x)x +χ0 − x +χ0 +Pδ +�ℓ + 0.5 +x +, x +� +× +� +W(j)(x)δD +ia + W(i)(x)δD +ja +� +. +(7) +Integrating the perturbed lensing spectrum then gives the total per- +turbation: +∆C(ij) +ℓ,a ≡ +� +dχ0∆C(ij) +ℓ,a (χ0) . +(8) +So far we have treated the function n(i)(z) as being completely free. +However, the functional derivative needs to respect the constraint +MNRAS 000, 1–9 (2022) + +functional photo-z +3 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +redshift z +1 +2 +3 +4 +n(i) +s (z) +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +tomographic bin index +Figure 1. Allowed perturbation for EUCLID to the SRD of the ten tomo- +graphic source bins. Solid lines show the fiducial SRD, while the bands +show the allowed perturbation to it. +given in Equation (3), thus limiting the possible variations of n(i)(z). +The normalization condition itself is again a function and we write +N[n(i) +s ] � 1 − +� +dz n(i) +s (z) = 0 , +(9) +this constraint can be implemented by first defining +n(i) +s (z) � +f(z) +� +dx′ f(x′) +(10) +which will be normalized by construction. n(i) +s (z) is a functional of +f and we can now evaluate the functional derivative of C[n[ f]] as +an unconstrained derivative but evaluated at f = n. To avoid clutter +we ignore the sub- and superscripts in this part +�δC[n[ f]] +δf(x) +� ������ f=n += +� +dx′ δC[n] +δn(x′) +δn(x′) +δ f(x) +������ f=n +. +(11) +With +δn(x′) +δf(x) = δD(x′ − x) +� +dy f(y) +− +f(x′) +�� +dy f(y) +�2 , +(12) +one finds +δC[n] +δ1n(x) ≡ +�δC[n[f]] +δf(x) +� ������ f=n += δC[n] +δn(x) − +� +dy δC[n] +δn(y) n(y) , +(13) +where we denote that we want to keep the normalization fixed by +the variation δ1. This is a very intuitive expression: the first term +evaluates the standard functional derivative, while the second term +corrects this variation to respect the normalization. +2.3 +Fisher forecast +The next step is to set some requirement on the lensing power spec- +tra. Here we will look at the difference in the χ2, assuming a Gaus- +sian likelihood and thus setting a lower limit on the required accu- +racy of n(i) +s (z). For modes aℓm with zero mean and covariance Cℓ, +the ∆χ2 between multipoles ℓmin and ℓmax can be written as +∆χ2(ℓmin, ℓmax) = fsky +ℓmax +� +ℓ=ℓmin +2ℓ + 1 +2 +tr +� +∆CℓC−1 +ℓ ∆CℓC−1 +ℓ +� +, +(14) +1 +2 +3 +4 +5 +6 +7 +8 +9 +order of central moment n +10−2 +100 +102 +104 +106 +108 +relative change in % +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +tomographic bin index +Figure 2. Allowed relative change in per-cent of the central moment of the +SRD in each tomographic bin. The changes are calculated from the per- +turbed SRD distributions as shown in Figure 1. +note that Cℓ is the matrix with the components C(ij). The factor fsky +takes into account the observed sky fraction. Using Equation (8) we +rewrite the previous equation as a Riemann sum +∆χ2(ℓmin, ℓmax) = fsky +ℓmax +� +ℓ=ℓmin +2ℓ + 1 +2 +× +� +r,s,i,j +tr +� +δCℓ +δ1n(i)(χr)C−1 +ℓ +δCℓ +δ1n(j)(χs)C−1 +ℓ +� +× DχrDχs∆n(i)(χr)∆n(j)(χs) , +(15) +with the measure Dχr. If we define the Fisher matrix in this case +as: +Fαβ = fsky +ℓmax +� +ℓ=ℓmin +2ℓ + 1 +2 +tr +� δCℓ +δ1nα +C−1 +ℓ +δCℓ +δ1nβ +C−1 +ℓ +� +Dχr(α)Dχs(β) , +(16) +where we labeled n(i)(χr) → nα, we recover for a difference in χ2 +using a scalar product on the finite dimensional Hilbert space of +shifts in the redshift distribution where the Fisher matrix acts as a +norm-inducing metric +∆χ2 = F(∆n, ∆n) ≡ ∆nT F∆n , +(17) +where ∆n is the vector containing shifts of the components nα. +The Fisher matrix, Equation (16), describes, how well the +shifts nα can be determined by a measurement of the angular power +spectra Cα given certain survey settings. Clearly, if one would try to +measure all possible perturbations, neighbouring δn(χ) are strongly +correlated. This is, however not the question we would like to ask +in this work. Instead, we want to look at the situation that we allow +any perturbation ∆n, irrespective of the correlation. Therefore, by +turning this argument around, we only use the diagonal part of the +Fisher matrix. +Lastly one should note that the functional derivative is strictly +defined as a limiting process for infinitesimally small perturbation +to the function at hand. The relation in general can be non-linear, +but as long as relative perturbations to the function are small with +respect to unity, these non-linear contributions are sub-dominant. +Especially for surveys with tight requirements on the SRDs this is +essentially always fulfilled. +MNRAS 000, 1–9 (2022) + +4 +Reischke +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +redshift z +1 +2 +3 +4 +5 +6 +n(i) +s (z) +KV450 +1 +2 +3 +4 +5 +tomographic bin index +Figure 3. Allowed perturbation for KV450 to the SRD for the 5 tomo- +graphic source bins. Solid lines show the fiducial SRD, while the bands +show the allowed perturbation to it. +3 +RESULTS +3.1 +Allowed Perturbations to the Source Redshift +Distribution +First we will look at the allowed perturbations to the SRD by as- +suming allowing for a total ∆χ2 of unity, corresponding to a one +σ shift of a linear model parameter. Clearly, there are many differ- +ent solutions ∆n that satisfy ∆χ2 = 1 subject to Equation (17). To +show the structure of the Fisher matrix we therefore distribute the +allowed ∆χ2 per ∆nα equally. +We will assume EUCLID specifications for the survey as +given in Blanchard et al. (2020) and assume ntomo = 10 tomo- +graphic bins, a sky fraction of 0.3. Furthermore, we will collect +multipoles between ℓmin = 10 and ℓmax = 3000. We then calculate +the diagonal Fisher matrix from Equation (16) and distribute the er- +rors equally as described above. This results into a possible realisa- +tion of ∆n yielding ∆χ2 = 1 subject to the constraint Equation (3). +Figure 1 shows the resulting perturbed SRDs. The solid lines show +the fiducial SRD, while the shaded areas show the allowed pertur- +bations to not cause a bias of more than 1 σ for a linear model pa- +rameter. Lastly, the tomographic bin index is shown as a colour-bar. +The general trend is very clear, the allowed perturbations become +very large around a small interval ∆χ around the mean of the distri- +butions. For most tomographic bins this coincides with the peak of +the distribution as they are very close to Gaussian. Only for the first +and the last bin these spikes are a bit offset since the distributions +are a bit more asymmetric. This already confirms that the most im- +portant part about the SRDs in cosmic shear measurements is to +calibrate the mean redshift of each tomographic bin very well. Fur- +thermore, we observe that the spikes tend to be narrower at higher +redshifts, indicating that the uncertainty on the mean of the SRD +is more important at higher redshifts. We want to stress again, that +this is just one realization of ∆n that produces a ∆χ2 = 1, but by +distributing the errors equally, it is possible to see, which pertur- +bations the final measurement is most sensitive to. However, the +uncertainties should not be used at literally value and are extreme +values, they just give a general trend. +Next, we use the perturbed SRDs to calculate their central mo- +ments µn: +µn � E[(X − E[X])n] = +� +p(x)(x − µ)ndx , +(18) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +∆χ2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +KV450 +68th +50th +95th percentile +Figure 4. ∆χ2 for 106 realisations of ∆n from the CKV450 +n(χ) +. We also show +the 50, 68 and 95 percentiles. +for a probability distribution function p(x) with mean µ. The per- +turbed SRDs are used to calculate the change in the central mo- +ments relative to the fiducial SRD. Figure 2 shows the resulting +relative change for all tomographic bins as a function of the order +of the central moment. Clearly, the first moment is most important +and while the second one still needs to be known at a 10% level, +all higher order moments are essentially unimportant. This is of +course reminiscent of the behaviour observed in Figure 1, where +the perturbations are such that they essentially fix the mean. It is of +course entirely possible, that we alter the shape of the distribution +in a different way but still achieve the desired accuracy. +Nonetheless, the results show that for the SRD for cosmic +shear only the mean redshift and the width are important with the +former influencing the result way stronger (by more than an order +of magnitude). In Appendix C we sample from the allowed changes +in the SRD and show the relative difference of the first two mo- +ments to illustrate their scatter. +3.2 +Propagating Redshift Errors +In this section we will revisit the KV450 data for the SRD (Hilde- +brandt et al. 2020). This data set is used since it includes a covari- +ance matrix from the direct calibration (DIR). For the clustering +redshifts (van den Busch et al. 2020) or the self-organising maps +(Wright et al. 2020) no bootstrap covariance was estimated so far. +For completeness the allowed perturbations are shown in Fig- +ure 3. Due to the lower signal-to-noise ratio of the measurement, +the allowed perturbations are much larger than in the previous case. +The features, however, are very similar. +Since we are expressing everything in co-moving distance, the +covariance matrix needs to be transformed accordingly. Let CKV450 +n(z) +be the covariance matrix in n(z) space, the transformed covariance +is then +CKV450 +n(χ) += JTCKV450 +n(z) +J , +(19) +where J is the Jacobian with components Ji +j = δi +jdz/dχ. Alterna- +tively, the Fisher matrix of the SRD perturbations can be expressed +in redshift space by the inverse transform. +Perturbations ∆n are now sampled from CKV450 +n(χ) +and propa- +gated to obtain ∆χ2 according to Equation (14). If the redshift errors +as given in CKV450 +n(χ) +are sufficiently small to not produce a significant +bias in the cosmological parameters such as S 8 we expect most +MNRAS 000, 1–9 (2022) + +functional photo-z +5 +0.200 +0.225 +0.250 +0.275 +0.300 +0.325 +0.350 +0.375 +0.400 +Ωm0 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +σ8 +KV450 +1 +2 +3 +4 +5 +6 +∆χ2 +Figure 5. The black histogram shows the induced shifts by the photo-z un- +certainty in the Ωm0-σ8-plane, derived from the ∆χ2 of Figure 4. In red +we show the contour from the Fisher matrix for KV450 enclosing the 1σ +confidence interval. +realisations (i.e. 68% Hildebrandt et al. 2020) to yield ∆χ2 < 1. +Figure 4 shows the resulting distribution in ∆χ2 for the 106 real- +izations of ∆n for KV450. The vertical dashed lines show the 50th, +68th and 95th percentile. It is clear from this plot that the precision +of the SRD used in KV450 is high enough to not yield any spurious +detection in the final parameter constraints since the 68th percentile +is still well below unity. +One could now further propagate these uncertainties into cos- +mological parameters using the corresponding Fisher matrix. For a +given shift in the SRD ∆n, the corresponding shifts in the cosmo- +logical parameters, ∆θ can be calculated: +∆θi = −(F−1)i +αFα +β∆nβ , +(20) +where Greek indices run over the perturbations in the SRD, while +Latin indices label cosmological parameters. Here we assumed the +sum convention. Fi +α hence is the mixed pseudo Fisher matrix: +Fi +α = −E +�∂ ln L +∂θi +δ ln L +δnα Dχr(α) +� +(21) +and it’s inverse is a pseudo inverse. Since the inversion of this ma- +trix is not necessarily stable we choose to go another route here. +Since the distribution of ∆χ2 is known, we are interested in sam- +ples of cosmological parameters with the same ∆χ2 with respect +to the best fit value. For a Gaussian posterior in one dimension +this would amount to a distribution such that the absolute value of +each sample is fixed to +√ +∆θ2. We sample from a standard Gaussian +distribution and modify its width by +√ +∆θ2. This Gaussian is then +mapped into the frame of the cosmological parameters under con- +sideration via the Cholesky decomposition of the Fisher matrix of +the cosmological parameters. In Figure 5 we apply this procedure +to the ∆χ2 distribution of KV450 (Figure 4). Each dot represents +one sample of the ∆χ2 distribution with its value shown as a colour +bar. It can be seen as the geodesic distance to the fiducial value +for the cosmological parameters in the parameter manifold (Giesel +et al. 2021). The red contours depict the expected 1, 2, 3σ confi- +dence regions from the Fisher forecast for KV450. Since in the +original analysis more than the two parameters here where used, +we re-scale the ∆χ2 accordingly, in particular by the χ2 quantile +function χ2 +k(p), where k = 10 is the number of parameters in the +actual analysis analysis (Hildebrandt et al. 2017) and p = 0.68. +This is done in order to obtain a fair comparison. It is clear from +0.794 +0.796 +0.798 +0.800 +0.802 +0.804 +0.806 +S8 +0 +25 +50 +75 +100 +125 +150 +175 +200 +KV450 +Figure 6. Induced scatter on the S 8 = σ8 +√Ωm0/0.3 parameter. This is +directly derived from the samples of Figure 5. The scatter is roughly 15 +per-cent of the statistical error budget reported in Hildebrandt et al. (2017, +2020). +the plot, that all samples for the photometric redshift distribution lie +well within the 1σ contour. Furthermore, it should be noted that we +are considering a very idealised forecast with two free parameters +and no systematics here. The procedure, however, can be general- +ized to any number of parameters. Furthermore, one can apply the +same analysis to a full Monte-Carlo-Markov-Chain (MCMC) by +matching those samples which are ∆χ2 away from the maximum +likelihood of the MCMC. Lastly, the samples from Figure 5 can be +mapped to S 8 = σ8 +√Ωm0/0.3. Figure 6 shows the resulting his- +togram of the scatter due to the photo-z uncertainties. Comparing +this to ∆S 8 = 0.076 at 68% confidence (Hildebrandt et al. 2020) +shows that the scatter induced by the redshift uncertainties as sam- +pled from the KV450 SRD covariance have a small effect on the +overall error budget. In Hildebrandt et al. (2017) a Fisher matrix +method for the shifts of the mean of the SRDs was investigated as +a a source of systematics, which found similar results to the once +presented here. The main difference between the two methods is +that we allow for general perturbations to the redshift distribution +(provided there correlation is given). Generalizing the procedure +in Hildebrandt et al. (2017) to moments higher than the variance +is bound to fail (see Appendix A). However, we would also con- +clude that even for EUCLID, the analysis of the first two moments +is probably sufficient. +In appendix C the mean and standard deviation of each SRD +in the five tomographic bins are shown for the realisations used in +this section as sampled from the DIR covariance matrix. Figure C1 +shows a very similar behaviour to what we found in fig. 2. In par- +ticular this is that the mean scatters less at higher redshifts, while +the standard deviation scatters roughly equally for most of the bins. +We close the section with a general discussion about the usage +of ∆χ2 or directly uncertainties in the parameters. It is in general +advantageous to make accuracy assessments for the SRD using the +∆χ2 and not by inverting the Fisher matrix for the parameters of +interests to obtain the shift values for those. The reason for this is +that ∆χ2 is an invariant quantity, while shifts in parameter space are +dependent on the specific model choice. The only caveat in the ∆χ2 +is that the number of parameter must be taken into account, this is, +however, much easier than calculating the Fisher matrix. +MNRAS 000, 1–9 (2022) + +6 +Reischke +4 +CONCLUSIONS +In this paper we have analysed the dependence of the cosmic shear +angular power spectrum on the SRD. This has be done by employ- +ing functional derivatives of the cosmic shear Cℓ with respect to +the SRD at a fixed co-moving distance χ0. By integrating over the +introduced error we estimated the ∆χ2 introduced by arbitrary un- +certainties in the SRD. We applied our method to a cosmic shear +survey with EUCLID specifications and KV450 since a covariance +of the SRD estimate was given. Our main findings can be sum- +marised as follows: +(i) Allowed perturbations of the SRD are such that they preserve +the mean of the underlying distribution. If they do, they can be +rather larger, even for a survey like EUCLID. This is in line with +the common practice of using only shifted means of the underlying +redshift distribution. +(ii) In order to achieve the accuracy required for EUCLID, the +mean of the redshift distribution needs to be determined between +1 and 0.01 per-cent, depending on the tomographic bin under con- +sideration. The variance of the SRD is still important at the 10 per- +cent level. There is still some sensitivity left in the skewness, but +all other moments are not relevant. +(iii) We performed a simplistic analysis of the KV450 SRDs to +check whether they fulfill the requirements and found that the un- +certainties, in this very idealised scenario, only yield biases up to +1σ in the final constraints. In a full analysis, this bias would be even +smaller. Thus confirming the redshift calibration used in KV450. +(iv) Even for EUCLID it is most likely not necessary to inves- +tigate moments of the redshift distribution n > 2. This conclusion +could change for different settings and self-calibration methods. +(v) The procedure outlined here has the advantage to be very +cheap computationally, since the functional derivatives only need +to be computed once. It is then only a matter of sampling from the +underlying SRD and to propagate these perturbations with the pre- +viously calculated functional derivative. It is hence not necessary +to push thousands of realisations of the SRD through the analysis +pipeline. +The method outlined here can thus be used to analyse whether a +perturbation in the SRD still fulfills the requirements of a given +experiment so that no biases of model parameters are introduced. It +allows for arbitrary perturbations to the SRD without requiring a fit +to the actual distribution. We intend to apply the presented method +to the updated SRDs of KiDS in the future. +For the interested reader the appendices Appendix A - Ap- +pendix E discuss various aspects of the analysis which could be +refined in future work. In particular we look at the Edgeworth ex- +pansion of the SRD in Appendix A, i.e. an expansion in the cu- +mulants of the underlying SRDs. However, we find that, even for +a realistic setting, the Edgeworth expansion cannot reproduce the +original SRDs if cumulants n > 2 are considered. +Data Availability: The data underlying this article will be +shared on reasonable request to the corresponding author. +ACKNOWLEDGMENTS +RR would like to thank Hendrik Hildebrandt and Björn Malte +Schäfer for insightful discussions and comments on the manuscript. +RR is supported by the European Research Council (Grant No. +770935). +REFERENCES +Abbott T. M. C., et al., 2018, Phys. Rev. D, 98, 043526 +Abbott T. M. C., et al., 2022, Phys. Rev. D, 105, 023520 +Alarcon A., Sánchez C., Bernstein G. M., Gaztañaga E., 2020, Monthly +Notices of the Royal Astronomical Society, 498, 2614 +Asgari M., et al., 2021, Astron. Astrophys., 645, A104 +Blanchard A., et al., 2020, Astron. Astrophys., 642, A191 +Blinnikov S., Moessner R., 1998, Astron. Astrophys. Suppl. 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B., 2017, Monthly Notices of the Royal +Astronomical Society, 466, 3558 +Mead A. J., Peacock J. A., Heymans C., Joudaki S., Heavens A. F., 2015, +MNRAS, 454, 1958 +Myles J., et al., 2021, MNRAS, 505, 4249 +Newman J. A., 2008, The Astrophysical Journal, 684, 88 +Rau M. M., Wilson S., Mandelbaum R., 2020, Monthly Notices of the Royal +Astronomical Society, 491, 4768 +Schaan E., Ferraro S., Seljak U., 2020, J. Cosmol. Astropart. Phys., 2020, +001 +Stölzner B., Joachimi B., Korn A., Hildebrandt H., Wright A. H., 2021, +Astron. Astrophys., 650, A148 +Sánchez C., Bernstein G. M., 2019, Monthly Notices of the Royal Astro- +nomical Society, 483, 2801 +Wright A. H., Hildebrandt H., Busch J. L. v. d., Heymans C., 2020, A&A, +637, A100 +van den Busch J. L., et al., 2020, A&A, 642, A200 +APPENDIX A: EDGEWORTH EXPANSION +In this section we employ an Edgeworth expansion for the photo- +z distribution. The Edgeworth expansion is an asymptotic expan- +sion (in contrast to the Gram-Charlier expansion). Starting from +the characteristic function (Blinnikov & Moessner 1998). +ϕ(j) +Z (t) = En( j)(z) +� +eitZ� +, +(A1) +i.e. the Fourier transform of the probability density n( j)(z). With the +definition of the moments ˜µn, the Taylor expansion of the charac- +teristic function is +ϕ(j) +Z (t) = 1 + +∞ +� +n1 +˜µn +n! (it)n . +(A2) +The logarithm of the characterstic function is the cumulant, κn, gen- +erating function +κn = 1 +in +dn +dtn log ϕ( j) +Z (t) +�����t=0 +. +(A3) +MNRAS 000, 1–9 (2022) + +functional photo-z +7 +Using this definition one can relate the cumulants to the moments +κn = n! +� +{km} +(−1)r−1(r − 1)! +n +� +m=1 +1 +km! +� ˜µm +m! +�km +, +(A4) +where {km} denotes the set of all solutions to the Diophantine equa- +tion +n +� +a=1 +aka − n = 0 . +(A5) +If a distribution is then expanded as a asymptotic series around a +normal distribution one finds +n(z) = +1 +√2πκ2 +exp +� +−(z − κ1)2 +2κ2 +� +× +� +1 + +∞ +� +s=1 +κs/2 +2 +� +{km} +Hes+2r +������� +z +κ1/2 +2 +������� +s +� +m=1 +1 +km! +� +λm+2 +(m + 2)! +�km � +≡ nG(z)(1 + Eg(z)), +(A6) +where λn � κn/κn/2 +2 . We are now interested in the sensitivity of the +distribution with respect to its cumulants. Here the cases n = 1, 2 +are a bit special: +∂n(z) +∂κ1 += n(z)z − κ1 +κ2 +(A7) +and for κ2 +∂n(z) +∂κ2 += +1 +2κ1/2 +2 +�������n(z) +�(z − κ1)2 +κ2 +− 1 +� ++ nG(z)∂Eg(z) +∂κ1/2 +2 +������� , +(A8) +where +∂Eg(z) +∂κ1/2 +2 += +∞ +� +s=1 +� +{km} +κs/2 +2 P(s, {km}) +� +He2r+s +������� +z +κ1/2 +2 +������� +× +������� +s +κ1/2 +2 +− +s +� +a +ka(2a + 2) +κka(a+1)+1/2 +2 +������� − (2r + s)He2r+s−1 +������� +z +κ3/2 +2 +������� +z +κ2 +� +, +(A9) +where we also defined the product: +P(s, {km}) � +s +� +m=1 +1 +km! +� +κm+2 +(m + 2)!κ2m+2 +2 +�km +. +(A10) +For all cumulants with n ≥ 3 one finds: +∂n(z) +∂κn += nG(z) +∞ +� +s=1 +� +{km} +κs/2 +2 P(s, {km})He2r+s +������� +z +κ1/2 +2 +������� +kn−2 +κn +. +(A11) +It should be noted, however, that the Edgeworth expansion is not a +convergent series but rather an asymptotic expansion. One therefore +needs to check whether the expansion is a good approximation of +the underlying distribution. +In this case one can define the ordinary Fisher matrix using +partial derivatives: +Fκ(i) +m κ( j) +n = fsky +ℓmax +� +ℓ=ℓmin +2ℓ + 1 +2 +tr +� ∂Cℓ +∂κ(i) +m +C−1 +ℓ +∂Cℓ +∂κ(j) +n +C−1 +ℓ +� +, +(A12) +where κ(i) +m is the m-th cumulant of the source-redshift distribution in +the i-th tomographic bin. +Figure A1 shows the fiducial redshift distributions for EU- +CLID and their Edgeworth expanded approximations as solid and +dashed lines respectively. The top plot uses the expansion up to κ3, +while the bottom plot sums contributions up to κ6. For all but the +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +z +0 +1 +2 +3 +4 +p(z) +order = 3 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +tomographic bin index +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +z +0 +1 +2 +3 +4 +p(z) +order = 6 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +tomographic bin index +Figure A1. SRD for EUCLID in all 10 tomographic bins. Solid lines rep- +resent the fiducial SRD, while dashed lines represent their respective Edge- +worth expansion. Cumulants up to order n = 3 , 6 are used respectively. +first and last tomographic bin, the Edgeworth series is a good ap- +proximation. This is expected as they are essentially Gaussian and +therefore κn ≈ 0 for n > 2. The first tomographic bin experiences +boundary effects at z = 0 and is therefore slightly skewed. This ef- +fect is even larger for the last tomographic bin, which has a very +long tail to high redshifts. While the first bin can still be described +by the Edgeworth expansion and the series converges, the 10th bin +shows negative probability in the Edgeworth series already at third +order. The situation becomes worst if higher order cumulants are +included. This goes to show that even for such an idealized case as +the EUCLID forecast, the use of the Edgeworth expansion can be +very dangerous. +For the case n = 2 we show the Pearson correlation coeffi- +cient of the joint covariance matrix between the first three cumu- +lants in each tomographic bin and four cosmological parameters in +Figure A2. We observe some correlations between the first and sec- +ond moment of each tomographic bin. There is a very strong cor- +relation between first and second moment of two different redshift +bins. Furthermore, one can see that parameters controlling the am- +plitude of the lensing spectrum are anti-correlated with the mean. +We want to stress again, however, that the expansion, even in this +case, is not convergent and results obtained with n > 2 have thus to +be taken with care. +MNRAS 000, 1–9 (2022) + +8 +Reischke +κ(0) +1 +κ(0) +2 +κ(1) +1 +κ(1) +2 +κ(2) +1 +κ(2) +2 +κ(3) +1 +κ(3) +2 +κ(4) +1 +κ(4) +2 +κ(5) +1 +κ(5) +2 +κ(6) +1 +κ(6) +2 +κ(7) +1 +κ(7) +2 +κ(8) +1 +κ(8) +2 +κ(9) +1 +κ(9) +2 +Ωm0 +σ8 +w0 +wa +κ(0) +1 +κ(0) +2 +κ(1) +1 +κ(1) +2 +κ(2) +1 +κ(2) +2 +κ(3) +1 +κ(3) +2 +κ(4) +1 +κ(4) +2 +κ(5) +1 +κ(5) +2 +κ(6) +1 +κ(6) +2 +κ(7) +1 +κ(7) +2 +κ(8) +1 +κ(8) +2 +κ(9) +1 +κ(9) +2 +Ωm0 +σ8 +w0 +wa +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +rij +Figure A2. Pearson correlation coefficient for the joint covariance matrix of the first two cumulants of the EUCLID like survey and four cosmological +parameters. +APPENDIX B: PHOTOMETRIC GALAXY CLUSTERING +For photometric galaxy clustering, the procedure can be simply +adopted by changing the weight function (up to galaxy bias, which +we absorb in the power spectrum). Again by using the Limber pro- +jection: +C +gig j +ℓ += +� χH +0 +dχ +χ2 W(i) +g (χ)W(j) +g (χ)Pgg +�ℓ + 0.5 +χ +, χ +� +, +(B1) +with the galaxy power spectrum Pgg and corresponding weights +given by: +W(i) +g (χ) = n(i) +g (χ) , +(B2) +therefore the functional derivative takes the very simple form +δC +gig j +ℓ +δna(χ0) = +Pgg +� +ℓ+0.5 +χ , χ +� +χ2 +� +n(j)(x)δD +ia + n(i)(x)δD +ja +� +. +(B3) +APPENDIX C: DISTRIBUTION OF THE MEAN AND +VARIANCE +We show the relative difference of the mean redshift and the stan- +dard deviation of the SRD for each tomographic. As before we dis- +tinguish between the EUCLID’s survey settings and KV450. In par- +ticular we sample from the diagonal covariance obtained from the +functional Fisher matrix as described in Section 3 for the former, +while we use the DIR covariance for the latter. +The top plots of Figure C1 show the distribution of the mean +and the standard deviation and show generally good agreement with +Figure 2, that is that the mean must be known below the per-cent +level for most bins, while the standard deviation needs to be de- +termined by roughly 10 per-cent. It should be noted that Figure 2 +considers the extreme case where we exactly look at the envelope +shown in Figure 1. +Finally, the buttom two plots show the same for KV450, where +we find much wider errors on mean and standard deviation, few per- +cent and a few ten per-cent respectively. The general trend, how- +ever, is the same - high redshift bins are more important than lower +redshift bins. +APPENDIX D: RELATIONSHIPS TO OBSERVABLES +Real surveys usually do not use the angular power spectra as a fi- +nal statistic. This is for example due to incomplete sky coverage, +masking effects, variable depth or simply the dimensionality of the +data vector. All these factors require a sufficient summary statis- +tic. Very commonly used ones are the correlation function or band +powers (or similarly pseudo-Cℓ). All of these are essentially linear +transformations of the pure angular power spectrum Cℓ and assume +the following general form: +O[Cℓ] = +� +dℓCℓWO(ℓ) , +(D1) +MNRAS 000, 1–9 (2022) + +functional photo-z +9 +−0.004 +−0.002 +0.000 +0.002 +0.004 +¯z/⟨¯z⟩ − 1 +0 +100 +200 +300 +400 +500 +600 +700 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +tomographic bin index +−0.20 −0.15 −0.10 −0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +σz/⟨σz⟩ − 1 +0 +10 +20 +30 +40 +50 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +tomographic bin index +−0.15 +−0.10 +−0.05 +0.00 +0.05 +0.10 +0.15 +¯z/⟨¯z⟩ − 1 +0 +10 +20 +30 +40 +50 +KV450 +1 +2 +3 +4 +5 +tomographic bin index +−0.4 +−0.3 +−0.2 +−0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +σz/⟨σz⟩ − 1 +0 +2 +4 +6 +8 +10 +12 +14 +16 +KV450 +1 +2 +3 +4 +5 +tomographic bin index +Figure C1. Distribution of the relative deviation of the mean and the variance of the SRD, n(i) +s (z). Top: For the EUCLID survey settings with realisations from +the inverse of the diagonal Fisher matrix used in Figure 1. Bottom: For KV450 using the samples generated from the DIR bootstrap covariance. +where O is some observable of interest and WO(ℓ) is the associ- +ated kernel defining the transformation. Again by the chain rule, +the functional derivative of this new observable with respect to the +SRD is readily available: +δO[Cℓ] +δn(χ0) = +� +dx +δO +δCℓ(x) +δCℓ(x) +δn(χ0) , +(D2) +where we dropped all the indices for less clutter. For band powers, +Cl, this would for example assume the following form: +δCl[Cℓ] +δn(χ0) = 1 +Nl +� +dℓℓS ℓ +δCℓ +δn(χ0) , +(D3) +where S ℓ is the band power response function and Nl is the normal- +isation. For the two-point correlation function ξ± one finds: +δξ±(θ) +δn(χ0) = 1 +2π +� +dℓℓJ0,4(ℓθ) δCℓ +δn(χ0) . +(D4) +APPENDIX E: NON-LIMBER Cℓ +The Limber projection used for the Cℓ is not valid on large angu- +lar scales, where it must be replaced by the full expression. In full +generality, for any tracers i and j of the matter density +Ci j +ℓ = 2 +π +� +dk k2Iℓ,k,i[ni]Iℓ,k,j[ni] , +(E1) +where the functional Ik,i[ni] is given by +Iℓ,k,i[ni] = +� +dχWi[ni] +� +Pii(k, χ)jℓ(χk) . +(E2) +Here Pii is the auto power spectrum of the tracer i and Wi is its +associated weight. Thus we find: +δCij +ℓ +δna = +� +dk k2 +� +Iℓ,k,i +δIℓ,k,j +δna δD +ja + Iℓ,k,j +δIℓ,k,i +δna δD +ia +� +, +(E3) +where +δIℓ,k,i +δni += +� +dχδWi +δn +� +Pii(k, χ) jℓ(χk) . +(E4) +The derivative of the weight function is calculated as before. +APPENDIX F: INTRINSIC ALGINMENTS +In this work, we have ignored intrinsic alignments (IA). Its in- +clusion is, however,straight forward by noting that the IA angular +power spectrum is simply given by +CII +ℓ = +� χH +0 +dχ +χ2 n(i) +s (χ)n(j) +s (χ)PII +�ℓ + 0.5 +χ +, χ +� +, +(F1) +where PII is the IA power spectrum, which summarises the reac- +tion of galaxy shapes to the ambient LSS on the two-point level. +The functional derivative there proceeds in the same way as in Ap- +pendix B. For the GI term of intrinsic alignments, one proceeds as +before for cosmic shear (compare Section 2). +MNRAS 000, 1–9 (2022) + diff --git a/YNE2T4oBgHgl3EQfvAgF/content/tmp_files/load_file.txt b/YNE2T4oBgHgl3EQfvAgF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a39cc62d86080c2ab2823bb4bfa03ad76b48eec5 --- /dev/null +++ b/YNE2T4oBgHgl3EQfvAgF/content/tmp_files/load_file.txt @@ -0,0 +1,609 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf,len=608 +page_content='MNRAS 000, 1–9 (2022) Preprint 11 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 Propagating photo-z uncertainties: a functional derivative approach Robert Reischke⋆1 1 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing, 44780 Bochum, Germany 11 January 2023 ABSTRACT Photometric redshifts are a key ingredient in the analysis and interpretation of large-scale structure (LSS) surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The accuracy and precision of these redshifts estimates is directly linked to the constraining power of photometric surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' It is hence necessary to define preci- sion and accuracy requirements for the redshift calibration to not to infer biased results in the final analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' For weak gravitational lensing of the LSS the photometry culminates in the es- timation of the source redshift distribution (SRD) in each of the tomographic bins used in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The focus has been on shifts of the mean of the SRDs and how well the calibration must be able to recover those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Since the estimated SRDs are usually given as a normalized histogram with corresponding errors, it would be advantageous to propagate these uncertain- ties accordingly to see whether the requirements of the given survey are indeed fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Here we propose the use of functional derivatives to calculate the sensitivity of the final observ- ables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' the lensing angular power spectrum, with respect to the SRD at a specific redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This allows the propagation of arbitrarily shaped small perturbations to the SRD, without hav- ing to run the whole analysis pipeline for each realization again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We apply our method to a EUCLID survey and demonstrate it with SRDs of the KV450 data set, recovering previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Lastly, we note that for cosmic shear moments of order larger than two will probably be not relevant when propagating redshift uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Key words: cosmology: theory, large-scale structure of Universe, surveys, galaxies: photom- etry 1 INTRODUCTION Cosmic shear, the weak gravitational lensing effect imprinted on distant galaxies by the large-scale structure (LSS), is one of the pri- mal science goals for EUCLID and Rubin-LSST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The blueprint for these missions has been set by current stage-3 surveys, including the Kilo-Degree Survey (Kuijken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Asgari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2021, KiDS), the Dark Energy Survey (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Gatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2022, DES) or the Subaru Hyper Suprime-Cam (Hamana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2020, HST), yielding tight constraints ob the matter distribution in the late Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The cosmic shear signal is estimated by measuring the coher- ent distortion of background galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Since the intrinsic elliptic- ity of galaxies is much larger than the lensing effect, millions of galaxies are required to measure a significant signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This casts a complete spectroscopic survey unfeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Hence, one has to rely on redshift estimates from photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In order to interpret the ob- served ellipticity correlations, the potometric redshifts have to be calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' There are different approaches for the calibration pro- cedure on the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' These include the calibration with a spec- troscopic reference sample (possibly with re-weighting) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Lima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Newman 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Matthews & Newman 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Masters ⋆ E-mail: reischke@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='ruhr-uni-bochum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='de et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Bonnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' McLeod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Myles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2021), using photome- try measurements in conjunction with clustering measurements of tracer populations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Sánchez & Bernstein 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' van den Busch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Alarcon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2020) and self-organising maps (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' It is also possible to partially self-calibrate the pho- tometric redshifts in weak lensing data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Schaan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In order to account for general shapes of the source-redshift dis- tributions (SRDs) different mixture models have been employed (see for example Rau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' These Gaussian processes are non-parametric, but they are by definition non-linear, which makes their implementation in cosmology pipelines in general very diffi- cult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Stölzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (2021) used linear fit parameters to circumvent this problem to self-calibrate the data, as it can be implemented in existing pipelines very easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Currently it is best practice to propagate the redshift uncer- tainty in the SRDs by introducing shift parameters in the mean of the distribution (Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Hikage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' As the sensitivity of surveys rises, however, the re- quirements on the SRD uncertainties become larger as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' There- fore, the contributions from higher order cumulants of the SRD be- come important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' As discussed above, previous works have focused on Gaussian mixture models to self-calibrate the cosmic shear mea- surement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In this paper we investigate the general sensitivity of the © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='04085v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='CO] 10 Jan 2023 2 Reischke lensing power spectrum to perturbations in the SRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In particu- lar we are calculating the functional derivative of the cosmic shear angular power spectrum with respect to the SRD at a particular co-moving distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This can then be mapped to a total error in the cosmic shear power spectrum if a perturbation in the SRD in a co-moving interval is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We take the constraint of the nor- malisation of the SRD into account when calculating the functional derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Therefore we can propagate arbitrary perturbations to the SRDs (subject to some underlying covariance) and propagate them into the Cℓ of cosmic shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This allows us to estimate the dif- ference in χ2 induced by the uncertainty in the SRD, without hav- ing to run thousands of realizations of the analysis pipeline used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' By using a Fisher matrix for the cosmological parameters, this ∆χ2 can then be mapped to potential biases in cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Here we studied a rather idealised scenario by working in Fourier space, assuming a Gaussian likelihood and ignoring intrinsic align- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The method, however, easily generalises and including these effects is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We structure the paper as follows: In Section 2 we briefly review cosmic shear basics and introduce the methodology used by calculating the functional derivative of the weak lensing angu- lar power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The results are presented in Section 3, where we apply the procedure to a survey with EUCLIDs specifications and to KiDS-VIKING-450 (KV450).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We conclude in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In the appendices we also investigate the possibility of an Edge- worth expansion of the SRD (Appendix A), discuss photometric galaxy clustering (Appendix B), the distribution of the mean and standard deviation of the SRD in Appendix C, the general relation- ship to observables (Appendix D), the functional derivative of the non-Limber projection in Appendix E and inrinsic alignments (Ap- pendix F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2 METHODOLOGY In this section we present the basic methodology of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In particular we describe the basics of cosmic shear and derive the function derivative of the lensing angular power spectrum with re- spect to the SRDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='1 Cosmic shear basics The equation for the cosmic shear power spectrum in tomographic bins i and j in the Limber proejction is (Limber 1954;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Loverde & Afshordi 2008) C κiκj ℓ = � χH 0 dχ χ2 W(i) κ (χ)W(j) κ (χ)Pδ �ℓ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 χ , χ � , (1) where Pδ is the matter power spectrum, for which we use the em- ulated spectrum from Mead et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' W(i) κ (χ) is the lensing weight of the i-th tomographic bin as given by: W(i) κ (χ) = 3Ωm0 2χ2 H χ a(χ) � χH χ dχ′n(i) s (χ′)χ′ − χ χ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (2) Here χ is the co-moving distance, a the scale factor, Ωm0 the matter density parameter today, χH the Hubble radius and n(i) s is the SRD in the i-th tomographic bin which builds on photo-z measurements and its calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' It is normalized in each tomographic bin such that � dz n(i) s (z) = 1 = � dχ n(i) s (z(χ)) dz dχ ≡ � dχ n(i) s (χ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (3) Since photo-z is just an estimate of the true redshift, the estimated source-redshift distribution, n(i) s , is not exactly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Here we in- vestigate two approaches: i) Use functional derivatives to evaluate the change of the lens- ing power spectrum when perturbing the n(i) s at different redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Given specific survey settings and precision goals, limits on the al- lowed change of the n(i) s can be determined, which in turn can be mapped to changes in the cumulants or moments of the underlying distribution (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' ii) We expand the underlying source-redshift distribution in an asymptotic Edgeworth series and investigate the requirements on the cumulants directly in a Fisher analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The second approach is not feasible for realistic SRDs (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='2 Functional derivative of the lensing power spectrum Here we wish to investigate the sensitivity of the weak lensing power spectrum to the full shape of the source-redshift distribution using functional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In particular we start by perturbing n(i) s (χ(z)) at a certain redshift z0, such that χ0 = χ(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The corre- sponding perturbed lensing weight is thus ∆W(i) κ (χ, χ0) = δW(i) κ (χ) δn(i) s (χ0) ∆n(i) s (χ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (4) This expression evaluates, how the lensing weight changes if the source-redshift distribution is perturbed by an amount ∆n(i) s at the co-moving distance χ0 corresponding to the redshift z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Ultimately, we are interested in the change of the lensing power spectrum, Equation (B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' First, by applying the Leibniz rule δC(ij) ℓ δn(a)(χ0) = � dx δC(ij) ℓ δW(a)(x) δW(a)(x) δn(a)(χ0) = � dx δW(a)(x) δn(a)(χ0) Pδ � ℓ+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 x , x � x2 � W(j)(x)δD ia + W(i)(x)δD ja � , (5) The missing ingredient is the functional derivative of the lensing kernel, for which we find δW(i)(x) δn(j) s (χ0) = 3Ωm0 2χ2 H x a(x) χ0 − x χ0 δD ijΘ(χ0 − x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (6) Θ(x) is the Heaviside function to ensure that the functional deriva- tive vanishes if the SRD is perturbed outside the integration bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Using Equation (4) and Equation (5) we can write the change in angular power spectrum ∆C(ij) ℓ (χ′) due to a change in the source- redshift distribution at co-moving distance χ0 as ∆C(ij) ℓ,a (χ0) ≡ δC(ij) ℓ δn(a)(χ0)∆n(a)(χ0) = 3Ωm0 2χ2 H ∆n(χ0) � dx a(x)x χ0 − x χ0 Pδ �ℓ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 x , x � × � W(j)(x)δD ia + W(i)(x)δD ja � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (7) Integrating the perturbed lensing spectrum then gives the total per- turbation: ∆C(ij) ℓ,a ≡ � dχ0∆C(ij) ℓ,a (χ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (8) So far we have treated the function n(i)(z) as being completely free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' However, the functional derivative needs to respect the constraint MNRAS 000, 1–9 (2022) functional photo-z 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 redshift z 1 2 3 4 n(i) s (z) 1 2 3 4 5 6 7 8 9 10 tomographic bin index Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Allowed perturbation for EUCLID to the SRD of the ten tomo- graphic source bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Solid lines show the fiducial SRD, while the bands show the allowed perturbation to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' given in Equation (3), thus limiting the possible variations of n(i)(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The normalization condition itself is again a function and we write N[n(i) s ] � 1 − � dz n(i) s (z) = 0 , (9) this constraint can be implemented by first defining n(i) s (z) � f(z) � dx′ f(x′) (10) which will be normalized by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' n(i) s (z) is a functional of f and we can now evaluate the functional derivative of C[n[ f]] as an unconstrained derivative but evaluated at f = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' To avoid clutter we ignore the sub- and superscripts in this part �δC[n[ f]] δf(x) � ������ f=n = � dx′ δC[n] δn(x′) δn(x′) δ f(x) ������ f=n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (11) With δn(x′) δf(x) = δD(x′ − x) � dy f(y) − f(x′) �� dy f(y) �2 , (12) one finds δC[n] δ1n(x) ≡ �δC[n[f]] δf(x) � ������ f=n = δC[n] δn(x) − � dy δC[n] δn(y) n(y) , (13) where we denote that we want to keep the normalization fixed by the variation δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This is a very intuitive expression: the first term evaluates the standard functional derivative, while the second term corrects this variation to respect the normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='3 Fisher forecast The next step is to set some requirement on the lensing power spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Here we will look at the difference in the χ2, assuming a Gaus- sian likelihood and thus setting a lower limit on the required accu- racy of n(i) s (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' For modes aℓm with zero mean and covariance Cℓ, the ∆χ2 between multipoles ℓmin and ℓmax can be written as ∆χ2(ℓmin, ℓmax) = fsky ℓmax � ℓ=ℓmin 2ℓ + 1 2 tr � ∆CℓC−1 ℓ ∆CℓC−1 ℓ � , (14) 1 2 3 4 5 6 7 8 9 order of central moment n 10−2 100 102 104 106 108 relative change in % 1 2 3 4 5 6 7 8 9 10 tomographic bin index Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Allowed relative change in per-cent of the central moment of the SRD in each tomographic bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The changes are calculated from the per- turbed SRD distributions as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' note that Cℓ is the matrix with the components C(ij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The factor fsky takes into account the observed sky fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Using Equation (8) we rewrite the previous equation as a Riemann sum ∆χ2(ℓmin, ℓmax) = fsky ℓmax � ℓ=ℓmin 2ℓ + 1 2 × � r,s,i,j tr � δCℓ δ1n(i)(χr)C−1 ℓ δCℓ δ1n(j)(χs)C−1 ℓ � × DχrDχs∆n(i)(χr)∆n(j)(χs) , (15) with the measure Dχr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' If we define the Fisher matrix in this case as: Fαβ = fsky ℓmax � ℓ=ℓmin 2ℓ + 1 2 tr � δCℓ δ1nα C−1 ℓ δCℓ δ1nβ C−1 ℓ � Dχr(α)Dχs(β) , (16) where we labeled n(i)(χr) → nα, we recover for a difference in χ2 using a scalar product on the finite dimensional Hilbert space of shifts in the redshift distribution where the Fisher matrix acts as a norm-inducing metric ∆χ2 = F(∆n, ∆n) ≡ ∆nT F∆n , (17) where ∆n is the vector containing shifts of the components nα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The Fisher matrix, Equation (16), describes, how well the shifts nα can be determined by a measurement of the angular power spectra Cα given certain survey settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Clearly, if one would try to measure all possible perturbations, neighbouring δn(χ) are strongly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This is, however not the question we would like to ask in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Instead, we want to look at the situation that we allow any perturbation ∆n, irrespective of the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Therefore, by turning this argument around, we only use the diagonal part of the Fisher matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Lastly one should note that the functional derivative is strictly defined as a limiting process for infinitesimally small perturbation to the function at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The relation in general can be non-linear, but as long as relative perturbations to the function are small with respect to unity, these non-linear contributions are sub-dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Especially for surveys with tight requirements on the SRDs this is essentially always fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' MNRAS 000, 1–9 (2022) 4 Reischke 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='00 redshift z 1 2 3 4 5 6 n(i) s (z) KV450 1 2 3 4 5 tomographic bin index Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Allowed perturbation for KV450 to the SRD for the 5 tomo- graphic source bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Solid lines show the fiducial SRD, while the bands show the allowed perturbation to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 3 RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='1 Allowed Perturbations to the Source Redshift Distribution First we will look at the allowed perturbations to the SRD by as- suming allowing for a total ∆χ2 of unity, corresponding to a one σ shift of a linear model parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Clearly, there are many differ- ent solutions ∆n that satisfy ∆χ2 = 1 subject to Equation (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' To show the structure of the Fisher matrix we therefore distribute the allowed ∆χ2 per ∆nα equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We will assume EUCLID specifications for the survey as given in Blanchard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (2020) and assume ntomo = 10 tomo- graphic bins, a sky fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Furthermore, we will collect multipoles between ℓmin = 10 and ℓmax = 3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We then calculate the diagonal Fisher matrix from Equation (16) and distribute the er- rors equally as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This results into a possible realisa- tion of ∆n yielding ∆χ2 = 1 subject to the constraint Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Figure 1 shows the resulting perturbed SRDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The solid lines show the fiducial SRD, while the shaded areas show the allowed pertur- bations to not cause a bias of more than 1 σ for a linear model pa- rameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Lastly, the tomographic bin index is shown as a colour-bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The general trend is very clear, the allowed perturbations become very large around a small interval ∆χ around the mean of the distri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' For most tomographic bins this coincides with the peak of the distribution as they are very close to Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Only for the first and the last bin these spikes are a bit offset since the distributions are a bit more asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This already confirms that the most im- portant part about the SRDs in cosmic shear measurements is to calibrate the mean redshift of each tomographic bin very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Fur- thermore, we observe that the spikes tend to be narrower at higher redshifts, indicating that the uncertainty on the mean of the SRD is more important at higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We want to stress again, that this is just one realization of ∆n that produces a ∆χ2 = 1, but by distributing the errors equally, it is possible to see, which pertur- bations the final measurement is most sensitive to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' However, the uncertainties should not be used at literally value and are extreme values, they just give a general trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Next, we use the perturbed SRDs to calculate their central mo- ments µn: µn � E[(X − E[X])n] = � p(x)(x − µ)ndx , (18) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 ∆χ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='2 KV450 68th 50th 95th percentile Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' ∆χ2 for 106 realisations of ∆n from the CKV450 n(χ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We also show the 50, 68 and 95 percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' for a probability distribution function p(x) with mean µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The per- turbed SRDs are used to calculate the change in the central mo- ments relative to the fiducial SRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Figure 2 shows the resulting relative change for all tomographic bins as a function of the order of the central moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Clearly, the first moment is most important and while the second one still needs to be known at a 10% level, all higher order moments are essentially unimportant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This is of course reminiscent of the behaviour observed in Figure 1, where the perturbations are such that they essentially fix the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' It is of course entirely possible, that we alter the shape of the distribution in a different way but still achieve the desired accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Nonetheless, the results show that for the SRD for cosmic shear only the mean redshift and the width are important with the former influencing the result way stronger (by more than an order of magnitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In Appendix C we sample from the allowed changes in the SRD and show the relative difference of the first two mo- ments to illustrate their scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='2 Propagating Redshift Errors In this section we will revisit the KV450 data for the SRD (Hilde- brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This data set is used since it includes a covari- ance matrix from the direct calibration (DIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' For the clustering redshifts (van den Busch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2020) or the self-organising maps (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2020) no bootstrap covariance was estimated so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' For completeness the allowed perturbations are shown in Fig- ure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Due to the lower signal-to-noise ratio of the measurement, the allowed perturbations are much larger than in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The features, however, are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Since we are expressing everything in co-moving distance, the covariance matrix needs to be transformed accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Let CKV450 n(z) be the covariance matrix in n(z) space, the transformed covariance is then CKV450 n(χ) = JTCKV450 n(z) J , (19) where J is the Jacobian with components Ji j = δi jdz/dχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Alterna- tively, the Fisher matrix of the SRD perturbations can be expressed in redshift space by the inverse transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Perturbations ∆n are now sampled from CKV450 n(χ) and propa- gated to obtain ∆χ2 according to Equation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' If the redshift errors as given in CKV450 n(χ) are sufficiently small to not produce a significant bias in the cosmological parameters such as S 8 we expect most MNRAS 000, 1–9 (2022) functional photo-z 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='400 Ωm0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='00 σ8 KV450 1 2 3 4 5 6 ∆χ2 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The black histogram shows the induced shifts by the photo-z un- certainty in the Ωm0-σ8-plane, derived from the ∆χ2 of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In red we show the contour from the Fisher matrix for KV450 enclosing the 1σ confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' realisations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 68% Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2020) to yield ∆χ2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Figure 4 shows the resulting distribution in ∆χ2 for the 106 real- izations of ∆n for KV450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The vertical dashed lines show the 50th, 68th and 95th percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' It is clear from this plot that the precision of the SRD used in KV450 is high enough to not yield any spurious detection in the final parameter constraints since the 68th percentile is still well below unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' One could now further propagate these uncertainties into cos- mological parameters using the corresponding Fisher matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' For a given shift in the SRD ∆n, the corresponding shifts in the cosmo- logical parameters, ∆θ can be calculated: ∆θi = −(F−1)i αFα β∆nβ , (20) where Greek indices run over the perturbations in the SRD, while Latin indices label cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Here we assumed the sum convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Fi α hence is the mixed pseudo Fisher matrix: Fi α = −E �∂ ln L ∂θi δ ln L δnα Dχr(α) � (21) and it’s inverse is a pseudo inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Since the inversion of this ma- trix is not necessarily stable we choose to go another route here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Since the distribution of ∆χ2 is known, we are interested in sam- ples of cosmological parameters with the same ∆χ2 with respect to the best fit value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' For a Gaussian posterior in one dimension this would amount to a distribution such that the absolute value of each sample is fixed to √ ∆θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We sample from a standard Gaussian distribution and modify its width by √ ∆θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This Gaussian is then mapped into the frame of the cosmological parameters under con- sideration via the Cholesky decomposition of the Fisher matrix of the cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In Figure 5 we apply this procedure to the ∆χ2 distribution of KV450 (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Each dot represents one sample of the ∆χ2 distribution with its value shown as a colour bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' It can be seen as the geodesic distance to the fiducial value for the cosmological parameters in the parameter manifold (Giesel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The red contours depict the expected 1, 2, 3σ confi- dence regions from the Fisher forecast for KV450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Since in the original analysis more than the two parameters here where used, we re-scale the ∆χ2 accordingly, in particular by the χ2 quantile function χ2 k(p), where k = 10 is the number of parameters in the actual analysis analysis (Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2017) and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This is done in order to obtain a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' It is clear from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='796 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='806 S8 0 25 50 75 100 125 150 175 200 KV450 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Induced scatter on the S 8 = σ8 √Ωm0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='3 parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This is directly derived from the samples of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The scatter is roughly 15 per-cent of the statistical error budget reported in Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (2017, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' the plot, that all samples for the photometric redshift distribution lie well within the 1σ contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Furthermore, it should be noted that we are considering a very idealised forecast with two free parameters and no systematics here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The procedure, however, can be general- ized to any number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Furthermore, one can apply the same analysis to a full Monte-Carlo-Markov-Chain (MCMC) by matching those samples which are ∆χ2 away from the maximum likelihood of the MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Lastly, the samples from Figure 5 can be mapped to S 8 = σ8 √Ωm0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Figure 6 shows the resulting his- togram of the scatter due to the photo-z uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Comparing this to ∆S 8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='076 at 68% confidence (Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2020) shows that the scatter induced by the redshift uncertainties as sam- pled from the KV450 SRD covariance have a small effect on the overall error budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (2017) a Fisher matrix method for the shifts of the mean of the SRDs was investigated as a a source of systematics, which found similar results to the once presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The main difference between the two methods is that we allow for general perturbations to the redshift distribution (provided there correlation is given).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Generalizing the procedure in Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (2017) to moments higher than the variance is bound to fail (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' However, we would also con- clude that even for EUCLID, the analysis of the first two moments is probably sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In appendix C the mean and standard deviation of each SRD in the five tomographic bins are shown for the realisations used in this section as sampled from the DIR covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Figure C1 shows a very similar behaviour to what we found in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In par- ticular this is that the mean scatters less at higher redshifts, while the standard deviation scatters roughly equally for most of the bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We close the section with a general discussion about the usage of ∆χ2 or directly uncertainties in the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' It is in general advantageous to make accuracy assessments for the SRD using the ∆χ2 and not by inverting the Fisher matrix for the parameters of interests to obtain the shift values for those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The reason for this is that ∆χ2 is an invariant quantity, while shifts in parameter space are dependent on the specific model choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The only caveat in the ∆χ2 is that the number of parameter must be taken into account, this is, however, much easier than calculating the Fisher matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' MNRAS 000, 1–9 (2022) 6 Reischke 4 CONCLUSIONS In this paper we have analysed the dependence of the cosmic shear angular power spectrum on the SRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This has be done by employ- ing functional derivatives of the cosmic shear Cℓ with respect to the SRD at a fixed co-moving distance χ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' By integrating over the introduced error we estimated the ∆χ2 introduced by arbitrary un- certainties in the SRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We applied our method to a cosmic shear survey with EUCLID specifications and KV450 since a covariance of the SRD estimate was given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Our main findings can be sum- marised as follows: (i) Allowed perturbations of the SRD are such that they preserve the mean of the underlying distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' If they do, they can be rather larger, even for a survey like EUCLID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This is in line with the common practice of using only shifted means of the underlying redshift distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (ii) In order to achieve the accuracy required for EUCLID, the mean of the redshift distribution needs to be determined between 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='01 per-cent, depending on the tomographic bin under con- sideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The variance of the SRD is still important at the 10 per- cent level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' There is still some sensitivity left in the skewness, but all other moments are not relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (iii) We performed a simplistic analysis of the KV450 SRDs to check whether they fulfill the requirements and found that the un- certainties, in this very idealised scenario, only yield biases up to 1σ in the final constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In a full analysis, this bias would be even smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Thus confirming the redshift calibration used in KV450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (iv) Even for EUCLID it is most likely not necessary to inves- tigate moments of the redshift distribution n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This conclusion could change for different settings and self-calibration methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (v) The procedure outlined here has the advantage to be very cheap computationally, since the functional derivatives only need to be computed once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' It is then only a matter of sampling from the underlying SRD and to propagate these perturbations with the pre- viously calculated functional derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' It is hence not necessary to push thousands of realisations of the SRD through the analysis pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The method outlined here can thus be used to analyse whether a perturbation in the SRD still fulfills the requirements of a given experiment so that no biases of model parameters are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' It allows for arbitrary perturbations to the SRD without requiring a fit to the actual distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We intend to apply the presented method to the updated SRDs of KiDS in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' For the interested reader the appendices Appendix A - Ap- pendix E discuss various aspects of the analysis which could be refined in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In particular we look at the Edgeworth ex- pansion of the SRD in Appendix A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' an expansion in the cu- mulants of the underlying SRDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' However, we find that, even for a realistic setting, the Edgeworth expansion cannot reproduce the original SRDs if cumulants n > 2 are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Data Availability: The data underlying this article will be shared on reasonable request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' ACKNOWLEDGMENTS RR would like to thank Hendrik Hildebrandt and Björn Malte Schäfer for insightful discussions and comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' RR is supported by the European Research Council (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 770935).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' REFERENCES Abbott T.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=', 2020, A&A, 637, A100 van den Busch J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=', 2020, A&A, 642, A200 APPENDIX A: EDGEWORTH EXPANSION In this section we employ an Edgeworth expansion for the photo- z distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The Edgeworth expansion is an asymptotic expan- sion (in contrast to the Gram-Charlier expansion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Starting from the characteristic function (Blinnikov & Moessner 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' ϕ(j) Z (t) = En( j)(z) � eitZ� , (A1) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' the Fourier transform of the probability density n( j)(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' With the definition of the moments ˜µn, the Taylor expansion of the charac- teristic function is ϕ(j) Z (t) = 1 + ∞ � n1 ˜µn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (it)n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (A2) The logarithm of the characterstic function is the cumulant, κn, gen- erating function κn = 1 in dn dtn log ϕ( j) Z (t) �����t=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (A3) MNRAS 000, 1–9 (2022) functional photo-z 7 Using this definition one can relate the cumulants to the moments κn = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' � {km} (−1)r−1(r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' n � m=1 1 km!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' � ˜µm m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' �km , (A4) where {km} denotes the set of all solutions to the Diophantine equa- tion n � a=1 aka − n = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (A5) If a distribution is then expanded as a asymptotic series around a normal distribution one finds n(z) = 1 √2πκ2 exp � −(z − κ1)2 2κ2 � × � 1 + ∞ � s=1 κs/2 2 � {km} Hes+2r ������� z κ1/2 2 ������� s � m=1 1 km!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' � λm+2 (m + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' �km � ≡ nG(z)(1 + Eg(z)), (A6) where λn � κn/κn/2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We are now interested in the sensitivity of the distribution with respect to its cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Here the cases n = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' 2 are a bit special: ∂n(z) ∂κ1 = n(z)z − κ1 κ2 (A7) and for κ2 ∂n(z) ∂κ2 = 1 2κ1/2 2 �������n(z) �(z − κ1)2 κ2 − 1 � + nG(z)∂Eg(z) ∂κ1/2 2 ������� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (A8) where ∂Eg(z) ∂κ1/2 2 = ∞ � s=1 � {km} κs/2 2 P(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' {km}) � He2r+s ������� z κ1/2 2 ������� × ������� s κ1/2 2 − s � a ka(2a + 2) κka(a+1)+1/2 2 ������� − (2r + s)He2r+s−1 ������� z κ3/2 2 ������� z κ2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (A9) where we also defined the product: P(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' {km}) � s � m=1 1 km!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' � κm+2 (m + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='κ2m+2 2 �km .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (A10) For all cumulants with n ≥ 3 one finds: ∂n(z) ∂κn = nG(z) ∞ � s=1 � {km} κs/2 2 P(s, {km})He2r+s ������� z κ1/2 2 ������� kn−2 κn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (A11) It should be noted, however, that the Edgeworth expansion is not a convergent series but rather an asymptotic expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' One therefore needs to check whether the expansion is a good approximation of the underlying distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In this case one can define the ordinary Fisher matrix using partial derivatives: Fκ(i) m κ( j) n = fsky ℓmax � ℓ=ℓmin 2ℓ + 1 2 tr � ∂Cℓ ∂κ(i) m C−1 ℓ ∂Cℓ ∂κ(j) n C−1 ℓ � , (A12) where κ(i) m is the m-th cumulant of the source-redshift distribution in the i-th tomographic bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Figure A1 shows the fiducial redshift distributions for EU- CLID and their Edgeworth expanded approximations as solid and dashed lines respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The top plot uses the expansion up to κ3, while the bottom plot sums contributions up to κ6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' For all but the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 z 0 1 2 3 4 p(z) order = 3 1 2 3 4 5 6 7 8 9 10 tomographic bin index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 z 0 1 2 3 4 p(z) order = 6 1 2 3 4 5 6 7 8 9 10 tomographic bin index Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' SRD for EUCLID in all 10 tomographic bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Solid lines rep- resent the fiducial SRD, while dashed lines represent their respective Edge- worth expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Cumulants up to order n = 3 , 6 are used respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' first and last tomographic bin, the Edgeworth series is a good ap- proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This is expected as they are essentially Gaussian and therefore κn ≈ 0 for n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The first tomographic bin experiences boundary effects at z = 0 and is therefore slightly skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This ef- fect is even larger for the last tomographic bin, which has a very long tail to high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' While the first bin can still be described by the Edgeworth expansion and the series converges, the 10th bin shows negative probability in the Edgeworth series already at third order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The situation becomes worst if higher order cumulants are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This goes to show that even for such an idealized case as the EUCLID forecast, the use of the Edgeworth expansion can be very dangerous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' For the case n = 2 we show the Pearson correlation coeffi- cient of the joint covariance matrix between the first three cumu- lants in each tomographic bin and four cosmological parameters in Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We observe some correlations between the first and sec- ond moment of each tomographic bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' There is a very strong cor- relation between first and second moment of two different redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Furthermore, one can see that parameters controlling the am- plitude of the lensing spectrum are anti-correlated with the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' We want to stress again, however, that the expansion, even in this case, is not convergent and results obtained with n > 2 have thus to be taken with care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' MNRAS 000, 1–9 (2022) 8 Reischke κ(0) 1 κ(0) 2 κ(1) 1 κ(1) 2 κ(2) 1 κ(2) 2 κ(3) 1 κ(3) 2 κ(4) 1 κ(4) 2 κ(5) 1 κ(5) 2 κ(6) 1 κ(6) 2 κ(7) 1 κ(7) 2 κ(8) 1 κ(8) 2 κ(9) 1 κ(9) 2 Ωm0 σ8 w0 wa κ(0) 1 κ(0) 2 κ(1) 1 κ(1) 2 κ(2) 1 κ(2) 2 κ(3) 1 κ(3) 2 κ(4) 1 κ(4) 2 κ(5) 1 κ(5) 2 κ(6) 1 κ(6) 2 κ(7) 1 κ(7) 2 κ(8) 1 κ(8) 2 κ(9) 1 κ(9) 2 Ωm0 σ8 w0 wa −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='00 rij Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Pearson correlation coefficient for the joint covariance matrix of the first two cumulants of the EUCLID like survey and four cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' APPENDIX B: PHOTOMETRIC GALAXY CLUSTERING For photometric galaxy clustering, the procedure can be simply adopted by changing the weight function (up to galaxy bias, which we absorb in the power spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Again by using the Limber pro- jection: C gig j ℓ = � χH 0 dχ χ2 W(i) g (χ)W(j) g (χ)Pgg �ℓ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 χ , χ � , (B1) with the galaxy power spectrum Pgg and corresponding weights given by: W(i) g (χ) = n(i) g (χ) , (B2) therefore the functional derivative takes the very simple form δC gig j ℓ δna(χ0) = Pgg � ℓ+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 χ , χ � χ2 � n(j)(x)δD ia + n(i)(x)δD ja � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (B3) APPENDIX C: DISTRIBUTION OF THE MEAN AND VARIANCE We show the relative difference of the mean redshift and the stan- dard deviation of the SRD for each tomographic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' As before we dis- tinguish between the EUCLID’s survey settings and KV450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In par- ticular we sample from the diagonal covariance obtained from the functional Fisher matrix as described in Section 3 for the former, while we use the DIR covariance for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The top plots of Figure C1 show the distribution of the mean and the standard deviation and show generally good agreement with Figure 2, that is that the mean must be known below the per-cent level for most bins, while the standard deviation needs to be de- termined by roughly 10 per-cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' It should be noted that Figure 2 considers the extreme case where we exactly look at the envelope shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Finally, the buttom two plots show the same for KV450, where we find much wider errors on mean and standard deviation, few per- cent and a few ten per-cent respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The general trend, how- ever, is the same - high redshift bins are more important than lower redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' APPENDIX D: RELATIONSHIPS TO OBSERVABLES Real surveys usually do not use the angular power spectra as a fi- nal statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' This is for example due to incomplete sky coverage, masking effects, variable depth or simply the dimensionality of the data vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' All these factors require a sufficient summary statis- tic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Very commonly used ones are the correlation function or band powers (or similarly pseudo-Cℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' All of these are essentially linear transformations of the pure angular power spectrum Cℓ and assume the following general form: O[Cℓ] = � dℓCℓWO(ℓ) , (D1) MNRAS 000, 1–9 (2022) functional photo-z 9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='004 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='004 ¯z/⟨¯z⟩ − 1 0 100 200 300 400 500 600 700 1 2 3 4 5 6 7 8 9 10 tomographic bin index −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='20 σz/⟨σz⟩ − 1 0 10 20 30 40 50 1 2 3 4 5 6 7 8 9 10 tomographic bin index −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='15 ¯z/⟨¯z⟩ − 1 0 10 20 30 40 50 KV450 1 2 3 4 5 tomographic bin index −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='4 σz/⟨σz⟩ − 1 0 2 4 6 8 10 12 14 16 KV450 1 2 3 4 5 tomographic bin index Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Distribution of the relative deviation of the mean and the variance of the SRD, n(i) s (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Top: For the EUCLID survey settings with realisations from the inverse of the diagonal Fisher matrix used in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Bottom: For KV450 using the samples generated from the DIR bootstrap covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' where O is some observable of interest and WO(ℓ) is the associ- ated kernel defining the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Again by the chain rule, the functional derivative of this new observable with respect to the SRD is readily available: δO[Cℓ] δn(χ0) = � dx δO δCℓ(x) δCℓ(x) δn(χ0) , (D2) where we dropped all the indices for less clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' For band powers, Cl, this would for example assume the following form: δCl[Cℓ] δn(χ0) = 1 Nl � dℓℓS ℓ δCℓ δn(χ0) , (D3) where S ℓ is the band power response function and Nl is the normal- isation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' For the two-point correlation function ξ± one finds: δξ±(θ) δn(χ0) = 1 2π � dℓℓJ0,4(ℓθ) δCℓ δn(χ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (D4) APPENDIX E: NON-LIMBER Cℓ The Limber projection used for the Cℓ is not valid on large angu- lar scales, where it must be replaced by the full expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' In full generality, for any tracers i and j of the matter density Ci j ℓ = 2 π � dk k2Iℓ,k,i[ni]Iℓ,k,j[ni] , (E1) where the functional Ik,i[ni] is given by Iℓ,k,i[ni] = � dχWi[ni] � Pii(k, χ)jℓ(χk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (E2) Here Pii is the auto power spectrum of the tracer i and Wi is its associated weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Thus we find: δCij ℓ δna = � dk k2 � Iℓ,k,i δIℓ,k,j δna δD ja + Iℓ,k,j δIℓ,k,i δna δD ia � , (E3) where δIℓ,k,i δni = � dχδWi δn � Pii(k, χ) jℓ(χk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' (E4) The derivative of the weight function is calculated as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' APPENDIX F: INTRINSIC ALGINMENTS In this work, we have ignored intrinsic alignments (IA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' Its in- clusion is, however,straight forward by noting that the IA angular power spectrum is simply given by CII ℓ = � χH 0 dχ χ2 n(i) s (χ)n(j) s (χ)PII �ℓ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content='5 χ , χ � , (F1) where PII is the IA power spectrum, which summarises the reac- tion of galaxy shapes to the ambient LSS on the two-point level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' The functional derivative there proceeds in the same way as in Ap- pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' For the GI term of intrinsic alignments, one proceeds as before for cosmic shear (compare Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} +page_content=' MNRAS 000, 1–9 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE2T4oBgHgl3EQfvAgF/content/2301.04085v1.pdf'} diff --git a/_NE1T4oBgHgl3EQfCwKv/content/tmp_files/2301.02869v1.pdf.txt b/_NE1T4oBgHgl3EQfCwKv/content/tmp_files/2301.02869v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4e9c0d9d962f713aaf62888ae8ee837e04f41aff --- /dev/null +++ b/_NE1T4oBgHgl3EQfCwKv/content/tmp_files/2301.02869v1.pdf.txt @@ -0,0 +1,453 @@ + + +The 42nd Asian Conference on Remote Sensing (ACRS2021) +22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam + +Deep Learning-Based UAV Aerial Triangulation without Image Control Points + +Jiageng Zhong1, Ming Li1, Jiangying Qin1, Hanqi Zhang1 +1State Key Laboratory of Information Engineering in Surveying Mapping and Remote Sensing, +Wuhan University, Wuhan 430079 China, +Email: zhongjiageng@whu.edu.cn, lisouming@whu.edu.cn, jy_qin@whu.edu.cn, hqzhang@whu.edu.cn + +KEY WORDS: aerial triangulation; Unmanned Aerial Vehicle; Convolutional Neural Network; image matching + +ABSTRACT: The emerging drone aerial survey has the advantages of low cost, high efficiency, and flexible use. +However, UAVs are often equipped with cheap POS systems and non-measurement cameras, and their flight attitudes +are easily affected. How to realize the large-scale mapping of UAV image-free control supported by POS faces many +technical problems. The most basic and important core technology is how to accurately realize the absolute orientation +of images through advanced aerial triangulation technology. In traditional aerial triangulation, image matching +algorithms are constrained to varying degrees by preset prior knowledge. In recent years, deep learning has developed +rapidly in the field of photogrammetric computer vision. It has surpassed the performance of traditional handcrafted +features in many aspects. It has shown stronger stability in image-based navigation and positioning tasks, especially +it has better resistance to unfavorable factors such as blur, illumination changes, and geometric distortion. Based on +the introduction of the key technologies of aerial triangulation without image control points, this paper proposes a +new drone image registration method based on deep learning image features to solve the problem of high mismatch +rate in traditional methods. It adopts SuperPoint as the feature detector, uses the superior generalization performance +of CNN to extract precise feature points from the UAV image, thereby achieving high-precision aerial triangulation. +Experimental results show that under the same pre-processing and post-processing conditions, compared with the +traditional method based on the SIFT algorithm, this method achieves suitable precision more efficiently, which can +meet the requirements of UAV aerial triangulation without image control points in large-scale surveys. + +1. INTODUCTION + +The UAV-based aerial triangulation using POS (position and orientation system) is extremely rapid and can easily +cover the survey area. The POS provides the measurement of the position and orientation of the camera so that each +image and pixel can be georeferenced to the Earth without the need for image control points, and the most important +and most commonly used data is the position data collected from Global Navigation Satellite Systems (GNSS). +Although the drone aerial survey has the advantages of low cost and high efficiency, it is still a problem to achieve +high accuracy. One important factor that would affect global accuracy is the precision of feature matching which +directly influences the precision of the entire registration. Therefore, accurate features extraction is the basic and key +technique in aerial triangulation. + +The feature extraction consists of keypoint detection and description and has been used in computer vision task for a +long time. The keypoint or feature can be described as a specific meaningful structure, but it is not clear what are the +relevant keypoints for an arbitrary input image (Mukherjee and et al., 2015). The function of a feature detector is to +detect keypoints and their corresponding descriptors. In the past decades, feature detectors have been an activate area +of research. Among the many detectors, SIFT (Scale-Invariant Feature Transform) (Lowe, 2004) is the most +representative and influential one. SIFT aims to solve the image rotation, affine transformations, intensity, and +viewpoint change in matching features (Karami and et al., 2017). It generally includes two major steps. It firstly +convolves the image with Gaussian filters at various scales and finds scale invariant keypoints via estimating a scale +space extreme. Then, for each keypoint, the local image descriptor is computed based on image gradient magnitude +and orientation (Lowe, 2004). And there are many other SIFT-like detectors such as SURF (Speed up Robust Feature) +(Bay and et al., 2008) and ORB (Oriented FAST and Rotated BRIEF) (Rublee and et al., 2011) which are more +efficient than SIFT. + +With the rapid development of deep learning methods and the increasing demand for better feature detection, new +feature detectors emerged, most of which are based on convolutional neural network. Different from the classical +algorithms, deep learning approaches can learn abstract image features from high-dimensional data in an end-to-end +fashion instead of relying on handcrafted features such as distinctive corners (Zeiler and Fergus, 2014). As +convolutional neural networks learn features based on supervision, their performance heavily relies on ground truth +information (Bojanić and et al.,2019). In other word, the key is usually a large dataset of 2D ground truth locations +labeled by human annotators. Unlike these approaches, a novel detector named SuperPoint (DeTone and et al., 2018) +is supervised by itself and works well for matching tasks. + + + +The 42nd Asian Conference on Remote Sensing (ACRS2021) +22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam + + +In this paper, a new aerial survey method without image control points is proposed, and SuperPoint is applied to +replace traditional feature detector in the aerial triangulation flow. Through the multiple perspectives experiment, it +is shown that the new method is able to achieve suitable precision more efficiently, compared to those based on classic +feature detectors. + +2. METHODOLOGY + +2.1 Big Picture + +Referring COLMAP’s incremental Structure-from-Motion pipeline (Schonberger and Frahm, 2016), our method can +be divided into three stages as shown in Figure 1. The first stage is to prepare UAV images and corresponding position +data which contains latitude and longitude location information. Note that the position data should be converted in +Gauss-Kruger Projection. + + +Figure 1. A flow chart of our aerial survey method + +The second stage is correspondence search which finds overlap in the input images and identifies projections of the +same points in overlapping images. For each image, the first step is to extract features which are invariant under +radiometric and geometric changes. In traditional aerial triangulation, SIFT (Lowe, 2004) is applied mostly, here it is +replaced by SuperPoint which can also output L2-normalized fixed length descriptors. As for feature matching, a +matcher that combines Lowe’s ratio test matcher (Lowe, 2004) and Nearest Neighbor search is adopted to improve +the matching accuracy. Based on matched features, images that cover the same scene part are discovered. So the +output of the second stage is a set of potentially overlapping image pairs and their associated feature correspondences +(Schonberger and Frahm, 2016). In addition, there are typically geometric verification that uses projective geometry +to verify the matches. + +The third stage is mainly carried out in four steps. Based on the output of the second stage, new images can be +registered. Next, as newly registered images lead to increase scene coverage, new scene points can be triangulated +and added to the scene structure. Then, considering the possible problem of error accumulation in reconstruction, BA +(Bundle Adjustment) (Triggs and et al., 1999) is applied to refine camera parameters and point parameters via +minimizing the reprojection error. Finally, the outliers are filtered. This iterative strategy can significantly improve +completeness and accuracy (Schonberger and Frahm, 2016). + +Through the workflows above, the aerial survey without image control points is finished. Specifically, all the images +are aligned and a set of sparse point cloud of the survey area is formed. + + +Images +GNSSMeasurement +SuperPointFeatureExtraction +Feature Matching +Features +ImageRegistration +OutlierFiltering +Reconstruction +Triangulation +Bundle Adjustment + +The 42nd Asian Conference on Remote Sensing (ACRS2021) +22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam + +2.2 SuperPoint Network + +SuperPoint is an encoder-decoder architecture and is a fully-convolutional neural network which operates on a full- +sized image. Its structure is shown in Figure 2. Its shared encoder processes the input image at first and branches into +two decoders, one for interest point detection and the other for interest point description. This strategy is quite different +from traditional system which first detects keypoints and then calculates the descriptors. + + +Figure 2. Structure of SuperPoint Network (DeTone and et al., 2018) + +It should be noted that Superpoint adopts self-supervised training strategy. It firstly trains a base detector called +MagicPoint based on supervision from a synthetic dataset, where keypoints can be determined unambiguously. Then +the detector capacity is expanded to real images using homographic adaption. Finally, a keypoint descriptor is +computed by an additional subnetwork. + +3. EXPERIMENTS + +3.1 Data Preparation + +In this section, we present experiment results of the traditional method (based on SIFT) and our method for +comparison. All experiments in this paper are based on two datasets collected from Chongqing. Each dataset contains +a UAV image sequence and corresponding POS data of a scene. The GSD (ground sample distance) of the aerial +images is 0.2 m, and the GNSS standard horizontal and vertical precision is 1 cm and 3 cm respectively. In addition, +there are also ground known points for precision check. To be more specific, for Scene 1 and Scene 2, there are 24 +images and 6 images respectively. The heading overlap and side overlap rate were set to 80% and 60% respectively. +These can meet the specification of topographic mapping at small scale. + +3.2 System Runtime + +The run-time of SuperPoint and SIFT is measured using a RTX 2060 GPU. The SuperPoint architecture is +implemented with Pytorch deep learning library (Paszke and et al., 2019). The average run-time of different +algorithms is measured as shown in Table 1. As the inference of the deep model is done in a single forward propagation +step, the run-time of a single forward pass is measured to be about 148 ms. And SIFT takes about 368 ms to process +one image. It can be seen that SuperPoint executes more efficiently than SIFT and may be applied in real time +surveying. + +Table 1. Mean execution times +Algorithm +SuperPoint +SIFT +Run-time +148 ms +368 ms + +3.3 Feature Extraction and Matching + +Several comparative experiments are carried out for qualitative and quantitative evaluation of the performance of the +keypoint detector and descriptor generator on the datasets. + +The appearance and distribution of keypoints from different detectors are intuitively demonstrated in Figure 3, and +the image is from the dataset of Scene 1. It can be observed that SuperPoint produces fewer feature points compared +to SIFT, and its location distribution is more dispersive. For instance, there is a tract of farmland in the top left of + +InterestPointDecoder +W +Conv +x +W/8 +Input +Encoder +Softmax +Reshape +W +H/8 +H +65 +1 +DescriptorDecoder +W +Conv +H +W/8 +D +Bi-Cubic +L2 +1 +Interpolate +Norm +H/8 +H +D +D + +The 42nd Asian Conference on Remote Sensing (ACRS2021) +22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam + +images, SIFT can hardly extract keypoints, and SuperPoint can extract more well-distributed keypoints. Instead, for +tree or residential areas that have rich texture information, there are more dense points produced by SIFT. + + + +(1) SuperPoint +(2) SIFT +Figure 3. The appearance and distribution of keypoints from different detectors + + +(1) SuperPoint + +(2) SIFT +Figure 4. Qualitative result of matching + +Then, to evaluate the performance of the descriptors, the extracted features are matched by a matcher that combines +Lowe’s ratio test matcher (Lowe, 2004) and Nearest Neighbor search. The ratio test checks if matches are ambiguous +and should be removed, because the probability that a match is correct can be determined by taking the ratio of +distance from the closest neighbor to the distance of the second closest (Lowe, 2004). A qualitative example of +SuperPoint versus SIFT is shown in Figure 4 and the distance ratio is set to 0.7. SuperPoint tends to produce a larger +number of correct matches which densely cover the image while there are several mismatches in the result of SIFT. + +The statistics analysis of the quality of descriptors is performed. Figure 5 shows the match rates and mismatch rates +under different distance ratios for real image data. The match rate is defined as the ratio of matched keypoints to all +keypoints, and the mismatch rate is defined as the ratio of keypoints which are matched falsely to matched keypoints. + + + +The 42nd Asian Conference on Remote Sensing (ACRS2021) +22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam + +Figure 5(a) and 5(b) shows that SuperPoint has higher match rates and lower mismatch rates in most cases. For SIFT +detector, there are too many mismatches to estimate the pose when the distance ratio is greater than 0.8. The ratio is +typically set between 0.5 to 0.8 in application, and in this range, SuperPoint can achieve zero mismatch. Therefore, it +is reasonable to suppose that SuperPoint is able to produce better descriptors. + + + +(a) Match rate +(b) Mismatch rate +Figure 5. The statistical results of feature matching + +Table 2. Relative orientation error +Algorithm +SuperPoint +SIFT +Reprojection Error +0.1454 pixel +0.1008 pixel + +A relative orientation process recreates relative translation and angular relationships between two successive +overlapping images (Tjahjadi and Agustina, 2019). In this paper, the reprojection error of relative orientation is used +as the metric for evaluating the quality of keypoints. Table 2 displays the errors of the image pair in Figure 4. SIFT +performs better on this metric as SuperPoint has a higher reprojection error. This is likely due to the fact that SIFT +performs extra sub-pixel localization, while SuperPoint does not perform this step. + +3.4 Aerial Triangulation + +Using our new aerial survey method described in Section 2, the aerial triangulation without image control points is +carried out on datasets of Scene 1 and Scene 2. The reprojection errors in bundle adjustment are displayed in Table 3, +and the camera position errors are displayed in Table 4. Due to extra sub-pixel localization, the SIFT-based method +reaches higher precision. As for camera position, our method has slightly higher precision, presumably because the +keypoints extracted by SuperPoint distribute more evenly. + +Table 3. Reprojection errors in bundle adjustment + +Our Method +SIFT +Scene 1 +0.387 pixel +0.332 pixel +Scene 2 +0.412 pixel +0.353 pixel + +Table 4. Camera position errors + +ERROR +Our Method +SIFT +Scene 1 +X error +0.603 m +0.530 m +Y error +0.716 m +1.004 m +Z error +0.202 m +0.166 m +XY error +0.936 m +1.136 m +XYZ error +0.958 m +1.148 m +Scene 2 +X error +0.451 m +0.425 m +Y error +0.570 m +0.422 m +Z error +0.791 m +1.043 m +XY error +0.727 m +0.599 m +XYZ error +1.074 m +1.203 m + +MatchRate +0.7 +SuperPoint +0.6 +SIFT +0.5 +0.2 +0.1 +0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Ratio of distances (closest/next closest)MismatchRate +0.14 +SuperPoint +0.12 +SIFT +0.1 +0.04 +0.02 +0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Ratio of distances (closest/next closest) + +The 42nd Asian Conference on Remote Sensing (ACRS2021) +22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam + +Table 5. Error of the checkpoint +ERROR +Our Method +SIFT +X error +-1.821 m +-2.444 m +Y error +-2.217 m +-1.635 m +Z error +-3.755 m +-3.925 m +XY error +2.870 m +2.940 m +XYZ error +4.726 m +4.904 m + +A known point in Scene 2 is used as the checkpoint for precision check, and Table 5 displays the checkpoint error. +The comparison result is consistent with Table 4. The results of experiments illustrate that our method is likely to +reach higher precision compared to traditional SIFT-based method, which confirms that learned representations for +descriptor matching outperform hand-tuned representations. + +4. CONCLUSION + +This paper presents a new aerial survey method without image control points, which adopts SuperPoint as feature +detector. A series of comparative experiments illustrate that our method has obvious advantage in efficiency, keypoint +distribution and matching quality, and it can achieve suitable precision. So, it can be concluded that our method is +capable to meet the application requirements of aerial triangulation. Future work will comprehensively evaluate the +performance of our method with more experiment. + +This paper has shown that the deep learning method outperforms traditional methods in many aspects, therefore, we +can consider that deep learning-based aerial survey would have an expected future. + +ACKNOWLEDGEMENTS + +This research was funded by the National Key R&D Program of China, grant numbers 2018YFB0505400, the +National Natural Science Foundation of China (NSFC), grant number 41901407 and the LIESMARS Special +Research Funding. + +REFERENCES + +Bay, H., Ess, A., Tuytelaars, T. and Van Gool, L. 2008. Speeded-up robust features (SURF). Computer vision and +image understanding, 110(3), pp.346-359. + +Bojanić, D., Bartol, K., Pribanić, T., Petković, T., Donoso, Y. D. and Mas, J. S. 2019. On the comparison of classic +and deep keypoint detector and descriptor methods. In: 2019 11th International Symposium on Image and Signal +Processing and Analysis (ISPA), pp. 64-69. + +DeTone, D., Malisiewicz, T. and Rabinovich, A. 2018. Superpoint: Self-supervised interest point detection and +description. In: Proceedings of the IEEE conference on computer vision and pattern recognition workshops. pp. 224- +236. + +Karami, E., Prasad, S. and Shehata, M. 2017. Image matching using SIFT, SURF, BRIEF and ORB: performance +comparison for distorted images. arXiv preprint arXiv:1710.02726. + +Lowe, D. G. 2004. Distinctive image features from scale-invariant keypoints. International journal of computer vision, +60(2), pp.91-110. + +Mukherjee, D., Wu, Q. J. and Wang, G. 2015. A comparative experimental study of image feature detectors and +descriptors. Machine Vision and Applications, 26(4), pp.443-466. + +Paszke, A., Gross, S. and et al. 2019. Pytorch: An imperative style, high-performance deep learning library. Advances +in neural information processing systems, 32, pp.8026-8037. + +Rublee, E., Rabaud, V., Konolige, K. and Bradski, G. 2011. ORB: An efficient alternative to SIFT or SURF. In: 2011 +International conference on computer vision. pp. 2564-2571. + + + + +The 42nd Asian Conference on Remote Sensing (ACRS2021) +22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam + +Schonberger, J. L. and Frahm, J. M. 2016. Structure-from-motion revisited. In: Proceedings of the IEEE conference +on computer vision and pattern recognition. pp. 4104-4113. + +Tjahjadi, M. E. and Agustina, F. 2019. Fast and stable direct relative orientation of UAV-based stereo pair. +International Journal of Advances in Intelligent Informatics, 5(1), pp.24-39. + +Triggs, B., McLauchlan, P. F., Hartley, R. I. and Fitzgibbon, A. W. 1999. Bundle adjustment—a modern synthesis. +In: International workshop on vision algorithms. pp. 298-372. + +Zeiler, M. D. and Fergus, R. 2014. Visualizing and understanding convolutional networks. In: European conference +on computer vision. pp. 818-833. + diff --git a/_NE1T4oBgHgl3EQfCwKv/content/tmp_files/load_file.txt b/_NE1T4oBgHgl3EQfCwKv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae36870695f2763b881a227d7021054209afe995 --- /dev/null +++ b/_NE1T4oBgHgl3EQfCwKv/content/tmp_files/load_file.txt @@ -0,0 +1,318 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf,len=317 +page_content='The 42nd Asian Conference on Remote Sensing (ACRS2021) 22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam Deep Learning-Based UAV Aerial Triangulation without Image Control Points Jiageng Zhong1, Ming Li1, Jiangying Qin1, Hanqi Zhang1 1State Key Laboratory of Information Engineering in Surveying Mapping and Remote Sensing, Wuhan University, Wuhan 430079 China, Email: zhongjiageng@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='cn, lisouming@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='cn, jy_qin@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='cn, hqzhang@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='cn KEY WORDS: aerial triangulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Unmanned Aerial Vehicle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Convolutional Neural Network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' image matching ABSTRACT: The emerging drone aerial survey has the advantages of low cost, high efficiency, and flexible use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' However, UAVs are often equipped with cheap POS systems and non-measurement cameras, and their flight attitudes are easily affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' How to realize the large-scale mapping of UAV image-free control supported by POS faces many technical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The most basic and important core technology is how to accurately realize the absolute orientation of images through advanced aerial triangulation technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' In traditional aerial triangulation, image matching algorithms are constrained to varying degrees by preset prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' In recent years, deep learning has developed rapidly in the field of photogrammetric computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' It has surpassed the performance of traditional handcrafted features in many aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' It has shown stronger stability in image-based navigation and positioning tasks, especially it has better resistance to unfavorable factors such as blur, illumination changes, and geometric distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Based on the introduction of the key technologies of aerial triangulation without image control points, this paper proposes a new drone image registration method based on deep learning image features to solve the problem of high mismatch rate in traditional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' It adopts SuperPoint as the feature detector, uses the superior generalization performance of CNN to extract precise feature points from the UAV image, thereby achieving high-precision aerial triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Experimental results show that under the same pre-processing and post-processing conditions, compared with the traditional method based on the SIFT algorithm, this method achieves suitable precision more efficiently, which can meet the requirements of UAV aerial triangulation without image control points in large-scale surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' INTODUCTION The UAV-based aerial triangulation using POS (position and orientation system) is extremely rapid and can easily cover the survey area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The POS provides the measurement of the position and orientation of the camera so that each image and pixel can be georeferenced to the Earth without the need for image control points, and the most important and most commonly used data is the position data collected from Global Navigation Satellite Systems (GNSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Although the drone aerial survey has the advantages of low cost and high efficiency, it is still a problem to achieve high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' One important factor that would affect global accuracy is the precision of feature matching which directly influences the precision of the entire registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Therefore, accurate features extraction is the basic and key technique in aerial triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The feature extraction consists of keypoint detection and description and has been used in computer vision task for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The keypoint or feature can be described as a specific meaningful structure, but it is not clear what are the relevant keypoints for an arbitrary input image (Mukherjee and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The function of a feature detector is to detect keypoints and their corresponding descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' In the past decades, feature detectors have been an activate area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Among the many detectors, SIFT (Scale-Invariant Feature Transform) (Lowe, 2004) is the most representative and influential one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' SIFT aims to solve the image rotation, affine transformations, intensity, and viewpoint change in matching features (Karami and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' It generally includes two major steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' It firstly convolves the image with Gaussian filters at various scales and finds scale invariant keypoints via estimating a scale space extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Then, for each keypoint, the local image descriptor is computed based on image gradient magnitude and orientation (Lowe, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' And there are many other SIFT-like detectors such as SURF (Speed up Robust Feature) (Bay and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=', 2008) and ORB (Oriented FAST and Rotated BRIEF) (Rublee and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=', 2011) which are more efficient than SIFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' With the rapid development of deep learning methods and the increasing demand for better feature detection, new feature detectors emerged, most of which are based on convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Different from the classical algorithms, deep learning approaches can learn abstract image features from high-dimensional data in an end-to-end fashion instead of relying on handcrafted features such as distinctive corners (Zeiler and Fergus, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' As convolutional neural networks learn features based on supervision, their performance heavily relies on ground truth information (Bojanić and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=',2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' In other word, the key is usually a large dataset of 2D ground truth locations labeled by human annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Unlike these approaches, a novel detector named SuperPoint (DeTone and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=', 2018) is supervised by itself and works well for matching tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The 42nd Asian Conference on Remote Sensing (ACRS2021) 22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam In this paper, a new aerial survey method without image control points is proposed, and SuperPoint is applied to replace traditional feature detector in the aerial triangulation flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Through the multiple perspectives experiment, it is shown that the new method is able to achieve suitable precision more efficiently, compared to those based on classic feature detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' METHODOLOGY 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='1 Big Picture Referring COLMAP’s incremental Structure-from-Motion pipeline (Schonberger and Frahm, 2016), our method can be divided into three stages as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The first stage is to prepare UAV images and corresponding position data which contains latitude and longitude location information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Note that the position data should be converted in Gauss-Kruger Projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' A flow chart of our aerial survey method The second stage is correspondence search which finds overlap in the input images and identifies projections of the same points in overlapping images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' For each image, the first step is to extract features which are invariant under radiometric and geometric changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' In traditional aerial triangulation, SIFT (Lowe, 2004) is applied mostly, here it is replaced by SuperPoint which can also output L2-normalized fixed length descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' As for feature matching, a matcher that combines Lowe’s ratio test matcher (Lowe, 2004) and Nearest Neighbor search is adopted to improve the matching accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Based on matched features, images that cover the same scene part are discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' So the output of the second stage is a set of potentially overlapping image pairs and their associated feature correspondences (Schonberger and Frahm, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' In addition, there are typically geometric verification that uses projective geometry to verify the matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The third stage is mainly carried out in four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Based on the output of the second stage, new images can be registered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Next, as newly registered images lead to increase scene coverage, new scene points can be triangulated and added to the scene structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Then, considering the possible problem of error accumulation in reconstruction, BA (Bundle Adjustment) (Triggs and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=', 1999) is applied to refine camera parameters and point parameters via minimizing the reprojection error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Finally, the outliers are filtered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' This iterative strategy can significantly improve completeness and accuracy (Schonberger and Frahm, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Through the workflows above, the aerial survey without image control points is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Specifically, all the images are aligned and a set of sparse point cloud of the survey area is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Images GNSSMeasurement SuperPointFeatureExtraction Feature Matching Features ImageRegistration OutlierFiltering Reconstruction Triangulation Bundle Adjustment The 42nd Asian Conference on Remote Sensing (ACRS2021) 22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='2 SuperPoint Network SuperPoint is an encoder-decoder architecture and is a fully-convolutional neural network which operates on a full- sized image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Its structure is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Its shared encoder processes the input image at first and branches into two decoders, one for interest point detection and the other for interest point description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' This strategy is quite different from traditional system which first detects keypoints and then calculates the descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Structure of SuperPoint Network (DeTone and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=', 2018) It should be noted that Superpoint adopts self-supervised training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' It firstly trains a base detector called MagicPoint based on supervision from a synthetic dataset, where keypoints can be determined unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Then the detector capacity is expanded to real images using homographic adaption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Finally, a keypoint descriptor is computed by an additional subnetwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' EXPERIMENTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='1 Data Preparation In this section, we present experiment results of the traditional method (based on SIFT) and our method for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' All experiments in this paper are based on two datasets collected from Chongqing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Each dataset contains a UAV image sequence and corresponding POS data of a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The GSD (ground sample distance) of the aerial images is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='2 m, and the GNSS standard horizontal and vertical precision is 1 cm and 3 cm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' In addition, there are also ground known points for precision check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' To be more specific, for Scene 1 and Scene 2, there are 24 images and 6 images respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The heading overlap and side overlap rate were set to 80% and 60% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' These can meet the specification of topographic mapping at small scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='2 System Runtime The run-time of SuperPoint and SIFT is measured using a RTX 2060 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The SuperPoint architecture is implemented with Pytorch deep learning library (Paszke and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The average run-time of different algorithms is measured as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' As the inference of the deep model is done in a single forward propagation step, the run-time of a single forward pass is measured to be about 148 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' And SIFT takes about 368 ms to process one image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' It can be seen that SuperPoint executes more efficiently than SIFT and may be applied in real time surveying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Mean execution times Algorithm SuperPoint SIFT Run-time 148 ms 368 ms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='3 Feature Extraction and Matching Several comparative experiments are carried out for qualitative and quantitative evaluation of the performance of the keypoint detector and descriptor generator on the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The appearance and distribution of keypoints from different detectors are intuitively demonstrated in Figure 3, and the image is from the dataset of Scene 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' It can be observed that SuperPoint produces fewer feature points compared to SIFT, and its location distribution is more dispersive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' For instance, there is a tract of farmland in the top left of InterestPointDecoder W Conv x W/8 Input Encoder Softmax Reshape W H/8 H 65 1 DescriptorDecoder W Conv H W/8 D Bi Cubic L2 1 Interpolate Norm H/8 H D D The 42nd Asian Conference on Remote Sensing (ACRS2021) 22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam images, SIFT can hardly extract keypoints, and SuperPoint can extract more well-distributed keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Instead, for tree or residential areas that have rich texture information, there are more dense points produced by SIFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' (1) SuperPoint (2) SIFT Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The appearance and distribution of keypoints from different detectors (1) SuperPoint (2) SIFT Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Qualitative result of matching Then, to evaluate the performance of the descriptors, the extracted features are matched by a matcher that combines Lowe’s ratio test matcher (Lowe, 2004) and Nearest Neighbor search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The ratio test checks if matches are ambiguous and should be removed, because the probability that a match is correct can be determined by taking the ratio of distance from the closest neighbor to the distance of the second closest (Lowe, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' A qualitative example of SuperPoint versus SIFT is shown in Figure 4 and the distance ratio is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' SuperPoint tends to produce a larger number of correct matches which densely cover the image while there are several mismatches in the result of SIFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The statistics analysis of the quality of descriptors is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Figure 5 shows the match rates and mismatch rates under different distance ratios for real image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The match rate is defined as the ratio of matched keypoints to all keypoints, and the mismatch rate is defined as the ratio of keypoints which are matched falsely to matched keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The 42nd Asian Conference on Remote Sensing (ACRS2021) 22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam Figure 5(a) and 5(b) shows that SuperPoint has higher match rates and lower mismatch rates in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' For SIFT detector, there are too many mismatches to estimate the pose when the distance ratio is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The ratio is typically set between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='8 in application, and in this range, SuperPoint can achieve zero mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Therefore, it is reasonable to suppose that SuperPoint is able to produce better descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' (a) Match rate (b) Mismatch rate Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The statistical results of feature matching Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Relative orientation error Algorithm SuperPoint SIFT Reprojection Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='1454 pixel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='1008 pixel A relative orientation process recreates relative translation and angular relationships between two successive overlapping images (Tjahjadi and Agustina, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' In this paper, the reprojection error of relative orientation is used as the metric for evaluating the quality of keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Table 2 displays the errors of the image pair in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' SIFT performs better on this metric as SuperPoint has a higher reprojection error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' This is likely due to the fact that SIFT performs extra sub-pixel localization, while SuperPoint does not perform this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='4 Aerial Triangulation Using our new aerial survey method described in Section 2, the aerial triangulation without image control points is carried out on datasets of Scene 1 and Scene 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The reprojection errors in bundle adjustment are displayed in Table 3, and the camera position errors are displayed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Due to extra sub-pixel localization, the SIFT-based method reaches higher precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' As for camera position, our method has slightly higher precision, presumably because the keypoints extracted by SuperPoint distribute more evenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Reprojection errors in bundle adjustment Our Method SIFT Scene 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='387 pixel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='332 pixel Scene 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='412 pixel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='353 pixel Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Camera position errors ERROR Our Method SIFT Scene 1 X error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='603 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='530 m Y error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='716 m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='004 m Z error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='202 m 0.' metadata={'source': 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+page_content='425 m Y error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='570 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='422 m Z error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='791 m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='043 m XY error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='727 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='599 m XYZ error 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='074 m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='203 m MatchRate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='7 SuperPoint 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='6 SIFT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='1 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='9 1 Ratio of distances (closest/next closest)MismatchRate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='14 SuperPoint 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='12 SIFT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='02 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='9 Ratio of distances (closest/next closest) The 42nd Asian Conference on Remote Sensing (ACRS2021) 22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Error of the checkpoint ERROR Our Method SIFT X error -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='821 m -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='444 m Y error -2.' metadata={'source': 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4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content='904 m A known point in Scene 2 is used as the checkpoint for precision check, and Table 5 displays the checkpoint error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The comparison result is consistent with Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' The results of experiments illustrate that our method is likely to reach higher precision compared to traditional SIFT-based method, which confirms that learned representations for descriptor matching outperform hand-tuned representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' CONCLUSION This paper presents a new aerial survey method without image control points, which adopts SuperPoint as feature detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' A series of comparative experiments illustrate that our method has obvious advantage in efficiency, keypoint distribution and matching quality, and it can achieve suitable precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' So, it can be concluded that our method is capable to meet the application requirements of aerial triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' Future work will comprehensively evaluate the performance of our method with more experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' This paper has shown that the deep learning method outperforms traditional methods in many aspects, therefore, we can consider that deep learning-based aerial survey would have an expected future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This research was funded by the National Key R&D Program of China, grant numbers 2018YFB0505400, the National Natural Science Foundation of China (NSFC), grant number 41901407 and the LIESMARS Special Research Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'} +page_content=' REFERENCES Bay, H.' metadata={'source': 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USA +TAUHEED KHAN MOHD, Augustana College, USA +In recent times, voice assistants have become a part of our day-to-day lives, allowing information retrieval by voice synthesis, voice +recognition, and natural language processing. These voice assistants can be found in many modern-day devices such as Apple, Amazon, +Google, and Samsung. This project is primarily focused on Virtual Assistance in Natural Language Processing. Natural Language +Processing is a form of AI that helps machines understand people and create feedback loops. This project will use deep learning +to create a Voice Recognizer and use Commonvoice and data collected from the local community for model training using Google +Colaboratory. After recognizing a command, the AI assistant will be able to perform the most suitable actions and then give a response. +The motivation for this project comes from the race and gender bias that exists in many virtual assistants. The computer industry +is primarily dominated by the male gender, and because of this, many of the products produced do not regard women. This bias has an +impact on natural language processing. This project will be utilizing various open-source projects to implement machine learning +algorithms and train the assistant algorithm to recognize different types of voices, accents, and dialects. Through this project, the goal +to use voice data from underrepresented groups to build a voice assistant that can recognize voices regardless of gender, race, or accent. +Increasing the representation of women in the computer industry is important for the future of the industry. By representing +women in the initial study of voice assistants, it can be shown that females play a vital role in the development of this technology. In +line with related work, this project will use first-hand data from the college population and middle-aged adults to train voice assistant +to combat gender bias. +Additional Key Words and Phrases: Voice Assistance, Machine Learning, Virtual Assistance, Artificial Intelligence, Selection Bias, +Sample Population, Python 3.10, Pyttsx3, PyTorch, JSON +1 +INTRODUCTION +The first-ever voice-activated consumer product was released to the public in 1922. It was known as “Radio Rex.” This +product was a toy that had a doghouse with a dog inside it. When someone said “Rex” next to the dog house, the dog +would jump out of the doghouse. This voice-activated toy was created even before modern computers existed.[1] +Authors’ addresses: Kashav Piya, kashavpiya19@augustana.edu, Augustana College, 639 38th St, Rock Island, Illinois, USA, 61201; Srijal Shrestha, +srijalshrestha18@augustana.edu, Augustana College, 639 38th St, Rock Island, Illinois, USA, 61201; Cameran Frank, cameranfrank18@augustana.edu, +Augustana College, 639 38th St, Rock Island, Illinois, USA, 61201; Estephanos Jebessa, estephanosjebessa19@augustana.edu, Augustana College, 639 38th +St, Rock Island, Illinois, USA, 61201; Tauheed Khan Mohd, tauheedkhanmohd@augustana.edu, Augustana College, 639 38th St, Rock Island, Illinois, USA, +61201. +Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not +made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components +of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to +redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. +© 2023 Association for Computing Machinery. +Manuscript submitted to ACM +Manuscript submitted to ACM +1 +arXiv:2301.00646v1 [eess.AS] 20 Dec 2022 + +2 +Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd +Since the development of that toy, there has been a considerable amount of development in voice recognition, natural +language processing, and machine learning. A voice assistant, also known as an intelligent personal assistant or a +connected speaker, is based on natural language speech recognition. Recently it has had a rise in popularity and has +been marketed and used by Apple, Amazon, Google, and Samsung. Now voice assistants are widely found in most +modern-day devices that a person would use. +Voice assistants are multi-purposed one of their main purposes was for a search to be carried out using a voice +command entered by the user as an input. They are also known to be used for information retrieval by voice synthesis. +They use a variety of voice recognition techniques, language processing algorithms, and voice synthesis to listen to +specific voice commands that may include wake words, tasks, and queries, and return relevant information or perform +a specific function as requested by the user. These assistants can be software-based which allows them to be integrated +into a wide range of devices such as laptops, mobile devices, and speakers, or can be specifically designed into a +standalone device like Amazon Echo or Amazon Alexa Wall Clock. [2] +These voice assistants work like a charm and are quite fascinating, making one might ask themselves what goes on +behind the hood of these amazing innovations or how do they work the way they do? +To answer the above query, in short voice assistants use artificial intelligence and voice recognition to deliver the +result that the user is looking for efficiently, and precisely. The user provides a command to the voice assistant that +is called intent. Through voice recognition, these intentions can be understood by our virtual assistants. Here, voice +recognition allows the speaker to speak into a device that takes the analog signal from the speaker and changes it into +a digital signal which is then processed by the computer to match it with words or phrases and then recognize the +command. Machine learning also has a huge part to play in this as the computer needs to be taught to recognize the +speaker’s words by feeding it a database of words and syllables in each language to match it with digital signals. This +process is known as pattern recognition. Additionally, these devices gather a lot of information from the commands +that they received previously to improve upon themselves using machine learning.[3] +There are multiple approaches to voice assistants, specifically two types: task-oriented and knowledge-oriented. +Most voice assistants these days can combine both the task-oriented as well as a knowledge-oriented workflow to +complete all the tasks that a user may ask the voice assistant to carry out. A task-oriented approach will most likely ask +something as simple as filling out a form, whereas a knowledge-based approach may include answering questions such +as who the President of the United States of America is or finding out what engine is in a Ford F50 which is a technical +specification of a product. [4] +The task-oriented approach/workflow is pretty much self-explanatory as it uses goals and tasks to achieve whatever +the user wants or needs. This approach usually requires the voice assistant to use a different application such as time, +weather, web browser, and music apps, to help complete its tasks. Some examples would be, asking a voice assistant to +set a reminder to take medicine at 6 PM, playing music using Spotify, etc. This approach does not require the virtual +voice assistant to search massive databases for knowledge. These tasks are often known as skills. And various assistants +allow for different skills to be installed according to the user’s preferences. [5] +Whereas a knowledge-oriented approach/workflow requires the use of analytical data to complete the tasks and help +the users to complete their tasks. [6] Unlike a task-oriented approach, this approach focuses on using online databases +to get related information in addition to already recorded knowledge to help users to complete tasks. An example of a +knowledge-based approach would be if a user asked for a question that would require searching the internet such as +what is the capital of the state of Illinois or who invented the telephone? +Manuscript submitted to ACM + +Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection +3 +Furthermore, there are two types of artificial intelligence (AI); In general, there is a weak AI and there is a strong +AI.[7] There are many types of machines such as Siri, Alexa, Cortana, and Bixby that can only perform certain tasks +that have been defined by the user while making the AI. These types of AIs are called Weak AI. And some machines or +systems have a mind of their own and can make decisions or take actions on their own without human interference. +These types of machines are called Strong AI. After learning the differences between strong and weak AI, the voice +assistant this project is opting for is an example of weak AI. +This projects Virtual Voice Assistant will include a variety of features such as greeting the users, fetching information +about a person, an object, or anything else in general from the internet, providing the time, opening web browsers, +playing music, and so on. It might also include additional features such as opening the web camera to take pictures, +forecasting the weather, logging off from your personal computer, telling you a joke, and many other features. +The field of Virtual Assistance has many avenues to consider from providing help in technology to connecting +people through the usage of technology. When studying what exactly a Virtual Assistant is, the field that was decided +on was Virtual Assistance in natural language processing, which means the technology can understand people more +accurately. Natural Language Processing is a form of AI that gives machines the ability to not just read but to understand +and interpret human language. With NLP, machines can make sense of the written or spoken text and perform tasks +including speech recognition, sentiment analysis, and automatic text summarization [8]. Therefore, not only does +natural language processing help humans it also helps with machine learning, in the sense that NLP will continue to +provide more data to better that analysis of speech and create a feedback loop. +The English language is an extremely hard language to understand and speak, especially if English is not your first +language. Not only is the usage of English vernacular hard to comprehend and execute, but there is also a form of a +language barrier in the different dialects that people possess. A person’s dialect can be a communication inhibitor in +many languages, not just in English, but in this project, the focus is on the English language [9]. +For this project, synthetic voices were originally used instead of human voices, in which data was collected. A +synthetic voice is a pre-recorded voice produced through text to speech whereas the human voice is pre-recorded. +‘Its use involves recording, in advance, a text read aloud by a human being.’ The usage of a synthetic voice will give +flexibility as they have a high capacity in reading textual context and can generate voice constantly. But “there is a +disadvantage in expressing social signs as it cannot express emotions, intentions, and attitudes through modulation of +the voice” [10]. +The computer industry is primarily dominated by the male gender and because of this extreme one-sided representa- +tion in the field a multitude of the products that are produced do not regard women. One of the fields that this bias +impacts is natural language processing. Because of the lack of females in the industry, the identification percentage +for female voices is lower than that of male voices, therefore resulting in the analysis and research of this topic of +selection bias of voice assistance [11]. Data augmentation by controlling the gender attribute is an effective technique +in mitigating gender bias in NLP processes. +2 +RELATED WORK +2.1 +How Does Voice Assistant Work? +Voice assistance has now been defined is, but how does it work? A voice assistant uses speech recognition along with +other identification of speech components to help the machine process the voice. Then, the speech is rendered into its +textual representation based on extracted patterns. Following that, this program isolates the most important words +Manuscript submitted to ACM + +4 +Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd +Fig. 1. How Does Voice Assistant Work? +or the action also known as anticipated intent. If the intent is not clear, the voice assistant is programmed to ask +more questions. It then retrieves information by API calls to access the relevant knowledge base. Finally, it relays the +information back to the human user through text to speech or fulfills the necessary action. Voice assistants rely on +Natural Language Processing and other machine learning algorithms to perform the best and overcome the challenges +that it faces. [12] +2.2 +Speech Recognition +How does a voice assistant understand what a person is talking about? These voice assistants use fast decoding +algorithms that allow real-time continuous speech recognition systems to provide instant responses.[13] Speech +Recognition has been a part of our day-to-day life for more than 10 years now as it seems to become more and more +advanced. The beauty of speech-to-text models goes unnoticed in this process as the machines make it look so seamless. +Most speech recognition algorithms use a mix of Natural Language Processing and Deep Learning techniques to parse +through the user query, get an appropriate response and present it back to the user in whichever form the user desires. +All the big companies such as Google’s Assistant, Amazon’s Alexa, Apple’s Siri, and others use the same techniques but +just with some different variations.[14] +Manuscript submitted to ACM + +Voice Assistant +5. +Human User +NLP +What is the +APIs +weather for +3. +tomorrow? +4.Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection +5 +2.2.1 +Signal Processing for Speech Recognition. Audio signals are any object that vibrates to produce sound waves. +When an object vibrates, the air molecules oscillate to and from their rest position and transmits its energy to the +neighboring molecules. This results in the transmission of energy from one molecule to another which in turn produces +a sound wave. [15] +There are a few terms that should be familiar when talking about sound and signal processing. +• Amplitude: Amplitude refers to the maximum displacement of the air molecules from the rest position +• Crest and Trough: The crest is the highest point in the wave whereas trough is the lowest point +• Wavelength: The distance between two successive crests or troughs is known as a wavelength +• Cycle: Every audio signal traverses in the form of cycles. One complete upward movement and downward +movement of the signal form a cycle +• Frequency: Frequency refers to how fast a signal is changing over a period +Additionally, there are two different types of signals: Digital and Analog Signal. For digital signal is a discrete +representation of a signal over a period. Here, the finite number of samples exists between any two-time intervals +whereas the analog signal is a continuous representation of a signal over a period which implies that there will be an +infinite number of samples between any two given time intervals. [16] +For this project, audio signals are needed for the Voice Assistant, so there is a question about how it stores a +signal that has an infinite number of samples since they are analog signals. This project will incorporate changing the +memory-hogging analog signal using complex techniques to convert it into digital signals to make it more convenient +to work with them. +To convert a signal from analog to digital, a technique called sampling the signal should be used which selects a +certain number of samples per second from the analog signal which makes storing and processing the signal memory +efficient. In analog to digital sampling "an input signal is converted from some continuously varying physical value (e.g. +pressure in air, or frequency or wavelength of light), by some electro-mechanical device into a continuously varying +electrical signal." [17] +2.2.2 +Feature Extraction Techniques for Audio signals. For this projects model to use the audio signals, the features +must be extracted from the audio such as time domain and frequency domain. The audio signal is representing by the +amplitude as a function of time. The features here are the amplitudes which are recorded at different time intervals. On +the other hand, in the frequency domain, the audio signal is represents amplitude as a function of frequency where the +features are amplitudes that are recorded at different frequencies. +2.3 +Papers Regarding this Topic +In machine learning when one trains their model, there is always a chance for problems when applying the model to the +population. This problem arises in big data when the population is too large for machines with limited computability. +To deal with this problem of big data, a sample population can be used. A sample population refers to a subset of the +population that represents the whole population. [18] This project will apply machine learning to samples population. +But there is a chance for selection bias, “selection bias is a systematic error that results in differences between a study +population and a target population; selection bias primarily affects the external validity of the results of a study”. +This means that the results thought to be true for the sample population may not be true for the actual population. +Depending on the population sample, if there are too many true results it will just show true no matter what. Sample +population can technically be wrong from the population just because of the sample collected. +Manuscript submitted to ACM + +6 +Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd +Likewise, voice assistants use sample data to analyze a user’s speech. “Currently speech recognition has significant +race and gender biases. It is just another form of AI that performs worse for women and non-white people. Currently, it +is designed to understand white male voices well” [19]. This is a significant problem because these days voice assistants +are a crucial part of people’s lives from setting alarms to transportation. Low accuracy in voice recognition would mean +severe consequences in people’s life [19]. In the paper Empirical Analysis of Bias in Voice-based Personal Assistants, +they try to check the accuracy of relevant voice assistants such as Google Assistant and Siri. They look at the different +accents for the Brazilian Portuguese, and how accuracy was off for certain accents than other ones. They also found +variation in the quality of recognition based on gender. [20] +So, taking the two paper’s ideas, to tackle this problem and move forward in this topic, data was gathered and training +our model not only on prevalent male voice data sets but also gathering voices manually and from Common-Voice +data sets by Mozilla for female and minority voices. This allows us to create samples proportionally to demographic +indicators of the country. The process for machine learning will be as follows: +• Filter the words that the user says +• Digitize the user’s speech into a format that the machine can read +• Analyze the user’s speech for meaning +• Decide what the user needs based on previous input and algorithms” +As talked above, data sets for voice recognition must be sampled so that each voice has an equal probability to be +included in the sample. This allows for less racial and gender bias for the models to work with. So, this will result in +everyone having their voice heard. +Additionally the paper “Dangerous Skills: Understanding and Mitigating Security Risks of Voice-Controlled Third- +Party Functions on Virtual Personal Assistant Systems” [21] and “Your Voice Assistant is Mine: How to Abuse Speakers +to Steal Information and Control Your Phone” talks about the development of voice assistant in speech recognition and +IoTs and with its development comes more vulnerabilities. They talk about voice-based remote attacks and permission +bypassing. These problems are extremely dangerous and can be misused to expose people’s information. So, these +papers advise incorporating the ideas of not allowing zero permission and context-aware information collection and +analysis features to the voice assistant. Not allowing those ideas will allow focusing on forcing restrictions on specific +operations that a particular process can perform [22]. +3 +EXPERIMENTAL SETUP +The data used is partially from Common-Voice, which is an online open source of voice recorders in multiple languages. +However, since this project is specifically for the English language the data set collected was English. However, this +data set, as expected has more male voices than female voices. Therefore, to even out the distribution between the +voices represented the rest of the data was collected via outreach into the Augustana College community, contacting +over 200 female students (ages ranging from 18 - 24 years old) and receiving about 65 voices samples with 4-7 minutes +of voice recording from each sample. The data set collected was then doubled, converted, and combined into .wav files. +After the data collection and manipulated, it was then applied to a gender recognition model in order to numerically +identify via the frequency of each voice; according to ASHA (American Speech Language and Hearing Association) the +average range for an adult woman is 165 to 255 Hz and the average range for adult males is 85 to 155 Hz [23]. There is +inherent technical problem down to the fact that females generally have higher pitched voices. Female voices tend to be +Manuscript submitted to ACM + +Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection +7 +quieter and sound more “breathy”. Female more easily masked by noise, like a fan or traffic in the background, which +makes it harder for speech recognition systems. +In order to run the data through training model for speech recognition. The full data set from Common-Voice was +not needed in fact there only segments of the a statement is used, therefore it was necessary to parse the data into +smaller form. +4 +FRAMEWORK +For Machine Learning, Python is known to be the most common language used due to its versatility, readability, and +abundant packages. So, for this project, it will be using Python 3.10 as the main programming language. This project +will be using the following python packages or libraries: +• Speech Recognition (Will only be used initially to create and test the basic functionalities of the project and will +be later replaced by this projects own model): It helps understand what the human is saying and converts the +speech into text. +• Pyttsx3: It is a simple text to speech conversion library in python which will be used to give this projects Voice +Assistant a voice. +• Wikipedia: This package is a Python that extracts information and data from Wikipedia which is a multilingual +online encyclopedia used by many people. +• Capture: It helps to capture images from your camera. +• Datetime: It is an inbuilt module in python that works with date and time. +• Os: It provides functions to interact with the operating system. +• Web Browser: It is an in-build package from Python that allows you to extract information from the web. +• JSON: It is a module that helps to store and exchange data. +• PyJokes: It gives the user a random joke. +• Pyaudio: It allows python to play and record audio between different platforms. +• Pywhatkit: It can access YouTube in order to play a video. +• Librosa: It is a package for analysing sound, audio and music. +• Soundfile: This package can help read and write sound files. +• Numpy: Numpy provides a powerful N-dimensional array object, sophisticated functions, tolls for integrating +C/C++ and Fortran code, useful linear algebra, Fourier transform, and random number capabilities, and much +more. +• BeautifulSoup: Beautiful Soup is a Python library for pulling data out of HTML and XML files. +• Pandas: pandas is a fast, powerful, flexible and easy to use open source data analysis and manipulation tool, built +on top of the Python programming language. +This project will be using all these modules and packages to create basic functionalities of the Voice Assistant. For +example, this project will use the Wikipedia package to get information regarding a person or a company. This function +is used to grab the first few sentences from its corresponding Wikipedia page which is generally the broad introduction +to the subject. +Then this project will have a Voice Recognizer which will replace the existing Speech Recognition module as well +as the Google API. The program uses deep learning to create this projects voice recognition using PyTorch. The data +is from open sources Mozilla Common-Voice as well as a few more from Kaggle to train this projects model and add +Manuscript submitted to ACM + +8 +Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd +voices from volunteers to help balance the data to reduce the biases while training for the wake word as well as the +general voice recognition itself. This project utilizes Google Colaboratory to train this projects model. Finally, if time +permits, the goal is to build a simple User Interface either with Flask or with other python libraries itself for a visual +element to the program. +5 +METHODS +As stated in the experimental setup, the Common-Voice data set planned to be used for the training model has recognized +46% male voices and 16% female voices. Therefore, the goal of this project is to come up with an effective solution to +this disparity. +When training a speech recognition model, different types of voice data can be used. Depending on the type of +interaction one’s looking to build, and how robust that interaction should be, different types of voice data might +be required. Although there are several easily available sources of speech data, such as public speech corpora or +pre-packaged data sets, it is almost always necessary to cooperate with a data services provider to collect your own +speech data, either remotely or in person. When you gather one’s own data, it is easy to tailor the speech data set to +include variables such as language, speaker demographics, audio requirements, and collection size. [24] +The Bureau of Labor Statistics (BLS) projects computer science research jobs will grow 19 percent by 2026. However, +in the United States, the percentage of women that receive a bachelor’s degree in Computer Science is still only 18 +percent. There is currently a high demand for computer scientists in the professional industry but despite this fact, this +industry remains male-dominated, in the United States. For example, in this year’s Senior Inquire class for Computer +Science, there is a 1:6 female to male ratio [25]. +Computer technology first emerged during World War II and continuing into the 1960s, women made up most of the +computing workforce. However, by 1970 women only accounted for about 14 percent of bachelor’s in computer science. +In 1984 that number rose to 37 percent. The percentage of women in computer science has since declined to 18 percent. +It was around the same time personal computers started showing up in homes. According to NPR, personal computers +were marketed almost exclusively to men and families were more likely to buy computers for boys than girls. +Computers are now commonplace in both classrooms and on individuals as personal assistance. It is hard, however, to +explain the exact reason why females are not as present in this major. There are organizations now that are researching +and improving ways to increase the potential of more females in the computer science major. It is said that one of the +reasons why women tend to trend away from the computer science field is because of marketing of the industry in the +past tailored to those with the geek persona and the social innuendos of what being a geek used to mean. [26] +One of the reasons why females should be represented in the computer industry is increasing the inclusion of women +is a sound business strategy. “A study by Deloitte found that women’s choices account for up to 85 percent of buying +decisions nationwide, and that diversity drives innovation. Though it is still commonplace to find boards and project +teams without a female member, the integration of female perspectives will naturally lead to higher revenues and a +better understanding of consumer marketplaces.” [27] +Therefore, the purpose of this project is to strive to increase the attractiveness of the potential that computer science +avenues provide for women, by increasing the representation of females in the initial study of voice assistants. If females +can realize that they possess a more essential in the initial makeup that products that are generated there is a possibility +that females will be more inclined into joining the computer science industry, by showing them that they do in fact +play an essential role in the make of technology. +Speech recognition data can be classified into three categories: +Manuscript submitted to ACM + +Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection +9 +• Controlled: Scripted speech data +• Semi-controlled: Scenario-based speech data +• Natural: Unscripted or conversational speech data +For this project, the primary type of voice data used is Semi-controlled data and natural unscripted conversational +speech data. For semi-controlled data, When developers need a natural sampling of different ways to ask for the same +thing or a greater diversity of command intentions, scenario-based voice data is collected (i.e. asking for different +things). As a result, scenario-based speech data adds variety to what is said as well as how it is said. +On the other hand, The most "natural" kind of speech is unscripted or conversational speech data, which is a recording +of a conversation between two or more speakers. Unscripted speech data, like spontaneous speech, occurs in a variety +of formats in the real world. For example, this information could be captured in the form of phone conversations or +recordings of individuals conversing in a busy room. If a developer is looking for conversational data on a given topic +(for example, music), two speakers might be asked to conduct a conversation about it. +Fig. 2. Train-Loss Graph +To compensate for this gender disparity, data augmentation will be done on the voice data-set in order to artificially +increase the diversity of the data-set and to increase the data-set size. This technique is performed by changing the +pitch, speed, injecting noise, and/or adding reverb to the audio data. +The model was trained with about 30,000 separate lines of data and was trained over 7 epochs with a learning rate of +5e-1 and batch sizes of 15. +The train loss graph shows that the speech recognition model has more than enough data to train the model and is +using more than enough data making the model almost over-fit which is also supported by the test loss graph. +As, shown by the train loss graph, the test loss graph also gives the same conclusion that the speech recognition +model might be slightly over fitting since the train loss is very low, compared the test loss which is much higher than +the train loss itself. However, the test loss is no too huge which makes the model usable. +Finally, after half of the training, the learning rate started slowing down as the model kept getting more and more +training with each epochs. +Manuscript submitted to ACM + +train_loss +： +8 +6 +4 +2 +0 +0 + 5k +10k +15k +20k +25k +30k10 +Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd +Fig. 3. Test-Loss Graph +Fig. 4. Training Learning Graph +6 +RESULTS +The goal of this project is to create a voice-generated virtual assistant using python that can work similarly to the modern +popular voice assistants such as Siri, Alexa, or Bixby. This projects personal AI voice assistant can understand voice +commands using speech recognition in Python and will be able to perform multiple tasks, using the pre-programmed +functions built into it as well as training data. +At the end of the project, this project will have AI voice assistant to be able to recognize voices and their commands +and perform the most suitable actions. This process of voice recognition is done by breaking down audio into individual +sounds, then converting them into a digital format that will be using Machine Learning algorithms and models to find +the word for that sound. The words will then be used by the voice assistant and see if anything related to it has been +pre-programmed and if not, it will be able to perform a most suitable action. The assistant will be speech-enabled, so +Manuscript submitted to ACM + +test_loss +1.4 +1.2 +1 +0.8 +0.6 +5k +10k +15k +20k +25ktrain_learning_rate +500μ +400μ +300μ +200μ +100μ +0 +0 +10k +20k +30kAddressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection +11 +after recognizing a statement or a question, it will call the necessary function to execute the task then give a response +based on the algorithm. +However, the project will incorporate some features such as sending text messages, sending emails, opening songs, +predicting the time, telling jokes, surfing the internet for information, forecasting weather, and many more. The final +project will not be limited to the features listed above and will possess more as well as having a clearer vision of what +this projects personal voice AI can do. +Fig. 5. Cer graph +Fig. 6. Wer graph +For the model that was trained, the Wer(Word Error Rate) was reduced to about 50% which is not so great when +compared to other models made by large scale companies, such as Google’s 4.9% WER and Microsoft’s 5.1%. Also from +Manuscript submitted to ACM + +cer + : +0.4 +0.3 +0.2 +5k +10k +15k +20k +25kwer +： +0.9 +0.8 +0.7 +0.6 +0.5 +5k +10k +15k +20k +25k12 +Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd +the trained model the Cer(Character Error Rate) to about 20% which means one out of every 5 characters were predicted +incorrectly which is not the best. +Fig. 7. Pitch Estimator graph +Part of this project was to see if the computer could correctly analyze whether a voice is female. Using the data +set collected from Augustana College where every of the 62 voice samples are females. Using the frequency ranges +mentioned earlier in the paper reported by ASHA as the boundaries for determination the computer categorized that 34 +out the 62 voice samples are female voices meaning that the computer correctly estimated about 55% of the samples. +Figure 7 is a sample of one of the correctly categorized voices samples showing the frequency boundary ranges for both +male and female voices with the red line be the voice samples calculated frequency. This results proves the hypothesis +that computers simple have a hard time distinguishing female voice samples regardless of the frequency ranges. +One of the biggest challenges is training the assistant to recognize different types of voices, accents, and dialects. To +counteract this problem by researching effective machine learning algorithms to implement to train the necessary voice +data. This project will utilize various open-source projects to complete the voice recognizer which then can incorporate +into this project own voice assistant. +One of the primary goals for this project is to build and train a model that utilizes more voice data from underrepre- +sented groups, as there is a huge gender and race bias in most virtual assistants. This statement above can be implied +because the training data used in some of the original voice assistants consists mainly of Caucasian males and Asian +males. The female group as well as people from other backgrounds that might have different accents or dialects are +extremely underrepresented. These models store the public data to try and improve the accuracy of the voice recognizer +in their voice assistants, but will try to incorporate these data from the beginning. In the end, the voice assistant will be +able to recognize voices regardless of their gender, race, or accent. +Another goal is the addition of a two factor authentication for added security to the user and their data. The two +factor authentication will be set to access the history of all the commands provided by the user. The collection of data is +done for the functionality of the virtual assistant. As well as allowing the user to keep track of the commands they +have used. The goal is to have this authentication app ready for the final project, however this is now part of a future +endeavor. A downside is that the voice recognition may not be viable, because it requires each user to train their voice +specifically to the virtual assistant. This process will also require a lot more storage of data and processing on the +programmers part. +This project demonstrates that gender and race biases exist in a lot of virtual assistants and tries to fix this problem by +training more voice data from underrepresented groups. Males, especially white and Asian men, are disproportionately +Manuscript submitted to ACM + +pitch estimation +Measured Frequency + Female min Freqency +20 +Female max Freqency +--- Male min Freqency +Male max Freqency +5 +0 +100 +200 +OOE +400 +500 +frequency (Hz)Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection +13 +over represented in Computer Science education and careers. Because of the male-dominated employment milieu, many +women abandon Computer Science careers. Researchers have found many hurdles to women in Computer Science +courses at the academic level, although efforts to address these issues have differed depending on geographic region or +educational level. +To combat this over-representation, gender depiction standards, as expressed through look, name, and voice, as well +as a review procedure, are critical changes that must be introduced. Users are predisposed to draw gender from a voice, +and often ascribe one to voices that are supposed to be neutral, as the makers of the gender-neutral voice assistant Q +discovered [28]. +7 +DISCUSSION +Virtual Assistants take in voice commands and process that data into commands or useful information. In this age, many +developed countries can already be considered as aged societies [29]. This causes lots of changes such as a discrepancy +in the proportion of seniors with the rest of the population. So, assistance technology such as a virtual assistant can +help with the problems associated with it. In the article “Home-Assistant Robot for an Aging Society” they provide +information on how assistance technology helps in labor support, healthy lifestyle support, and household and care +support. In consideration to virtual assistants, they give a list of activities done by an IRT home assistant robot that +models 3D space in a vector space and uses voice recognition with reinforcement learning to do several tasks: +• Performing Chores in a Home Environment +• Deformable Object Detection through image-based learning +• Geometrical Object Modeling and Its Application to Position Estimation +• Manipulation of Daily Tools and Appliances +• Integration of Basic Tasks Into Sequential Behavior +• Failure Detection and Recovery +The IRT home assistant robot works with voice commands to learn tasks. One can guide it to perform physical +activities such as carrying items, pushing objects, collecting items, and sweeping. For example, it can take images of +clothing and with vector data, learn about wrinkles in clothing to pick up clothes. The robot with more data can do the +task without any commands. Likewise, it uses environment recognition to do similar tasks at several locations of the +house using commands. The implementation of this assistance technology can increase the ease and productivity of +performing home activities. These assistants not only help old people but also physically disabled and blind people as it +does not require learning and typing where they can just speak and ask questions to the assistant. +In the expected results, this project discussed some of the outcomes of this projects virtual assistant. Again, it is +still in the beginning phase, however, the main goal is to make it so that the assistant can do tasks like Siri or Alexa. +“Virtual assistants are regularly used to make online interfaces more user friendly. It also generates positive responses +from Internet users leading to a more interpersonal shopping experience, greater pleasure, and customer flow” [30]. +In line with the related work, this project will be aiming to tackle the problem of the under-representation of women’s +voices and dialects for training. This project will be collecting first-hand data through participatory surveys and training +our model based on that. Then this project will use data from online databases to further train our voice assistant. +By doing this, this project can avoid the gender bias present in voice assistants. To confront the bias for dialects, this +project will also be collecting voice command data from a diverse audience. This helps create a proportional sample +that will help represent the college population and middle-aged adults that is the target. +Manuscript submitted to ACM + +14 +Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd +Currently the project is not a proper application. It was presented with a GUI (Graphic User Interface) to access +this project. In the future the goal is to implement an startup application with a better GUI that will run as soon as +you turn your computer on. Since the application at the moment is only a GUI that will only receive commands there +are no security risks. The program was also implemented so that the voice assistant will only listen when you press +the listen button. In the future the plan is to make it so you can use your voice in order to activate it from the start by +using a wake word. Doing so will make it easier to use but the goal is for the voice assistant to not collect sensitive +information when the wake word is used by accident. To counter that the plan is to add a mute feature for the assistant +to not allow it to listen. The future plan is also to add an indicator for when the voice assistant is actively listening +as well as keeping logs of when listening occur ed and a text alert through a mobile device. But when developing the +full application there will be a lot of potential security risks to consider. The plan to add several measures to manage +the risks. One of the ways would be to personalize the voice assistant to one account unless one links another one. +The future plan also includes adding a feature to ask for two-factor authentication in order to access the device use +and history. This will help mitigate potential risks of identity theft or user impersonation. It will also add an extra +layer of security to protect sensitive data from theft. The two-factor authorization that is used will be a pattern and/or +pass-code and a token, or biometric data and a token. For the token, it will have a token generator or an authentication +app. For the pattern and/or pass-code, this project will make sure that people can input it through their phones. As for +biometric data, this project will add a part to the voice assistant so that it will learn the user’s tone and pitch if allowed. +As for the future of the project, the voice assistant will keep on building up with more features aimed for making +the day to day activities easier for the user. Our research on the speech recognition will continue to grow as well as +trying on new alternatives such as adding more layers to our neural network, using audio books to train our model, +and using various other architectures available such as ctcdecode which can only be used in Linux environment which +will improve our word error rate dramatically. The end goal of this project is ambitious considering that there are many +speech recognition systems out already that perform very well, and have resources that cannot be compared to ours, +but there will be improvements in the speech recognition system with more training and additional libraries. +Manuscript submitted to ACM + +Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection +15 +REFERENCES +[1] S. Subhash, P. N. Srivatsa, S. Siddesh, A. Ullas, and B. Santhosh, “Artificial intelligence-based voice assistant,” in 2020 Fourth World Conference on +Smart Trends in Systems, Security and Sustainability (WorldS4), pp. 593–596, IEEE, 2020. +[2] F. Nasirian, M. Ahmadian, and O.-K. D. Lee, “Ai-based voice assistant systems: evaluating from the interaction and trust perspectives,” 2017. +[3] O. Kudina, ““alexa, who am i?”: Voice assistants and hermeneutic lemniscate as the technologically mediated sense-making,” Human Studies, vol. 44, +no. 2, pp. 233–253, 2021. +[4] V. Chattaraman, W.-S. Kwon, J. E. Gilbert, and K. Ross, “Should ai-based, conversational digital assistants employ social-or task-oriented interaction +style? a task-competency and reciprocity perspective for older adults,” Computers in Human Behavior, vol. 90, pp. 315–330, 2019. +[5] M. Schmidt and P. Braunger, “Towards a speaking style-adaptive assistant for task-oriented applications,” Studientexte zur Sprachkommunikation: +Elektronische Sprachsignalverarbeitung 2018, pp. 143–150, 2018. +[6] A. Bernaras, “Problem-oriented and task-oriented models of design in the commonkads framework,” in Artificial Intelligence in Design’94, pp. 499–516, +Springer, 1994. +[7] S. Bringsjord and B. Schimanski, “What is artificial intelligence? psychometric ai as an answer,” in IJCAI, pp. 887–893, Citeseer, 2003. +[8] G. G. Hendrix, E. D. Sacerdoti, D. Sagalowicz, and J. Slocum, “Developing a natural language interface to complex data,” ACM Transactions on +Database Systems (TODS), vol. 3, no. 2, pp. 105–147, 1978. +[9] J. B. Wold, “Difficulties in learning english as a second or foreign language,” 2006. +[10] G. Beller and X. Rodet, “Content-based transformation of the expressivity in speech,” in Proceedings of the 16th ICPhS, pp. 2157–2160, Citeseer, 2007. +[11] A. Caliskan, “Detecting and mitigating bias in natural language processing,” 2021. +[12] K. Chowdhary, “Natural language processing,” Fundamentals of artificial intelligence, pp. 603–649, 2020. +[13] D. R. Reddy, “Speech recognition by machine: A review,” Proceedings of the IEEE, vol. 64, no. 4, pp. 501–531, 1976. +[14] J. Meyer, L. Dentel, and F. Meunier, “Speech recognition in natural background noise,” PloS one, vol. 8, no. 11, p. e79279, 2013. +[15] J. Laroche, “Time and pitch scale modification of audio signals,” in Applications of digital signal processing to audio and acoustics, pp. 279–309, +Springer, 2002. +[16] D.-S. Kim, S.-Y. Lee, and R. M. Kil, “Auditory processing of speech signals for robust speech recognition in real-world noisy environments,” IEEE +Transactions on speech and audio processing, vol. 7, no. 1, pp. 55–69, 1999. +[17] Crowcroft, “Analog to digital conversion: Sampling,” 1998. +[18] G. D. Israel, “Determining sample size,” 1992. +[19] J. P. Bajorek, “Voice recognition still has significant race and gender biases,” Harvard Business Review, vol. 10, 2019. +[20] L. Lima, V. Furtado, E. Furtado, and V. Almeida, “Empirical analysis of bias in voice-based personal assistants,” in Companion Proceedings of the 2019 +World Wide Web Conference, pp. 533–538, 2019. +[21] N. Zhang, X. Mi, X. Feng, X. Wang, Y. Tian, and F. Qian, “Dangerous skills: Understanding and mitigating security risks of voice-controlled third-party +functions on virtual personal assistant systems,” in 2019 IEEE Symposium on Security and Privacy (SP), pp. 1381–1396, IEEE, 2019. +[22] W. Diao, X. Liu, Z. Zhou, and K. Zhang, “Your voice assistant is mine: How to abuse speakers to steal information and control your phone,” in +Proceedings of the 4th ACM Workshop on Security and Privacy in Smartphones & Mobile Devices, pp. 63–74, 2014. +[23] S. Watson, “The unheard female voice: Women are more likely to be talked over and unheeded. but slps can help them speak up and be heard.,” 2019. +[24] “3 types of speech recognition data (and what they’re used for),” Mar 2022. +[25] A. Zilberman and L. Ice, “Why computer occupations are behind strong stem employment growth in the 2019–29 decade,” Computer, vol. 4, no. 5,164.6, +pp. 11–5, 2021. +[26] L. Carter, “Why students with an apparent aptitude for computer science don’t choose to major in computer science,” ACM SIGCSE Bulletin, vol. 38, +no. 1, pp. 27–31, 2006. +[27] C. staff, “Women in computer science: Getting involved in stem,” 2022. +[28] M. Robison, “Voice assistants have a gender bias problem. what can we do about it?,” 2020. +[29] K. Yamazaki, R. Ueda, S. Nozawa, M. Kojima, K. Okada, K. Matsumoto, M. Ishikawa, I. Shimoyama, and M. Inaba, “Home-assistant robot for an aging +society,” Proceedings of the IEEE, vol. 100, no. 8, pp. 2429–2441, 2012. +[30] M. Holzwarth, C. Janiszewski, and M. M. Neumann, “The influence of avatars on online consumer shopping behavior,” Journal of marketing, vol. 70, +no. 4, pp. 19–36, 2006. +Manuscript submitted to ACM + diff --git a/btAyT4oBgHgl3EQfwPm0/content/tmp_files/load_file.txt b/btAyT4oBgHgl3EQfwPm0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b8432c96e634bad68d3b4a4d46c184284a111851 --- /dev/null +++ b/btAyT4oBgHgl3EQfwPm0/content/tmp_files/load_file.txt @@ -0,0 +1,542 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf,len=541 +page_content='Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection KASHAV PIYA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Augustana College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' USA SRIJAL SHRESTHA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Augustana College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' USA CAMERAN FRANK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Augustana College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' USA ESTEPHANOS JEBESSA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Augustana College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' USA TAUHEED KHAN MOHD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Augustana College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' USA In recent times,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' voice assistants have become a part of our day-to-day lives,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' allowing information retrieval by voice synthesis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' voice recognition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' and natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' These voice assistants can be found in many modern-day devices such as Apple, Amazon, Google, and Samsung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This project is primarily focused on Virtual Assistance in Natural Language Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Natural Language Processing is a form of AI that helps machines understand people and create feedback loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This project will use deep learning to create a Voice Recognizer and use Commonvoice and data collected from the local community for model training using Google Colaboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' After recognizing a command, the AI assistant will be able to perform the most suitable actions and then give a response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The motivation for this project comes from the race and gender bias that exists in many virtual assistants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The computer industry is primarily dominated by the male gender, and because of this, many of the products produced do not regard women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This bias has an impact on natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This project will be utilizing various open-source projects to implement machine learning algorithms and train the assistant algorithm to recognize different types of voices, accents, and dialects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Through this project, the goal to use voice data from underrepresented groups to build a voice assistant that can recognize voices regardless of gender, race, or accent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Increasing the representation of women in the computer industry is important for the future of the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' By representing women in the initial study of voice assistants, it can be shown that females play a vital role in the development of this technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' In line with related work, this project will use first-hand data from the college population and middle-aged adults to train voice assistant to combat gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Additional Key Words and Phrases: Voice Assistance, Machine Learning, Virtual Assistance, Artificial Intelligence, Selection Bias, Sample Population, Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='10, Pyttsx3, PyTorch, JSON 1 INTRODUCTION The first-ever voice-activated consumer product was released to the public in 1922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' It was known as “Radio Rex.” This product was a toy that had a doghouse with a dog inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' When someone said “Rex” next to the dog house, the dog would jump out of the doghouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This voice-activated toy was created even before modern computers existed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [1] Authors’ addresses: Kashav Piya, kashavpiya19@augustana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='edu, Augustana College, 639 38th St, Rock Island, Illinois, USA, 61201;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Srijal Shrestha, srijalshrestha18@augustana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='edu, Augustana College, 639 38th St, Rock Island, Illinois, USA, 61201;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Cameran Frank, cameranfrank18@augustana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='edu, Augustana College, 639 38th St, Rock Island, Illinois, USA, 61201;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Estephanos Jebessa, estephanosjebessa19@augustana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='edu, Augustana College, 639 38th St, Rock Island, Illinois, USA, 61201;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Tauheed Khan Mohd, tauheedkhanmohd@augustana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='edu, Augustana College, 639 38th St, Rock Island, Illinois, USA, 61201.' metadata={'source': 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requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Manuscript submitted to ACM Manuscript submitted to ACM 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='00646v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='AS] 20 Dec 2022 2 Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd Since the development of that toy, there has been a considerable amount of development in voice recognition, natural language processing, and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' A voice assistant, also known as an intelligent personal assistant or a connected speaker, is based on natural language speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Recently it has had a rise in popularity and has been marketed and used by Apple, Amazon, Google, and Samsung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Now voice assistants are widely found in most modern-day devices that a person would use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Voice assistants are multi-purposed one of their main purposes was for a search to be carried out using a voice command entered by the user as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' They are also known to be used for information retrieval by voice synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' They use a variety of voice recognition techniques, language processing algorithms, and voice synthesis to listen to specific voice commands that may include wake words, tasks, and queries, and return relevant information or perform a specific function as requested by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' These assistants can be software-based which allows them to be integrated into a wide range of devices such as laptops, mobile devices, and speakers, or can be specifically designed into a standalone device like Amazon Echo or Amazon Alexa Wall Clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [2] These voice assistants work like a charm and are quite fascinating, making one might ask themselves what goes on behind the hood of these amazing innovations or how do they work the way they do?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' To answer the above query, in short voice assistants use artificial intelligence and voice recognition to deliver the result that the user is looking for efficiently, and precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The user provides a command to the voice assistant that is called intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Through voice recognition, these intentions can be understood by our virtual assistants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Here, voice recognition allows the speaker to speak into a device that takes the analog signal from the speaker and changes it into a digital signal which is then processed by the computer to match it with words or phrases and then recognize the command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Machine learning also has a huge part to play in this as the computer needs to be taught to recognize the speaker’s words by feeding it a database of words and syllables in each language to match it with digital signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This process is known as pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Additionally, these devices gather a lot of information from the commands that they received previously to improve upon themselves using machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [3] There are multiple approaches to voice assistants, specifically two types: task-oriented and knowledge-oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Most voice assistants these days can combine both the task-oriented as well as a knowledge-oriented workflow to complete all the tasks that a user may ask the voice assistant to carry out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' A task-oriented approach will most likely ask something as simple as filling out a form, whereas a knowledge-based approach may include answering questions such as who the President of the United States of America is or finding out what engine is in a Ford F50 which is a technical specification of a product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [4] The task-oriented approach/workflow is pretty much self-explanatory as it uses goals and tasks to achieve whatever the user wants or needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This approach usually requires the voice assistant to use a different application such as time, weather, web browser, and music apps, to help complete its tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Some examples would be, asking a voice assistant to set a reminder to take medicine at 6 PM, playing music using Spotify, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This approach does not require the virtual voice assistant to search massive databases for knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' These tasks are often known as skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' And various assistants allow for different skills to be installed according to the user’s preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [5] Whereas a knowledge-oriented approach/workflow requires the use of analytical data to complete the tasks and help the users to complete their tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [6] Unlike a task-oriented approach, this approach focuses on using online databases to get related information in addition to already recorded knowledge to help users to complete tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' An example of a knowledge-based approach would be if a user asked for a question that would require searching the internet such as what is the capital of the state of Illinois or who invented the telephone?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Manuscript submitted to ACM Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection 3 Furthermore, there are two types of artificial intelligence (AI);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' In general, there is a weak AI and there is a strong AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [7] There are many types of machines such as Siri, Alexa, Cortana, and Bixby that can only perform certain tasks that have been defined by the user while making the AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' These types of AIs are called Weak AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' And some machines or systems have a mind of their own and can make decisions or take actions on their own without human interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' These types of machines are called Strong AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' After learning the differences between strong and weak AI, the voice assistant this project is opting for is an example of weak AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This projects Virtual Voice Assistant will include a variety of features such as greeting the users, fetching information about a person, an object, or anything else in general from the internet, providing the time, opening web browsers, playing music, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' It might also include additional features such as opening the web camera to take pictures, forecasting the weather, logging off from your personal computer, telling you a joke, and many other features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The field of Virtual Assistance has many avenues to consider from providing help in technology to connecting people through the usage of technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' When studying what exactly a Virtual Assistant is, the field that was decided on was Virtual Assistance in natural language processing, which means the technology can understand people more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Natural Language Processing is a form of AI that gives machines the ability to not just read but to understand and interpret human language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' With NLP, machines can make sense of the written or spoken text and perform tasks including speech recognition, sentiment analysis, and automatic text summarization [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Therefore, not only does natural language processing help humans it also helps with machine learning, in the sense that NLP will continue to provide more data to better that analysis of speech and create a feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The English language is an extremely hard language to understand and speak, especially if English is not your first language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Not only is the usage of English vernacular hard to comprehend and execute, but there is also a form of a language barrier in the different dialects that people possess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' A person’s dialect can be a communication inhibitor in many languages, not just in English, but in this project, the focus is on the English language [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' For this project, synthetic voices were originally used instead of human voices, in which data was collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' A synthetic voice is a pre-recorded voice produced through text to speech whereas the human voice is pre-recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' ‘Its use involves recording, in advance, a text read aloud by a human being.’ The usage of a synthetic voice will give flexibility as they have a high capacity in reading textual context and can generate voice constantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' But “there is a disadvantage in expressing social signs as it cannot express emotions, intentions, and attitudes through modulation of the voice” [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The computer industry is primarily dominated by the male gender and because of this extreme one-sided representa- tion in the field a multitude of the products that are produced do not regard women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' One of the fields that this bias impacts is natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Because of the lack of females in the industry, the identification percentage for female voices is lower than that of male voices, therefore resulting in the analysis and research of this topic of selection bias of voice assistance [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Data augmentation by controlling the gender attribute is an effective technique in mitigating gender bias in NLP processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='1 How Does Voice Assistant Work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Voice assistance has now been defined is, but how does it work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' A voice assistant uses speech recognition along with other identification of speech components to help the machine process the voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Then, the speech is rendered into its textual representation based on extracted patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Following that, this program isolates the most important words Manuscript submitted to ACM 4 Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' How Does Voice Assistant Work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' or the action also known as anticipated intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' If the intent is not clear, the voice assistant is programmed to ask more questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' It then retrieves information by API calls to access the relevant knowledge base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Finally, it relays the information back to the human user through text to speech or fulfills the necessary action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Voice assistants rely on Natural Language Processing and other machine learning algorithms to perform the best and overcome the challenges that it faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [12] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='2 Speech Recognition How does a voice assistant understand what a person is talking about?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' These voice assistants use fast decoding algorithms that allow real-time continuous speech recognition systems to provide instant responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [13] Speech Recognition has been a part of our day-to-day life for more than 10 years now as it seems to become more and more advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The beauty of speech-to-text models goes unnoticed in this process as the machines make it look so seamless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Most speech recognition algorithms use a mix of Natural Language Processing and Deep Learning techniques to parse through the user query, get an appropriate response and present it back to the user in whichever form the user desires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' All the big companies such as Google’s Assistant, Amazon’s Alexa, Apple’s Siri, and others use the same techniques but just with some different variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [14] Manuscript submitted to ACM Voice Assistant 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Human User NLP What is the APIs weather for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' tomorrow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='1 Signal Processing for Speech Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Audio signals are any object that vibrates to produce sound waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' When an object vibrates, the air molecules oscillate to and from their rest position and transmits its energy to the neighboring molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This results in the transmission of energy from one molecule to another which in turn produces a sound wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [15] There are a few terms that should be familiar when talking about sound and signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Amplitude: Amplitude refers to the maximum displacement of the air molecules from the rest position Crest and Trough: The crest is the highest point in the wave whereas trough is the lowest point Wavelength: The distance between two successive crests or troughs is known as a wavelength Cycle: Every audio signal traverses in the form of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' One complete upward movement and downward movement of the signal form a cycle Frequency: Frequency refers to how fast a signal is changing over a period Additionally, there are two different types of signals: Digital and Analog Signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' For digital signal is a discrete representation of a signal over a period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Here, the finite number of samples exists between any two-time intervals whereas the analog signal is a continuous representation of a signal over a period which implies that there will be an infinite number of samples between any two given time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [16] For this project, audio signals are needed for the Voice Assistant, so there is a question about how it stores a signal that has an infinite number of samples since they are analog signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This project will incorporate changing the memory-hogging analog signal using complex techniques to convert it into digital signals to make it more convenient to work with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' To convert a signal from analog to digital, a technique called sampling the signal should be used which selects a certain number of samples per second from the analog signal which makes storing and processing the signal memory efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' In analog to digital sampling "an input signal is converted from some continuously varying physical value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' pressure in air, or frequency or wavelength of light), by some electro-mechanical device into a continuously varying electrical signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='" [17] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='2 Feature Extraction Techniques for Audio signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' For this projects model to use the audio signals, the features must be extracted from the audio such as time domain and frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The audio signal is representing by the amplitude as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The features here are the amplitudes which are recorded at different time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' On the other hand, in the frequency domain, the audio signal is represents amplitude as a function of frequency where the features are amplitudes that are recorded at different frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='3 Papers Regarding this Topic In machine learning when one trains their model, there is always a chance for problems when applying the model to the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This problem arises in big data when the population is too large for machines with limited computability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' To deal with this problem of big data, a sample population can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' A sample population refers to a subset of the population that represents the whole population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [18] This project will apply machine learning to samples population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' But there is a chance for selection bias, “selection bias is a systematic error that results in differences between a study population and a target population;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' selection bias primarily affects the external validity of the results of a study”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This means that the results thought to be true for the sample population may not be true for the actual population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Depending on the population sample, if there are too many true results it will just show true no matter what.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Sample population can technically be wrong from the population just because of the sample collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Manuscript submitted to ACM 6 Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd Likewise, voice assistants use sample data to analyze a user’s speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' “Currently speech recognition has significant race and gender biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' It is just another form of AI that performs worse for women and non-white people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Currently, it is designed to understand white male voices well” [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This is a significant problem because these days voice assistants are a crucial part of people’s lives from setting alarms to transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Low accuracy in voice recognition would mean severe consequences in people’s life [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' In the paper Empirical Analysis of Bias in Voice-based Personal Assistants, they try to check the accuracy of relevant voice assistants such as Google Assistant and Siri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' They look at the different accents for the Brazilian Portuguese, and how accuracy was off for certain accents than other ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' They also found variation in the quality of recognition based on gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [20] So, taking the two paper’s ideas, to tackle this problem and move forward in this topic, data was gathered and training our model not only on prevalent male voice data sets but also gathering voices manually and from Common-Voice data sets by Mozilla for female and minority voices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This allows us to create samples proportionally to demographic indicators of the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The process for machine learning will be as follows: Filter the words that the user says Digitize the user’s speech into a format that the machine can read Analyze the user’s speech for meaning Decide what the user needs based on previous input and algorithms” As talked above, data sets for voice recognition must be sampled so that each voice has an equal probability to be included in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This allows for less racial and gender bias for the models to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' So, this will result in everyone having their voice heard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Additionally the paper “Dangerous Skills: Understanding and Mitigating Security Risks of Voice-Controlled Third- Party Functions on Virtual Personal Assistant Systems” [21] and “Your Voice Assistant is Mine: How to Abuse Speakers to Steal Information and Control Your Phone” talks about the development of voice assistant in speech recognition and IoTs and with its development comes more vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' They talk about voice-based remote attacks and permission bypassing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' These problems are extremely dangerous and can be misused to expose people’s information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' So, these papers advise incorporating the ideas of not allowing zero permission and context-aware information collection and analysis features to the voice assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Not allowing those ideas will allow focusing on forcing restrictions on specific operations that a particular process can perform [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' 3 EXPERIMENTAL SETUP The data used is partially from Common-Voice, which is an online open source of voice recorders in multiple languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' However, since this project is specifically for the English language the data set collected was English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' However, this data set, as expected has more male voices than female voices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Therefore, to even out the distribution between the voices represented the rest of the data was collected via outreach into the Augustana College community, contacting over 200 female students (ages ranging from 18 - 24 years old) and receiving about 65 voices samples with 4-7 minutes of voice recording from each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The data set collected was then doubled, converted, and combined into .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='wav files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' After the data collection and manipulated, it was then applied to a gender recognition model in order to numerically identify via the frequency of each voice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' according to ASHA (American Speech Language and Hearing Association) the average range for an adult woman is 165 to 255 Hz and the average range for adult males is 85 to 155 Hz [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' There is inherent technical problem down to the fact that females generally have higher pitched voices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Female voices tend to be Manuscript submitted to ACM Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection 7 quieter and sound more “breathy”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Female more easily masked by noise, like a fan or traffic in the background, which makes it harder for speech recognition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' In order to run the data through training model for speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The full data set from Common-Voice was not needed in fact there only segments of the a statement is used, therefore it was necessary to parse the data into smaller form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' 4 FRAMEWORK For Machine Learning, Python is known to be the most common language used due to its versatility, readability, and abundant packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' So, for this project, it will be using Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='10 as the main programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This project will be using the following python packages or libraries: Speech Recognition (Will only be used initially to create and test the basic functionalities of the project and will be later replaced by this projects own model): It helps understand what the human is saying and converts the speech into text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Pyttsx3: It is a simple text to speech conversion library in python which will be used to give this projects Voice Assistant a voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Wikipedia: This package is a Python that extracts information and data from Wikipedia which is a multilingual online encyclopedia used by many people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Capture: It helps to capture images from your camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Datetime: It is an inbuilt module in python that works with date and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Os: It provides functions to interact with the operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Web Browser: It is an in-build package from Python that allows you to extract information from the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' JSON: It is a module that helps to store and exchange data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' PyJokes: It gives the user a random joke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Pyaudio: It allows python to play and record audio between different platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Pywhatkit: It can access YouTube in order to play a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Librosa: It is a package for analysing sound, audio and music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Soundfile: This package can help read and write sound files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Numpy: Numpy provides a powerful N-dimensional array object, sophisticated functions, tolls for integrating C/C++ and Fortran code, useful linear algebra, Fourier transform, and random number capabilities, and much more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' BeautifulSoup: Beautiful Soup is a Python library for pulling data out of HTML and XML files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Pandas: pandas is a fast, powerful, flexible and easy to use open source data analysis and manipulation tool, built on top of the Python programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This project will be using all these modules and packages to create basic functionalities of the Voice Assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' For example, this project will use the Wikipedia package to get information regarding a person or a company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This function is used to grab the first few sentences from its corresponding Wikipedia page which is generally the broad introduction to the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Then this project will have a Voice Recognizer which will replace the existing Speech Recognition module as well as the Google API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The program uses deep learning to create this projects voice recognition using PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The data is from open sources Mozilla Common-Voice as well as a few more from Kaggle to train this projects model and add Manuscript submitted to ACM 8 Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd voices from volunteers to help balance the data to reduce the biases while training for the wake word as well as the general voice recognition itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This project utilizes Google Colaboratory to train this projects model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Finally, if time permits, the goal is to build a simple User Interface either with Flask or with other python libraries itself for a visual element to the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' 5 METHODS As stated in the experimental setup, the Common-Voice data set planned to be used for the training model has recognized 46% male voices and 16% female voices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Therefore, the goal of this project is to come up with an effective solution to this disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' When training a speech recognition model, different types of voice data can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Depending on the type of interaction one’s looking to build, and how robust that interaction should be, different types of voice data might be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Although there are several easily available sources of speech data, such as public speech corpora or pre-packaged data sets, it is almost always necessary to cooperate with a data services provider to collect your own speech data, either remotely or in person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' When you gather one’s own data, it is easy to tailor the speech data set to include variables such as language, speaker demographics, audio requirements, and collection size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [24] The Bureau of Labor Statistics (BLS) projects computer science research jobs will grow 19 percent by 2026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' However, in the United States, the percentage of women that receive a bachelor’s degree in Computer Science is still only 18 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' There is currently a high demand for computer scientists in the professional industry but despite this fact, this industry remains male-dominated, in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' For example, in this year’s Senior Inquire class for Computer Science, there is a 1:6 female to male ratio [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Computer technology first emerged during World War II and continuing into the 1960s, women made up most of the computing workforce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' However, by 1970 women only accounted for about 14 percent of bachelor’s in computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' In 1984 that number rose to 37 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The percentage of women in computer science has since declined to 18 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' It was around the same time personal computers started showing up in homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' According to NPR, personal computers were marketed almost exclusively to men and families were more likely to buy computers for boys than girls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Computers are now commonplace in both classrooms and on individuals as personal assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' It is hard, however, to explain the exact reason why females are not as present in this major.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' There are organizations now that are researching and improving ways to increase the potential of more females in the computer science major.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' It is said that one of the reasons why women tend to trend away from the computer science field is because of marketing of the industry in the past tailored to those with the geek persona and the social innuendos of what being a geek used to mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' [26] One of the reasons why females should be represented in the computer industry is increasing the inclusion of women is a sound business strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' “A study by Deloitte found that women’s choices account for up to 85 percent of buying decisions nationwide, and that diversity drives innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Though it is still commonplace to find boards and project teams without a female member, the integration of female perspectives will naturally lead to higher revenues and a better understanding of consumer marketplaces.” [27] Therefore, the purpose of this project is to strive to increase the attractiveness of the potential that computer science avenues provide for women, by increasing the representation of females in the initial study of voice assistants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' If females can realize that they possess a more essential in the initial makeup that products that are generated there is a possibility that females will be more inclined into joining the computer science industry, by showing them that they do in fact play an essential role in the make of technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Speech recognition data can be classified into three categories: Manuscript submitted to ACM Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection 9 Controlled: Scripted speech data Semi-controlled: Scenario-based speech data Natural: Unscripted or conversational speech data For this project, the primary type of voice data used is Semi-controlled data and natural unscripted conversational speech data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' For semi-controlled data, When developers need a natural sampling of different ways to ask for the same thing or a greater diversity of command intentions, scenario-based voice data is collected (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' asking for different things).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' As a result, scenario-based speech data adds variety to what is said as well as how it is said.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' On the other hand, The most "natural" kind of speech is unscripted or conversational speech data, which is a recording of a conversation between two or more speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Unscripted speech data, like spontaneous speech, occurs in a variety of formats in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' For example, this information could be captured in the form of phone conversations or recordings of individuals conversing in a busy room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' If a developer is looking for conversational data on a given topic (for example, music), two speakers might be asked to conduct a conversation about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Train-Loss Graph To compensate for this gender disparity, data augmentation will be done on the voice data-set in order to artificially increase the diversity of the data-set and to increase the data-set size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This technique is performed by changing the pitch, speed, injecting noise, and/or adding reverb to the audio data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The model was trained with about 30,000 separate lines of data and was trained over 7 epochs with a learning rate of 5e-1 and batch sizes of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The train loss graph shows that the speech recognition model has more than enough data to train the model and is using more than enough data making the model almost over-fit which is also supported by the test loss graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' As, shown by the train loss graph, the test loss graph also gives the same conclusion that the speech recognition model might be slightly over fitting since the train loss is very low, compared the test loss which is much higher than the train loss itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' However, the test loss is no too huge which makes the model usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Finally, after half of the training, the learning rate started slowing down as the model kept getting more and more training with each epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Manuscript submitted to ACM train_loss ： 8 6 4 2 0 0 5k 10k 15k 20k 25k 30k10 Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Test-Loss Graph Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Training Learning Graph 6 RESULTS The goal of this project is to create a voice-generated virtual assistant using python that can work similarly to the modern popular voice assistants such as Siri, Alexa, or Bixby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This projects personal AI voice assistant can understand voice commands using speech recognition in Python and will be able to perform multiple tasks, using the pre-programmed functions built into it as well as training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' At the end of the project, this project will have AI voice assistant to be able to recognize voices and their commands and perform the most suitable actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This process of voice recognition is done by breaking down audio into individual sounds, then converting them into a digital format that will be using Machine Learning algorithms and models to find the word for that sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The words will then be used by the voice assistant and see if anything related to it has been pre-programmed and if not, it will be able to perform a most suitable action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The assistant will be speech-enabled, so Manuscript submitted to ACM test_loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='6 5k 10k 15k 20k 25ktrain_learning_rate 500μ 400μ 300μ 200μ 100μ 0 0 10k 20k 30kAddressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection 11 after recognizing a statement or a question, it will call the necessary function to execute the task then give a response based on the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' However, the project will incorporate some features such as sending text messages, sending emails, opening songs, predicting the time, telling jokes, surfing the internet for information, forecasting weather, and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The final project will not be limited to the features listed above and will possess more as well as having a clearer vision of what this projects personal voice AI can do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Cer graph Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Wer graph For the model that was trained, the Wer(Word Error Rate) was reduced to about 50% which is not so great when compared to other models made by large scale companies, such as Google’s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='9% WER and Microsoft’s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Also from Manuscript submitted to ACM cer : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='2 5k 10k 15k 20k 25kwer ： 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='5 5k 10k 15k 20k 25k12 Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd the trained model the Cer(Character Error Rate) to about 20% which means one out of every 5 characters were predicted incorrectly which is not the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Pitch Estimator graph Part of this project was to see if the computer could correctly analyze whether a voice is female.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Using the data set collected from Augustana College where every of the 62 voice samples are females.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Using the frequency ranges mentioned earlier in the paper reported by ASHA as the boundaries for determination the computer categorized that 34 out the 62 voice samples are female voices meaning that the computer correctly estimated about 55% of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Figure 7 is a sample of one of the correctly categorized voices samples showing the frequency boundary ranges for both male and female voices with the red line be the voice samples calculated frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This results proves the hypothesis that computers simple have a hard time distinguishing female voice samples regardless of the frequency ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' One of the biggest challenges is training the assistant to recognize different types of voices, accents, and dialects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' To counteract this problem by researching effective machine learning algorithms to implement to train the necessary voice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This project will utilize various open-source projects to complete the voice recognizer which then can incorporate into this project own voice assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' One of the primary goals for this project is to build and train a model that utilizes more voice data from underrepre- sented groups, as there is a huge gender and race bias in most virtual assistants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This statement above can be implied because the training data used in some of the original voice assistants consists mainly of Caucasian males and Asian males.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The female group as well as people from other backgrounds that might have different accents or dialects are extremely underrepresented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' These models store the public data to try and improve the accuracy of the voice recognizer in their voice assistants, but will try to incorporate these data from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' In the end, the voice assistant will be able to recognize voices regardless of their gender, race, or accent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Another goal is the addition of a two factor authentication for added security to the user and their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The two factor authentication will be set to access the history of all the commands provided by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The collection of data is done for the functionality of the virtual assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' As well as allowing the user to keep track of the commands they have used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The goal is to have this authentication app ready for the final project, however this is now part of a future endeavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' A downside is that the voice recognition may not be viable, because it requires each user to train their voice specifically to the virtual assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This process will also require a lot more storage of data and processing on the programmers part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This project demonstrates that gender and race biases exist in a lot of virtual assistants and tries to fix this problem by training more voice data from underrepresented groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Males, especially white and Asian men, are disproportionately Manuscript submitted to ACM pitch estimation Measured Frequency Female min Freqency 20 Female max Freqency --- Male min Freqency Male max Freqency 5 0 100 200 OOE 400 500 frequency (Hz)Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection 13 over represented in Computer Science education and careers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Because of the male-dominated employment milieu, many women abandon Computer Science careers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Researchers have found many hurdles to women in Computer Science courses at the academic level, although efforts to address these issues have differed depending on geographic region or educational level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' To combat this over-representation, gender depiction standards, as expressed through look, name, and voice, as well as a review procedure, are critical changes that must be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Users are predisposed to draw gender from a voice, and often ascribe one to voices that are supposed to be neutral, as the makers of the gender-neutral voice assistant Q discovered [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' 7 DISCUSSION Virtual Assistants take in voice commands and process that data into commands or useful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' In this age, many developed countries can already be considered as aged societies [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This causes lots of changes such as a discrepancy in the proportion of seniors with the rest of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' So, assistance technology such as a virtual assistant can help with the problems associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' In the article “Home-Assistant Robot for an Aging Society” they provide information on how assistance technology helps in labor support, healthy lifestyle support, and household and care support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' In consideration to virtual assistants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' they give a list of activities done by an IRT home assistant robot that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='models 3D space in a vector space and uses voice recognition with reinforcement learning to do several tasks: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='Performing Chores in a Home Environment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='Deformable Object Detection through image-based learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='Geometrical Object Modeling and Its Application to Position Estimation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='Manipulation of Daily Tools and Appliances ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='Integration of Basic Tasks Into Sequential Behavior ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='Failure Detection and Recovery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content='The IRT home assistant robot works with voice commands to learn tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' One can guide it to perform physical activities such as carrying items, pushing objects, collecting items, and sweeping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' For example, it can take images of clothing and with vector data, learn about wrinkles in clothing to pick up clothes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The robot with more data can do the task without any commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Likewise, it uses environment recognition to do similar tasks at several locations of the house using commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The implementation of this assistance technology can increase the ease and productivity of performing home activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' These assistants not only help old people but also physically disabled and blind people as it does not require learning and typing where they can just speak and ask questions to the assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' In the expected results, this project discussed some of the outcomes of this projects virtual assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Again, it is still in the beginning phase, however, the main goal is to make it so that the assistant can do tasks like Siri or Alexa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' “Virtual assistants are regularly used to make online interfaces more user friendly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' It also generates positive responses from Internet users leading to a more interpersonal shopping experience, greater pleasure, and customer flow” [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' In line with the related work, this project will be aiming to tackle the problem of the under-representation of women’s voices and dialects for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This project will be collecting first-hand data through participatory surveys and training our model based on that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Then this project will use data from online databases to further train our voice assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' By doing this, this project can avoid the gender bias present in voice assistants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' To confront the bias for dialects, this project will also be collecting voice command data from a diverse audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This helps create a proportional sample that will help represent the college population and middle-aged adults that is the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Manuscript submitted to ACM 14 Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd Currently the project is not a proper application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' It was presented with a GUI (Graphic User Interface) to access this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' In the future the goal is to implement an startup application with a better GUI that will run as soon as you turn your computer on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Since the application at the moment is only a GUI that will only receive commands there are no security risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The program was also implemented so that the voice assistant will only listen when you press the listen button.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' In the future the plan is to make it so you can use your voice in order to activate it from the start by using a wake word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Doing so will make it easier to use but the goal is for the voice assistant to not collect sensitive information when the wake word is used by accident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' To counter that the plan is to add a mute feature for the assistant to not allow it to listen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The future plan is also to add an indicator for when the voice assistant is actively listening as well as keeping logs of when listening occur ed and a text alert through a mobile device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' But when developing the full application there will be a lot of potential security risks to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The plan to add several measures to manage the risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' One of the ways would be to personalize the voice assistant to one account unless one links another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The future plan also includes adding a feature to ask for two-factor authentication in order to access the device use and history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' This will help mitigate potential risks of identity theft or user impersonation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' It will also add an extra layer of security to protect sensitive data from theft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The two-factor authorization that is used will be a pattern and/or pass-code and a token, or biometric data and a token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' For the token, it will have a token generator or an authentication app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' For the pattern and/or pass-code, this project will make sure that people can input it through their phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' As for biometric data, this project will add a part to the voice assistant so that it will learn the user’s tone and pitch if allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' As for the future of the project, the voice assistant will keep on building up with more features aimed for making the day to day activities easier for the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' Our research on the speech recognition will continue to grow as well as trying on new alternatives such as adding more layers to our neural network, using audio books to train our model, and using various other architectures available such as ctcdecode which can only be used in Linux environment which will improve our word error rate dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} +page_content=' The end goal of this project is ambitious considering that there are many speech recognition systems out 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btAyT4oBgHgl3EQfwPm0/content/2301.00646v1.pdf'} diff --git a/ctAzT4oBgHgl3EQfZ_wC/content/tmp_files/2301.01359v1.pdf.txt b/ctAzT4oBgHgl3EQfZ_wC/content/tmp_files/2301.01359v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5ff5fa8cb257e92ccdb60f1ccfb2a25fea34237a --- /dev/null +++ b/ctAzT4oBgHgl3EQfZ_wC/content/tmp_files/2301.01359v1.pdf.txt @@ -0,0 +1,1405 @@ +arXiv:2301.01359v1 [math.NT] 3 Jan 2023 +PROOFS OF MODULO 11 AND 13 CYLINDRIC KANADE-RUSSELL CONJECTURES +FOR A2 ROGERS–RAMANUJAN TYPE IDENTITIES +ALI KEMAL UNCU +Abstract. We present proofs of two new families of sum-product identities arising from the cylindric parti- +tions paradigm. Most of the presented expressions, the related sum-product identities, and the ingredients for +the proofs were first conjectured by Kanade–Russell in the spirit of Andrews–Schilling–Warnaar identities of +the A2 Rogers–Ramanujan type. We follow the footsteps of Kanade–Russell while we alter the computations +heavily to accomplish our goals. +1. Introduction +There is an ever-growing synergy between number theory, combinatorics, q-series, and affine Lie algebras +that led to groundbreaking techniques and beautiful mathematical discoveries. Among these are the Rogers– +Ramanujan type identities where an infinite q-series is equal to a infinite product with a modular structure. +First appeared at the intersection of number theory and combinatorics, the Rogers–Ramanujan identities +have been of great interest. These sum-product identities have been studied, proved and generalized in many +different ways over the years [3, 6, 13, 14, 17, 22, 24, 25, 37]. These identities also naturally arose in many +other fields including mathematical physics [10], representation theory of affine Lie algebras and vector operator +algebras [31, 32], knot theory in relation to the colored Jones polynomials [8], and algebraic geometry [15] over +the years. +For some non-negative integer L and formal variables a and q, let q-Pochhammer symbol be (a; q)L := +(1 − a)(1 − aq) . . . (1 − aqL−1), and (a; q)∞ := limL→∞(a; q)L, θ(a; q) := (a, q/a; q)∞, and for a1, . . . , ak some +formal variables, define the shorthand notation θ(a1, a2, . . . , ak; q) := θ(a1; q)θ(a2; q) . . . θ(ak; q). +The Rogers–Ramanujan identities are as follows [38]. +Theorem 1.1 (Rogers–Ramanujan identities). +(1.1) +� +n≥0 +qn2 +(q; q)n += +1 +θ(q; q5) +and +� +n≥0 +qn2+n +(q; q)n += +1 +θ(q2; q5). +The reciprocal q-Pochhammer products on the right-hand side of (1.1) has the ±1 and ±2 residue classes +modulo 5, respectively. We call these modulo 5 identities. +A composition c of n is a finite list of non-negative integers that sum up to n. A partition is a composition +where no element of the list (called parts) are zero and the list elements are ordered in a non-increasing order. +We define the size of a composition π as the sum of all its parts and denote this by |π|. We denote the number +of parts in a composition π by #(π). A composition (resp. partition) with size n is called “a composition +(resp. partition) of n.” The empty list is considered as the unique composition/partition of 0 with 0 parts. +For example, (2, 0, 2) is a composition with 3 parts and (1, 1, 1, 1), (4, 3, 1), and (2, 2) are partitions of 4, 8, +and 4, respectively. +MacMahon [33] and Schur [39] gave combinatorial interpretations to Rogers–Ramanujan identities inde- +pendently. +Date: January 5, 2023. +2010 Mathematics Subject Classification. Primary 05A15; Secondary 05A17, 05A19, 11B65, 11P84, 17B65, 68R05. +Key words and phrases. Cylindric partitions, Partition identities, Rogers–Ramanujan identities, Andrews–Schilling–Warnaar +identities. +Research of the author is partly supported by EPSRC grant number EP/T015713/1 and partly by FWF grant P-34501N. +1 + +2 +ALI KEMAL UNCU +Theorem 1.2 (Combinatorial interpretaton of Rogers–Ramanujan identities). Let i = 0 or 1. For every +natural number n, the number of partitions of n such that the difference between two consecutive parts is at +least 2 and the the smallest part is strictly greater than i is equal to the number of partitions of n into parts +congruent to ±(1 + i) mod 5. +Gordon [24] presented a wide generalization of Theorem 1.2 to all odd modulus ≥ 5. +Theorem 1.3 (Gordon’s identities, 1961). Let r and i be integers such that r ≥ 2 and 1 ≤ i ≤ r. The number +of partitions π = (π1, π2, . . . , πs) of n such that πj − πj+r−1 ≥ 2 for all j with at most i− 1 1s appears as parts +in π are equal to the number of partitions of n whose parts are not congruent to 0, ±i mod 2r + 1. +The Rogers–Ramanujan identities correspond to the cases r = i = 2 and r = 2, i = 1. +Andrews found the q-series counterpart to Gordon’s identities [3]. +Theorem 1.4 (Andrews–Gordon identities, 1974). Let r ≥ 2 and 1 ≤ i ≤ r be two integers. We have +(1.2) +� +n1≥···≥nr−1≥0 +qn2 +1+···+n2 +r−1+ni+···+nr−1 +(q; q)n1 +� +n1 +n1 − n2 +� +q +· · · +� +nr−2 +nr−2 − nr−1 +� +q += θ(qi; q2r+1)(q2r+1; q2r+1)∞ +(q; q)∞ +, +where for two integers n and m, +�m + n +m +� +q +:= + + + + + +(q; q)m+n +(q; q)m(q; q)n +for m, n ≥ 0, +0 +otherwise, +is the classical q-binomial coefficient. +Note that the Rogers–Ramanujan identities are the particular case of (1.2) where r = i = 2, and r = 2 and +i = 1. Interested readers can get a great overview of the history of the Rogers–Ramanujan identities, their +significance, and some generalizations in the recent book of Sills [40]. +The identities (1.2) can be proven by the Bailey machinery coming from the world of q-series. This powerful +mechanism starts with a pair of q-expressions, called a Bailey pair, that satisfies a pre-defined relation and +modifies this pair iteratively (using Bailey lemma or one of its generalizations) to make a new Bailey pair (see +[2, 4, 9, 40]). That way, by starting with the pair related to Rogers–Ramanujan identities, a whole infinite +chain of identities (1.2) can be acquired. The identities (1.2) are certain characters related to affine Lie algebra +A(1) +1 , and we thus refer to them as A1 Rogers–Ramanujan identities. The original Bailey mechanism was later +extended to An−1 for general n [34, 35]. However,these works did not yield An−1 Rogers–Ramanujan identities. +In their influential paper, Andrews, Schilling and Warnaar [7] were able to describe an A2 Bailey lemma +and the associated Bailey machinery. They found several infinite families of identities, One of their modulo 7 +identities is as follows. +Theorem 1.5 (Andrews–Schilling–Warnaar, 1999). +(1.3) +� +r1,s1≥0 +qr2 +1−r1s1+s2 +1+r1+s1 +(q; q)r1 +�2r1 +s1 +� +q += +1 +θ(q2, q3, q3; q7). +Andrews–Schilling–Warnaar found several very general families of sum-product identities. Of particular +interest to representation theory, the product-sides of these identities are character formulas of the W3 algebra +multiplied by an extra factor (q; q)−1 +∞ [21]. These formulas do not yield manifestly positive sum-sides for the +character formulas because of this extra factor. +For example, one of Andrews–Schilling–Warnaar’s modulo 10 identities after clearing the extra factor +(q; q)−1 +∞ is as follows. +Theorem 1.6 (Andrews–Schilling–Warnaar, 1999). +(1.4) +(q, q)∞ +� +r1≥r2≥0 +s1≥s2≥0 +qr2 +1−r1s1+s2 +1+r2 +2−r2s2+s2 +2+r1+r2+s1+s2 +(q; q)r1−r2(q; q)r2(q; q)s1−r2(q; q)s2(q; q)r2+s2+1 += +1 +θ(q2, q3, q3, q4, q4, q5; q10) + +MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES +3 +Recall the Euler’s Pentagonal Number Theorem [5] +(1.5) +(q, q)∞ = +∞ +� +i=−∞ +(−1)iqi(3i+1)/2. +Although it is easy to see that the right-hand side of (1.4) has positive coefficients, in light of (1.5) this is not +directly visible on the left-hand side. In contrast, both sides of (1.3) are manifestly positive. The manifestly +positive sum representations give insight to the structure of certain modules for the affine Lie algebra A(1) +2 . +These mentioned character of standard modules for the affine Lie algebra A(1) +2 . Interested readers can find +more on this connection in [7, 29, 28, 31, 32]. +Recently, the discovery of manifestly positive identities of these character formulas through a scheme with +combinatorial roots attracted the attention and led to many new Rogers–Ramanujan type identities. +In 1997, Gessel and Krattenthaler [23] defined cylindric partitions in context of non-intersecting lattice +paths. Borodin [11] gave univariate product formulas for the generating functions of the number of cylindric +partitions. Foda and Welsh [21] proved the A1 Rogers–Ramanujan identities using the combinatorics of cylin- +dric partitions. This led to Corteel’s combinatorial proof of the Rogers–Ramanujan identities using cylindric +partitions [17]. In 2019, Corteel and Welsh [18] derived functional equations for the bivariate generating func- +tions for the number the number of cylindric partitions using the largest part statistic. While doing so, they +also gave a new proof of Andrews–Schilling–Warnaar’s modulo 7 A2 Rogers–Ramanujan identities (including +(1.3)) and a fifth missing identity which was originally conjectured by Feigin–Foda–Welsh [20]. +All these +modulo 7 identities have manifestly positive sum sides. [18] has been the catalyst for the recent developments. +Ablinger and the author [1] implemented the Corteel–Welsh’s cylindric partitions related functional equations +in their symbolic computation implementation qFunctions to be able to exploit this combinatorial idea using +formal manipulation and computer algebra techniques. Corteel, Dousse and the author [19] later proved the +modulo 8 identities that arise from the cylindric partitions paradigm with the help of this implementation. +One of such identities is as follows (see Theorem 1.6 in [19]). +Theorem 1.7 (Corteel–Dousse–U., 2021). +(1.6) +� +r1≥s1≥r2≥0 +r1≥s2≥0 +qr2 +1−r1s1+s2 +1+r2 +2+s2 +2+s1s2+r1+r2+s1+s2 +(q; q)r1 +�r1 +s1 +� +q +�r1 +s2 +� +q +�s1 +r2 +� +q += +1 +θ(q2, q3, q3, q4, q4, q5; q10) +Unlike (1.4), (1.6) has a manifestly positive sum-side. Shortly after [19], in late 2021, Warnaar [42] come +up with many beautiful conjectures for manifestly positive sum-sides related to higher moduli (not divisible +by 3). In 2022, Tsuchioka [41] proved manifestly positive sum-sides for modulus 6 using finite-automata and +automated proofs. He was also able to analyze the structure of relevant level 3 standard modules for the affine +Lie algebra A(1) +2 . +In a different vein, Bridges and the author studied weighted versions of cylindric partitions as well as +cylindric partitions into distinct parts in [12]. +Earlier in 2022, Kanade and Russell [29] aimed (and succeeded) at conjecturing A2 Rogers–Ramanujan +type identities in the form of Andrews–Schilling–Warnaar instead of aiming for manifestly positive sum-sides. +They were able to make explicit claims for each modulus ≥ 5. They proved the cases for moduli 5, 6, 7, 8 and +10. Their exploration came to an end due to the increasing computational difficulties. +In this paper, we approach the conjectures of Kanade–Russell by changing the computational techniques +used. We prove all modulo 11 and 13 A2 Rogers–Ramanujan identities coming from the cylindric partitions +paradigm. Two such identities are as follows: + +4 +ALI KEMAL UNCU +Theorem 1.8. +� +r1≥r2≥r3≥0 +s1≥s2≥s3≥0 +qr2 +1−r1s1+s2 +1+r2 +2−r2s2+s2 +2+r2 +3+r3s3+s2 +3+r1+r2+r3+s1+s2+s3 +(q; q)r1−r2(q; q)r2−r3(q; q)r3(q; q)s1−s2(q; q)s2−s3(q; q)s3(q; q)r3+s3+1 += +1 +(q; q)∞ +1 +θ(q2, q3, q3, q4, q4, q5, q5; q11). +Theorem 1.9. +� +r1≥r2≥r3≥0 +s1≥s2≥s3≥0 +qr2 +1−r1s1+s2 +1+r2 +2−r2s2+s2 +2+r2 +3−r3s3+s2 +3+r1+r2+r3+s1+s2+s3 +(q; q)r1−r2(q; q)r2−r3(q; q)r3(q; q)s1−s2(q; q)s2−s3(q; q)s3(q; q)r3+s3+1 += +1 +(q; q)∞ +1 +θ(q2, q3, q3, q4, q4, q5, q5, q6, q6; q13). +The organization of this paper is as follows. In Section 2, we introduce cylindric partitions, the relevant +results and the conjectures of Kanade–Russell of which we prove some cases of. Section 3 is dedicated to +rewording the conjectures and the description ot the proof methodology. In Sections 4 and 5 we present +the proofs of the modulo 11 and 13 A2 Rogers–Ramanujan identities in Andrews–Schilling–Warnaar form, +respectively. We outline some natural questions and mathematical challenges that arise from this work in +Section 6. Section 7 is reserved for a discussion on how the computerized proofs have been carried in earlier +work [19, 29] and this paper and what future improvements can be done to take us further mathematically. +Acknowledgement +The author would like to thank the workshop on cylindric partitions group that came together in November +2022 in Linz for all the stimulating discussions. In particular, the author would like to thank Shashank Kanade +for suggesting that the researchers working on cylindric partitions should come together and join forces in the +first place, and for all his comments on this manuscript. +The author would also like to thank Christian +Koutschan his encouragement of the author in the necessary implementations. +Research of the author is partly supported by EPSRC grant number EP/T015713/1 and partly by FWF +grant P-34501N. +2. Necessary definitions +We shall start with the definition of a cylindric partition. +Definition 2.1. A cylindric partition is made up of a composition c = (c1, c2, . . . , cr) called profile with r +parts, and a vector π = (π(1), π(1), . . . , π(r)) consisting of r partitions π(i) = (π(i) +1 , π(i) +2 , . . . ), that satisfy the +inequalities +π(i) +j +≥ π(i+1) +j+ci+1 +and +π(r) +j +≥ π(1) +j+c1. +For example, the vector partition π = {(1, 1, 1, 1), (4, 3, 1), (2, 2)} together with the profile (2, 0, 2) is a +cylindric partition. Note that the same vector partition can also satisfy the cylindric partition inequalities +with different profiles. For example, π is also a cylindric partition for profiles (2, 0, 0), (2, 0, 1), etc. We can +define the total size of a cylindric partition π as the sum of all the sizes of the partitions included. We denote +the total size, once again, by |π|. Let c be a composition and let Pc be the set of all vector partitions that are +cylindric partitions with profile c. +For a given profile c, let Pc be the set of all cylindric partitions with profile c. Let +Fc(z, q) := +� +π∈Pc +zmax(π)q|π|, +the bivariate generating function for the number of cylindric partitions where the exponents of z and q are +keeping record of the largest parts size and the total of the parts in π, respectively. Borodin [11] showed that +when z = 1, Fc(z, q) generating functions have product formula. + +MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES +5 +Theorem 2.2 (Borodin, 2007). Let r and l be positive integers, and let c = (c1, c2, . . . , cr) be a composition +of l. Define m := r + l and s(i, j) := ci + ci+1 + · · · + cj. Then, +(2.1) +Fc(1, q) = +1 +(qm; qm)∞ +r +� +i=1 +r +� +j=i +ci +� +k=1 +1 +(qk+j−i+s(i+1,j); qm)∞ +r +� +i=2 +i� +j=2 +ci +� +k=1 +1 +(qm−k+j−i−s(j,i−1); qm)∞ +. +Focusing on replacing the largest part in a given cylindric partition, Corteel–Welsh [18] defined a q-difference +equation for Fc(z, q). This functional equation relates Fc(z, q) with other generating functions Fc∗(z, q) where +#(c) = #(c∗) and |c| = |c∗|. Let c = (c1, . . . , cr) (with the convention that c0 = cr) be a given composition +and define Ic to be the set of indices for the non-zero entries in c. Given a non-empty subset J ⊆ Ic, the +composition c(J) = (c1(J), . . . , cr(J)) is defined by: +(2.2) +ci(J) := + + + + + +ci − 1 +if i ∈ J and i − 1 /∈ J, +ci + 1 +if i /∈ J and i − 1 ∈ J, +ci +otherwise. +Then the explicit q-difference equation Fc(z, q) satisfies is as follows. +Theorem 2.3 (Corteel–Welsh, 2019). For any profile c, +(2.3) +Fc(z, q) = +� +∅⊂J⊆Ic +(−1)|J|−1 Fc(J)(zq|J|, q) +(1 − zq|J|) +, +with the initial conditions Fc(0, q) = Fc(z, 0) = 1. +Let c be a profile and c′ be a cyclic shift of c. There is a clear one-to-one correspondence between cylindric +partitions in Pc and Pc′ by cyclically shifting the vector of partitions counted in Pc. This is enough to see that +the generating functions for these sets of cylindric partitions are equal, i.e. Fc(z, q) = Fc′(z, q). Therefore, we +can cyclically shift the profiles and lower the number of (seemingly different) generating functions that appear +in the coupled system of q-difference equations. +We can also normalize (2.3) and get an equivalent q-difference equation. For example, let +Gc(z, q) := (zq; q)∞Fc(z, q). +The equation (2.3) is equivalent to +(2.4) +Gc(z, q) = +� +∅⊂J⊆Ic +(−1)|J|−1(zq; q)|J|−1Gc(J)(zq|J|, q), +with the initial conditions Gc(0, q) = Gc(z, 0) = 1. This q-difference equation (2.4) with polynomial coeffi- +cients, in practice, played a central role in the proofs of modulo 7 and modulo 8 cylindric partition with 3-part +profile identities in [18] and [19], respectively. Weighted versions of (2.3) and (2.4) are later presented in [12]. +In [29], Kanade–Russell decided to change the initial conditions of (2.4) slightly. While this does not change +the q-difference equations, this lead to the conjectural discovery of explicit formulas for most of these 3-part +profile cylindric partition generating functions. Let +(2.5) +Hc(z, q) := (zq; q)∞ +(q; q)∞ +Fc(z, q). +Then Hc(z, q) satisfies the same q-difference equation as Gc(z, q), namely +(2.6) +Hc(z, q) = +� +∅⊂J⊆Ic +(−1)|J|−1(zq; q)|J|−1Hc(J)(zq|J|, q), +with the initial conditions Hc(0, q) = 1/(q; q)∞ and Hc(z, 0) = 1. +From this point forward we only focus on cylindric partition profiles with 3-parts. +Let k ≥ 2, let +(2.7) +ρ = (ρ1, ρ2, . . . , ρk−1) ∈ Zk−1, and σ = (σ1, σ2, . . . , σk−1) ∈ Zk−1 + +6 +ALI KEMAL UNCU +and define +S3k−1(z; ρ|σ) = +� +r,s∈Zk−1 +≥0 +zr1 +q +�k−1 +i=1 r2 +i −risi+s2 +i +ρiri+σisi +�k−2 +i=1 (q; q)ri−ri+1(q; q)si−si+1 +q2rk−1sk−1 +(q; q)rk−1(q; q)sk−1(q; q)rk−1+sk−1+1 +, +(2.8) +S3k(z; ρ|σ) = +� +r,s∈Zk−1 +≥0 +zr1 +q +�k−1 +i=1 r2 +i −risi+s2 +i +ρiri+σisi +�k−2 +i=1 (q; q)ri−ri+1(q; q)si−si+1 +1 +(q; q)rk−1+sk−1(q; q)rk−1+sk−1+1 +�rk−1 + sk−1 +rk−1 +� +q3, +(2.9) +S3k+1(z; ρ|σ) = +� +r,s∈Zk−1 +≥0 +zr1 +q +�k−1 +i=1 r2 +i −risi+s2 +i +ρiri+σisi +�k−2 +i=1 (q; q)ri−ri+1(q; q)si−si+1 +1 +(q; q)rk−1(q; q)sk−1(q; q)rk−1+sk−1+1 +. +(2.10) +Let +(2.11) +ei = (0, 0, . . . , 0 +� +�� +� +i +, 1, 1, . . ., 1) ∈ Zk−1 +and +δi := (δij)1≤j≤k−1 ∈ Zk−1. +It is easy to see that +(2.12) +Sm(zqn; ρ|σ) = Sm(z; ρ + nδ1|σ) +for any n ∈ Z. +Kanade–Russell conjectured that for any fixed k ≥ 3, the H(c1,c2,c3)(z, q) can be expressed as linear combi- +nations of the Sm(z; ρ|σ) functions. Precisely they claimed the following. +Conjecture 2.4 (Kanade–Russell, 2022). Let k ≥ 3 and |c|+3 = m = 3k+{−1, 0, 1}. Using cyclis symmetries +asusme that c1 ≥ c2, c3. If c2, c3 ≤ k − 1, then +(2.13) +H(c1,c2,c3)(z, q) = + + + +Sm(z; ec2|ec3) − qSm(z; ec2−1|ec3−1), +c2, c3 > 0, +Sm(z; ec2|e0), +c3 = 0, +Sm(z; e0|ec3) − q(1 − z)Sm(z; e0 + δ0|ec3−1), +c2 = 0, c3 ̸= 0, +where ei and δi be defined as in (2.11). +The explicit claims that Conjecture 2.4 provide do not cover all the functions Hc(z, q) with |c| + 3 = m. +It does provide enough claims to recover explicit expression claims. How to find the conjectural Sm(z; ρ|σ) +equivalents of the other Hc(z, q) that appear in the coupled q-difference equation system is explained in [29]. +The profiles related to the functions to be recovered are called ”under-the-line” profiles by Kanade–Russell. +We will also call these profiles as such while we ignore to explain anything about the line. These under-the-line +profile related functions can have shifts of z in the Sm(z; ρ|σ) language. These shifts are inherited from the +q-difference equations (2.6). One can use (2.12) to clear all the shifts in z. Therefore, from now on in all our +expressions we will translate any z shift of Sm(z; ρ|σ) using (2.12) and this way ignore any and all shifts in z. +To further emphasize this moving forward on we suppress the variable z from our notation and write +Sm(ρ|σ) := Sm(z; ρ|σ). +Proof of Conjecture 2.4 (and its extension to all 3-part profiles with total m − 3) requires one to show that +the expressions in Sm(ρ|σ) are the correct expressions for the respective Hc(z; q) functions. This can be done +by showing that the expressions in Sm(ρ|σ) satisfies the same recurrence relation specified by (2.6) and the +initial conditions of the expression holds. In [29], it is already proven that for c2, c3 ≤ k − 1, the conjectural +formulas of (2.13) all satisfy the necessary initial conditions +(2.14) +Hc(z, 0) = 1 +and +Hc(0, z) = 1/(q; q)∞. +It was noted in the [29, Lemma 9.1, Lemma 9.2] that Sm(ρ|σ) functions satisfy the following list of recur- +rences. + +MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES +7 +Lemma 2.5 (Kanade–Russell, 2022). Let k ≥ 3, let m = 3k + {−1, 0, 1} and let δi := (δij)1≤j≤k−1 ∈ Zk−1, +where δij is the Kronecker delta function. The following recurrence relations follow for all 1 ≤ i ≤ k − 2, +Sm(ρ|σ) − Sm(ρ + δi − δi+1|σ) − zqi+�i +j=1 ρjSm(ρ + 2 +i +� +j=1 +δj|σ − +i +� +j=1 +δj) = 0, +(R(i) +1 (ρ|σ)) +Sm(ρ|σ) − Sm(ρ|σ + δi − δi+1) − zqi+�i +j=1 σjSm(ρ − +i +� +j=1 +δj|σ + 2 +i +� +j=1 +δj) = 0. +(R(i) +2 (ρ|σ)) +i. If m ≡ −1 (mod 3), +a) and if σk−1 = 0, then +(R3(ρ|σ)) +Sm(ρ|σ) − Sm(ρ|σ + δk−1) − qSm(ρ + δk−1|σ + δk−1) + qSm(ρ + δk−1|σ + δk−2 + δk−1) = 0. +b) and if ρk−1 = 0, then +(R4(ρ|σ)) +Sm(ρ|σ) − Sm(ρ + δk−1|σ) − qSm(ρ + δk−1|σ + δk−1) + qSm(ρ + δk−2 + δk−1|σ + δk−1) = 0. +ii. If m ≡ 0 (mod 3), then +Sm(ρ|σ) − (1 + q)Sm(ρ + δk−1|σ + δk−1) + qSm(ρ + 2δk−1|σ + 2δk−1) +(R3(ρ|σ)) +− zqk−1+�k−1 +j=1 ρjSm(ρ + 2 +k−1 +� +j=1 +δj|σ − +k−1 +� +j=1 +δj) +− qk−1+�k−1 +j=1 σjSm(ρ − +k−2 +� +j=1 +δj + 2δk−1|σ + 2 +k−1 +� +j=1 +δj) = 0. +Sm(ρ|σ) − (1 + q)Sm(ρ + δk−1|σ + δk−1) + qSm(ρ + 2δk−1|σ + 2δk−1) +(R4(ρ|σ)) +− zqk−1+�k−1 +j=1 ρjSm(ρ + 2 +k−1 +� +j=1 +δj|σ − +k−2 +� +j=1 +δj + 2δk−1) +− qk−1+�k−1 +j=1 σjSm(ρ − +k−1 +� +j=1 +δj|σ + 2 +k−1 +� +j=1 +δj) = 0. +iii. If m ≡ 1 (mod 3), then +Sm(ρ|σ) − Sm(ρ|σ + δk−1) − qSm(ρ + δk−1|σ + 2δk−1) +(R3(ρ|σ)) ++ qSm(ρ + δk−1|σ + 2δk−1) − qk−1+�k−1 +j=1 σjSm(ρ − +k−1 +� +j=1 +δj|σ + 2 +k−1 +� +j=1 +δj) = 0 +Sm(ρ|σ) − Sm(ρ + δk−1|σ) − qSm(ρ + δk−1|σ + δk−1) +(R4(ρ|σ)) ++ qSm(ρ + 2δk−1|σ + δk−1) − zqk−1+�k−1 +j=1 ρjSm(ρ + 2 +k−1 +� +j=1 +δj|σ − +k−1 +� +j=1 +δj) = 0. +Then they made the following claim (see [29, Conjecture 9.3]). +Conjecture 2.6 (Kanade–Russell, 2022). In each modulus m ≥ 5, the relations (R(i) +1 (ρ|σ))-(R4(ρ|σ)) are +enough to prove recurrences necessary for the proof of Conjecture 2.4. +We find this conjecture highly sensible. For all m ≥ 5, the explicit Sm’s are 2⌊m/3⌋-fold sums. Same is +true for the number of distinct functional equations (R(i) +1 (ρ|σ))-(R4(ρ|σ)). One can easily check that these +relations are distinct by comparing the first two terms in each left-hand side. Each second term corresponds to +a canonical shift in one of the summation variables. One would expect to see every relation that the Sm(ρ|σ) +functions satisfy to be translated and recovered as a combination the relations (R(i) +1 (ρ|σ))-(R4(ρ|σ)). Hence, + +8 +ALI KEMAL UNCU +if the claims of Conjecture 2.4 are correct, for any fixed profile c the coupled q-difference equations (2.6) +written using the explicit claims of (2.13) (together with the “under-the-line” expressions) can be recovered +as a combination of the relations (R(i) +1 (ρ|σ))-(R4(ρ|σ)). +3. Proof Methodology +Conjecture 2.6 can be rephrased as a set inclusion question. Let m = 3k + {−1, 0, 1} with k ≥ 3, ρ and σ +as in (2.7) and 1 ≤ i ≤ k − 2. Define +(3.1) +Im := ⟨R(i) +1 (ρ|σ), R(i) +2 (ρ|σ), R3(ρ|σ), R4(ρ|σ)⟩, +the ideal generated by the left-hand sides of the recurrences (R(i) +1 (ρ|σ))-(R4(ρ|σ)) as polynomials in the ring +Z((q, z))[Sm(ρ|σ)]. Here which R3(ρ|σ) and R4(ρ|σ) to be included in Im is to be understood by the residue class +of m modulo 3. Recall that ρ and σ are integer vectors with k −1 entries. Therefore the ring Z((q, z))[Sm(ρ|σ)] +is a formal polynomial ring defined on a countable set of variables. +For any given fixed m = 3k + {−1, 0, 1} with k ≥ 3, let the set of all the coupled system of q-difference +equations (2.6) for the profiles c with |c| + 3 = m be Hm. Any relation in Hm can be written in Sm(ρ|σ) +functions using the Conjecture 2.4 (and the paragraph below it). Let Sm be the set of all relations in Hm +written in the conjectural Sm(ρ|σ) form. +Now we can write Conjecture 2.6 in its equivalent form: +Conjecture 3.1. Let m = 3k + {−1, 0, 1} with k ≥ 3 be fixed. +∀h ∈ Sm, we have h ∈ Im. +The infinite set {R(i) +1 (ρ|σ), R(i) +2 (ρ|σ), R3(ρ|σ), R4(ρ|σ) : 1 ≤ i ≤ k − 2, ρ, σ ∈ Zk−1} that spans Im has +non-trivial relations within itself and not all the elements of this set are generators of Im. However, we do not +know an exact pattern of which elements are related at the moment. Nevertheless, it is easy to understand +that Im is generated by infinitely many elements since ρ and σ ∈ Zk−2. +On the other hand, for any fixed m, the Sm(ρ|σ) functions that appear within the formulas from Sm +make up a finite list. One can easily find explicit bounds for the entries of vectors ρ and σ such that every +Sm(ρ|σ) that appear in Sm is within the bounds. This observation suggests that instead of attempting to +prove Conjecture 3.1, we can instead go after a stronger conjecture that is more suitable for computations. +To that end, let [N] := {−N, . . . , −1, 0, 1, . . ., N} and we define +Im,N := ⟨{R(i) +1 (ρ|σ), R(i) +2 (ρ|σ), R3(ρ|σ), R4(ρ|σ) : ρ, σ ∈ [N]k−1}⟩ ⊂ Im. +With this definition we form the stronger conjecture: +Conjecture 3.2. Let m = 3k + {−1, 0, 1} with k ≥ 3 be fixed. There is some N ∈ N such that +∀h ∈ Sm, we have h ∈ Im,N. +Since Im,N ⊂ Im, it is clear that Conjecture 3.2 implies Conjecture 3.1. +Finally we transferred the open problems into a linear algebra setting, and we can approach it as such. +Let m and N be fixed. we can order all the Sm(ρ|σ) that appears in the spanning set of Im,N and write in +a column vector ⃗s. Then the matrix A is uniquely defined by +A⃗s = ⃗0A, +where ⃗0A is the colum vector with the same number of rows as A. Every row of A, corresponds to a functional +relation Rj(σ|ρ) ∈ {R(i) +1 (ρ|σ), R(i) +2 (ρ|σ), R3(ρ|σ), R4(ρ|σ) : ρ, σ ∈ [N]k−1} and every column of A corresponds +to the coefficients of Sm(ρ|σ). Also observe that A is a finite dimensional matrix with entries in Z[q, z]. +One can use Gaussian elimination on A. Any non-trivial relation within the functional relations (R(i) +1 (ρ|σ))- +(R4(ρ|σ)) within the defining bounds of A would yield 0 rows. Let B be the matrix consisting of non-zero +rows of A after the Gaussian elimination is performed. It should still be clear that +B⃗s = ⃗0B. + +MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES +9 +Moreover, the ideal Im,N is generated by the equations that appear in B⃗s = ⃗0. +Therefore, for any element of h ∈ Sm one can check whether that element is in Im,N by simply writing that +relation as a row vector ⃗h (with respect to the vector ⃗s, i.e. +vech is defined by h := [⃗h⃗s = 0]), add the row vector ⃗h to B and perform Gaussian elimination to this new +matrix. If the Gaussian elimination yields a zero row, this means that ⃗h is a linear combination of rows in B, +or equivalently this means h ∈ Im,N. If no zero row appears, then h ̸∈ Im,N. +This approach is clearly algorithmic. Furthermore, termination of the algorithm and a definitive answer +among the termination are both guaranteed. Top it all up, the explicit combination of (R(i) +1 (ρ|σ))-(R4(ρ|σ)) +functional equations that is equivalent to a given h ∈ Sm is also easy to find. One only needs to use an +augmented version of A where one more column is added to keep track of the name of the relations (R(i) +1 (ρ|σ))- +(R4(ρ|σ)) while doing the row reductions. +After these considerations, proof of Conjectures 3.2 (and consequently Conjectures 2.4, 2.6, and 3.1) comes +down to experimentally identifying an N and being able to perform the Gaussian elimination calculations. +4. Modulo 11 Identities +Let m = 11 (= 3k − 1 with k = 4), for this family of identities ρ and σ ∈ Z3. There are a total of 15 +essentially unique 3 part compositions of 8 that appear in the coupled q-difference system (2.6). Conjecture 2.4 +suggests that the following sum representations for Hc(z, q) hold for all but one of these: +(4.1) +H(8,0,0)(z, q) += S11((1, 1, 1) | (1, 1, 1)), +H(7,1,0)(z, q) += S11((0, 1, 1) | (1, 1, 1)), +H(7,0,1)(z, q) += S11((1, 1, 1) | (0, 1, 1)) − q(1 − z)S11((2, 1, 1) | (1, 1, 1)), +H(6,2,0)(z, q) += S11((0, 0, 1) | (1, 1, 1)), +H(6,1,1)(z, q) += S11((0, 1, 1) | (0, 1, 1)) − qS11((1, 1, 1) | (1, 1, 1)), +H(6,0,2)(z, q) += S11((1, 1, 1) | (0, 0, 1)) − q(1 − z)S11((2, 1, 1) | (0, 1, 1)), +H(5,3,0)(z, q) += S11((0, 0, 0) | (1, 1, 1)), +H(5,2,1)(z, q) += S11((0, 0, 1) | (0, 1, 1)) − qS11((0, 1, 1) | (1, 1, 1)), +H(5,1,2)(z, q) += S11((0, 1, 1) | (0, 0, 1)) − qS11((1, 1, 1) | (0, 1, 1)), +H(5,0,3)(z, q) += S11((1, 1, 1) | (0, 0, 0)) − q(1 − z)S11((2, 1, 1) | (0, 0, 1)), +H(4,3,1)(z, q) += S11((0, 0, 0) | (0, 1, 1)) − qS11((0, 0, 1) | (1, 1, 1)), +H(4,2,2)(z, q) += S11((0, 0, 1) | (0, 0, 1)) − qS11((0, 1, 1) | (0, 1, 1)), +H(4,1,3)(z, q) += S11((0, 1, 1) | (0, 0, 0)) − qS11((1, 1, 1) | (0, 0, 1)), +H(3,3,2)(z, q) += S11((0, 0, 0) | (0, 0, 1)) − qS11((0, 0, 1) | (0, 1, 1)). +Only H(4,4,0)(z, q) misses a claimed formula and that can be recovered by the q-difference equations (2.6). +We know that H(4,4,0)(z, q) satisfies +(4.2) +H(4,4,0)(z, q) + (1 − qz)H(4,1,3)(q2z, q) − H(4,3,1)(qz, q) − H(5,0,3)(qz, q) = 0. +Using the conjectured series equivalents (4.1) of H(4,1,3)(z, q), H(4,3,1)(z, q) and H(5,0,3)(z, q), we see that +H(4,4,0)(z, q) = −(S11(qz; (0, 0, 0)|(0, 1, 1)) − qS11(qz; (0, 0, 1)|(1, 1, 1))) ++ (1 − qz)(S11(q2z; (0, 1, 1)|(0, 0, 0)) − qS11(q2z; (1, 1, 1)|(0, 0, 1))) +(4.3) +− (S11(qz; (1, 1, 1)|(0, 0, 0)) − q(1 − z)S11(qz(2, 1, 1)|(0, 0, 1))). +Notice that we used the shifts in the variable z in (4.3). We clear these shifts by employing (2.12). This yields +an explicit claim for H(4,4,0)(z, q): +(4.4) +H(4,4,0)(z, q) = S11((1, 0, 0) | (0, 1, 1)) − qS11((1, 0, 1) | (1, 1, 1)) + qzS11(2, 1, 1) | (0, 0, 0)), +with no shifts in z, where the S11(ρ|σ) functions fit the forms in Lemma 2.5. +We can also see that H(4,4,0)(z, q) satisfies the necessary initial conditions (2.14). The initial condition +H(4,4,0)(z, 0) = 1 is immediate by (4.4) and (2.8). We can also see that H(4,4,0)(0, q) = 1/(q; q)∞ by plugging + +10 +ALI KEMAL UNCU +in z = 0 in (4.2) and using the initial conditions of the other proven initial conditions (2.14) for the functions +in (4.2). +Our proof routine explained in Section 3 can start once all the normalized generating functions Hc(z, q)’s +are (conjecturally) translated in the S11((a1, a2, a3)|(b1, b2, b3)) language. It is easy to see that the following +four recurrences, +H(8,0,0)(z, q) − H(7,1,0)(qz, q) = 0, +H(7,0,1)(z, q) − H(6,1,1)(qz, q) + (1 − qz)H(7,1,0)(q2z, q) − H(8,0,0)(qz, q) = 0, +H(6,0,2)(z, q) − H(5,1,2)(qz, q) + (1 − qz)H(6,1,1)(q2z, q) − H(7,0,1)(qz, q) = 0, +H(5,0,3)(z, q) − H(4,1,3)(qz, q) + (1 − qz)H(5,1,2)(q2z, q) − H(6,0,2)(qz, q) = 0, +trivializes to 0 = 0 once the terms on the left-hand sides are written in S11((a1, a2, a3)|(b1, b2, b3)) using (4.1) +and (2.12). Therefore, these relations are trivially in I11, the ideal generated by the functional relations of the +S11((a1, a2, a3)|(b1, b2, b3)) series. +Recall that we used the coupled q-difference equation (4.2) to make an explicit claim for H(4,4,0)(z, q). +Hence, the functional relation of H(4,4,0)(z, q) also trivializes to 0 = 0 once written in the claimed S11(ρ|σ) +forms. The very claim (4.4) is instrumental in proving that the q-difference equations satisfied by H(5,3,0)(z, q), +H(4,3,1)(z, q), and H(4,1,3)(z, q) in S11((a1, a2, a3)|(b1, b2, b3)) language are elements of I11. +Next, we look at the q-difference equation satisfied by H(7,1,0)(z, q) from (2.6): +H(7,1,0)(z, q) − H(7,0,1)(qz, q) − H(6,2,0)(qz, q) + (1 − qz)H(6,1,1)(q2z, q) = 0. +After the use of (4.1) and (2.12), we see that this q-difference equation is equivalent to the following conjectural +form +S11((0, 1, 1)|(1, 1, 1)) − S11((1, 0, 1)|(1, 1, 1)) − qzS11((2, 1, 1)|(0, 1, 1)) = 0. +This is nothing but the relation R(1) +1 (0, 1, 1)|(1, 1, 1) of (R(i) +1 (ρ|σ)) given in Lemma 2.5. Hence, this relation is +also within I11 and covered by the relations of S11((a1, a2, a3)|(b1, b2, b3)). +As a second explicit example, consider the q-difference equation satisfied by H(6,1,1)(z, q), +H(6,1,1)(z, q) − H(7,1,0)(qz, q) − H(6,0,2)(qz, q) − H(5,2,1)(qz, q) + (1 − qz)H(7,0,1)(q2z, q) ++ (1 − qz)H(6,2,0)(q2z, q) + (1 − qz)H(5,1,2)(q2z, q) − (1 − qz)(1 − q2z)H(6,1,1)(q3z, q) = 0. +Using employing (4.1) and (2.12), we see get the (conjecturally) equivalent form +S11((0, 1, 1)|(0, 1, 1)) − S11((1, 0, 1)|(0, 1, 1)) − S11((1, 1, 1)|(1, 1, 1)) ++ (1 − qz)S11((2, 0, 1)|(1, 1, 1)) − qzS11((2, 1, 1)|(0, 0, 1)) + q2z(1 − qz)S11((3, 1, 1)|(0, 1, 1)) = 0. +This relation can be checked to be the side-by-side additions of +R(1) +1 ((0, 1, 1)|(0, 1, 1)) − (1 − qz)R(1) +1 ((1, 1, 1)|(1, 1, 1)) + qzR(1) +2 ((2, 1, 1)|(−1, 1, 1)) ∈ I11. +We can one-by-one write down the remaining 8 recurrences, their S11(ρ|σ) equivalents, and what combina- +tion of (R(i) +1 (ρ|σ))-(R4(ρ|σ)) is equivalent to the functional equations in the S11(ρ|σ). This way we prove that +these relations are all included in the ideal I11. We need to say that these relations gets messier, pages long and +not hand-verifiable. Printing these would be a waste of page/paper and instead we keep these in the digital +realm for interested readers to check it easily, or print on paper as they wish. To that end, similar to how it was +handled in [29], we include text files M11RecHXYZ Explicit.txt in the ancillary files portion of ArXiv and on +the author’s website [36]. Here XYZ is to be replaced by the relevant profile’s digits such as 620 for the profile +(6, 2, 0). One can check that the elements of I11 given in these text files are equivalent to the q-difference +equations (2.6) satisfied by H(X,Y,Z)(z, q) after they are translated to S11(ρ|σ) form using (4.1), (4.4) and +(2.12). The functional equation names are reflected in the text as RX[{Y},{{a1,a2,a3},{b1,b2,b3}}] for X +and Y to be replaced by 1 or 2 to denote R(Y ) +X ((a1, a2, a3)|(b1, b2, b3)), or RZ[{{a1,a2,a3},{b1,b2,b3}}] for +Z to be replaced by 3 or 4 to denote R3((a1, a2, a3)|(b1, b2, b3)) and R4((a1, a2, a3)|(b1, b2, b3)), respectively. + +MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES +11 +The definitions of these functional equations can be found in Lemma 2.5 for m = 11. A guide document that +explicitly lists each R functional relation for modulo 10 is given in M11R text file. One also can see that the +largest entry within ρ = (a1, a2, a3) and σ = (b1, b2, b3) of the relations (R(i) +1 (ρ|σ))-(R4(ρ|σ)) for the modulo +11 case given in the additional documents is 6. This proves the following theorem and its corollary. +Theorem 4.1. Conjecture 3.2 is correct for m = 11 and N = 6. +Corollary 4.2. Conjecture 3.1 is correct for m = 11. +Corollary 4.2 is equivalent to the following theorem: +Theorem 4.3. The claimed expressions (4.1) and (4.4) hold. +Observe that Theorem 4.3 adds a new supporting case to Corollary 2.6. +Now that the main conjectures are proven for the modulus 11 cases, we can specialize z = 1 and see the 15 +sum-product identities coming from the cylindric partitions paradigm. +Theorem 4.4. The following identities hold +� +r1≥r2≥r3≥0 +s1≥s2≥s3≥0 +qr2 +1−r1s1+s2 +1+r2 +2−r2s2+s2 +2+r2 +3+r3s3+s2 +3 pc(r1, r2, r3, s1, s2, s3, q) +(q; q)r1−r2(q; q)r2−r3(q; q)r3(q; q)s1−s2(q; q)s2−s3(q; q)s3(q; q)r3+s3+1 +(4.5) += +1 +(q; q)∞ +1 +θ(qi1, qi2, qi3, qi4, qi5, qi6, qi7; q11), +where the polynomials pc(r1, r2, r3, s1, s2, s3, q) and the 7-tuples (i1, i2, i3, i4, i5, i6, i7) for each profile is given +in the following table: +Profile c +pc(r1, r2, r3, s1, s2, s3, q) +(i1, i2, i3, i4, i5, i6, i7) +(8, 0, 0) +qr1+r2+r3+s1+s2+s3 +(2, 3, 3, 4, 4, 5, 5) +(7, 1, 0) +qr2+r3+s1+s2+s3 +(1, 2, 3, 4, 4, 5, 5) +(7, 0, 1) +qr1+r2+r3+s2+s3 +(1, 2, 3, 4, 4, 5, 5) +(6, 2, 0) +qr3+s1+s2+s3 +(1, 2, 2, 3, 4, 5, 5) +(6, 1, 1) +qr2+r3+s2+s3(1 − qr1+s1+1) +(1, 1, 3, 3, 4, 5, 5) +(6, 0, 2) +qr1+r2+r3+s3 +(1, 2, 2, 3, 4, 5, 5) +(5, 3, 0) +qs1+s2+s3 +(1, 2, 2, 3, 3, 4, 5) +(5, 2, 1) +qr3+s2+s3(1 − qr2+s1+1) +(1, 1, 2, 3, 4, 4, 5) +(5, 1, 2) +qr2+r3+s3(1 − qr1+s2+1) +(1, 1, 2, 3, 4, 4, 5) +(5, 0, 3) +qr1+r2+r3 +(1, 2, 2, 3, 3, 4, 5) +(4, 3, 1) +qs2+s3(1 − qr3+s1+1) +(1, 1, 2, 3, 3, 4, 5) +(4, 2, 2) +qr3+s3(1 − qr2+s2+1) +(1, 1, 2, 2, 4, 4, 5) +(4, 1, 3) +qr2+r3(1 − qr1+s3+1) +(1, 1, 2, 3, 3, 4, 5) +(3, 3, 2) +qs3(1 − qr3+s2+1) +(1, 1, 2, 2, 3, 5, 5) +(4, 4, 0) +qr1(qs2+s3 − qr3+s1+s2+s3+1 + qr1+r2+r3+1) +(1, 2, 2, 3, 3, 4, 4) +In Theorem 4.4, we chose to put the profile (4, 4, 0) related sum-product identity under a line in the table. +This is to indicate that this identity is not a direct claim made by combining (2.4) with z = 1 and (2.1). We +first recovered a formula for H(4,4,0)(z, q) as a combination of S11((a1, a2, a3)|(b1, b2, b3)) series and then made +this claim. This line also has the added benefit that it aligns us with Kanade–Russell’s language as this is the +sum-product identity related to the under-the-line Hc(z, q) function, which we chose not to directly define. +The sum sides are the expressions (4.1) and (4.4) with z = 1 written explicitly using (2.8) with k = 4. The +product sides follow from (2.5) with z = 1 followed by (2.1). The product related to the first profile, (8, 0, 0), +on the table is presented in the introduction as Theorem 1.8. +Observe that the products that appear on the right-hand side of (4.5) related to the profiles (c1, c2, c3) and +(c1, c3, c2) are the same. The symmetry for the generating functions have been observed and noted before, for +example in [19, Corollary 2.2]. This symmetry is visible on the sum side of Theorem 4.4 too. One can get + +12 +ALI KEMAL UNCU +the “other” sum by merely replacing the variable ‘r’s and ‘s’s. Note that this is a byproduct of setting z = 1 +and this similarity does not exist on the sum side for generic z. In that light, this theorem consisting of 15 +sum-product identities actually provide a total of 10 essentially unique sum-product identities. +We also note that among these identities the ones related to profiles (8, 0, 0), (6, 1, 1), (4, 2, 2), (3, 3, 2) are +the i = 1, . . . , 4 cases of (5.28), respectively, and (5, 3, 0) and (6, 2, 0) are the σ = 0 and 1 cases of (5.29), +respectively, of [7, Theorem 5.3]. +5. Modulo 13 Identities +Similar to Section 4, we start by listing the explicit claims of Conjecture 2.4 for the modulus m = 13 family. +(5.1) +H(10,0,0)(z, q) += S13((1, 1, 1)|(1, 1, 1)), +H(9,1,0)(z, q) += S13((0, 1, 1)|(1, 1, 1)), +H(9,0,1)(z, q) += S13((1, 1, 1)|(0, 1, 1)) − q(1 − z)S13((2, 1, 1)|(1, 1, 1)), +H(8,2,0)(z, q) += S13((0, 0, 1)|(1, 1, 1)), +H(8,1,1)(z, q) += S13((0, 1, 1)|(0, 1, 1)) − qS13((1, 1, 1)|(1, 1, 1)), +H(8,0,2)(z, q) += S13((1, 1, 1)|(0, 0, 1)) − q(1 − z)S13((2, 1, 1)|(0, 1, 1)), +H(7,3,0)(z, q) += S13((0, 0, 0)|(1, 1, 1)), +H(7,2,1)(z, q) += S13((0, 0, 1)|(0, 1, 1)) − qS13((0, 1, 1)|(1, 1, 1)), +H(7,1,2)(z, q) += S13((0, 1, 1)|(0, 0, 1)) − qS13((1, 1, 1)|(0, 1, 1)), +H(7,0,3)(z, q) += S13((1, 1, 1)|(0, 0, 0)) − q(1 − z)S13((2, 1, 1)|(0, 0, 1)), +H(6,3,1)(z, q) += S13((0, 0, 0)|(0, 1, 1)) − qS13((0, 0, 1)|(1, 1, 1)), +H(6,2,2)(z, q) += S13((0, 0, 1)|(0, 0, 1)) − qS13((0, 1, 1)|(0, 1, 1)), +H(6,1,3)(z, q) += S13((0, 1, 1)|(0, 0, 0)) − qS13((1, 1, 1)|(0, 0, 1)), +H(5,3,2)(z, q) += S13((0, 0, 0)|(0, 0, 1)) − qS13((0, 0, 1)|(0, 1, 1)), +H(5,2,3)(z, q) += S13((0, 0, 1)|(0, 0, 0)) − qS13((0, 1, 1)|(0, 0, 1)), +H(4,3,3)(z, q) += S13((0, 0, 0)|(0, 0, 0)) − qS13((0, 0, 1)|(0, 0, 1)). +There are six profiles that are not covered by Conjecture 2.4. Once again, using (2.6) explicit claims for +the normalized generating functions related to the number of cylindric partitions with these profiles can be +recovered. We make the claims in the following succession. +First we look at the q-difference equation (2.6) that H(7,3,0): +(5.2) +H(7,3,0)(z, q) − H(7,2,1)(qz, q) − H(6,4,0)(qz, q) + (1 − qz)H(6,3,1)(q2z, q) = 0. +By writing the S13((a1, a2, a3)|(b1, b2, b3)) equivalents for the functions in (5.1) and using (2.12), we get +H(6,4,0)(z, q) = S13((−1, 0, 0)|(1, 1, 1)) − S13((0, 0, 1)|(0, 1, 1)) + qS13((0, 1, 1)|(1, 1, 1)) +(5.3) ++ (1 − z)S13((1, 0, 0)|(0, 1, 1)) − q(1 − z)S13((1, 0, 1)|(1, 1, 1)). +Note that we did not use the q-difference equation of H(6,4,0)(z, q) to make a claim for its formula. In +Section 4, there was only a single missing formula. That allowed us to use the q-difference equation for that +very function and get a formula in S11((a1, a2, a3)|(b1, b2, b3))’s with no backwards shifts (i.e. z �→ z/q, which +also reflects as negative indices in the first variable a1). This may not be possible in general. The q-difference +equation H(6,4,0)(z, q) satisfies is +(5.4) +H(6,4,0)(z, q) − H(6,3,1)(qz, q) − H(5,5,0)(qz, q) + (1 − qz)H(5,4,1)(q2z, q) = 0. +The conjectural formulas (5.1) does not cover H(5,5,0)(z, q). Hence, we cannot fully translate H(6,4,0)(z, q) to +a formula made up of S13((a1, a2, a3)|(b1, b2, b3)) series. Nevertheless, as also noted in [29], we can recover +formulas for all the missing functions using other recurrences and backwards shifts in a1. + +MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES +13 +In fact, the recurrence (5.4) and (5.3) can be put together to claim a formula for H(5,5,0)(z, q). After the +similar considerations we claim +H(5,5,0)(z, q) = S13((−2, 0, 0)|(1, 1, 1)) − S13((−1, 0, 1)|(0, 1, 1)) + qS13((−1, 1, 1)|(1, 1, 1)) +(5.5) ++ (1 − z/q − z)S13((0, 0, 0)|(0, 1, 1)) − (q − z − qz)S13((0, 0, 1)|(1, 1, 1)) +− q(1 − z)zS13((1, 0, 0)|(1, 1, 1)) − (1 − z)S13((1, 0, 1)|(0, 0, 1)) ++ q(1 − z)S13((1, 1, 1)|(0, 1, 1)) + (1 − z)(1 − qz)S13((2, 0, 0)|(0, 0, 1)) +− q(1 − z)(1 − qz)S13((2, 0, 1)|(0, 1, 1)). +We point out that the coefficients of the claimed H(5,5,0)(z, q) formula now can be seen to have a Laurent +polynomial. This is a byproduct of the backwards shifts in z. +Using the q-difference equation for H(6,3,1)(z, q), +H(6,3,1)(z, q) − H(7,3,0)(qz, q) − H(6,2,2)(qz, q) − H(5,4,1)(qz, q) + (1 − qz)H(7,2,1)(q2z, q) +(5.6) ++ (1 − qz)H(6,4,0)(q2z, q) + (1 − qz)H(5,3,2)(q2z, q) − (1 − qz)(1 − q2z)H(6,3,1)(q3z, q) = 0, +(5.1) and (5.3) we claim that +H(5,4,1)(z, q) = S13((−1, 0, 0)|(0, 1, 1)) − qS13((−1, 0, 1)|(1, 1, 1)) − zS13((0, 0, 0)|(1, 1, 1)) +(5.7) +− S13((0, 0, 1)|(0, 0, 1)) + qS13((0, 1, 1)|(0, 1, 1)) + (1 − z)S13((1, 0, 0)|(0, 0, 1)) +− q(1 − z)S13((1, 0, 1)|(0, 1, 1)). +Using the q-difference equation for H(5,3,2)(z, q), +H(5,3,2)(z, q) − H(6,3,1)(qz, q) − H(5,2,3)(qz, q) − H(4,4,2)(qz, q) + (1 − qz)H(6,2,2)(q2z, q) +(5.8) ++ (1 − qz)H(5,4,1)(q2z, q) + (1 − qz)H(4,3,3)(q2z, q) − (1 − qz)(1 − q2z)H(5,3,2)(q3z, q) = 0, +(5.1) and (5.7) we claim that +H(4,4,2)(z, q) = S13((−1, 0, 0)|(0, 0, 1)) − qS13((−1, 0, 1)|(0, 1, 1)) − zS13((0, 0, 0)|(0, 1, 1)) +(5.9) +− S13((0, 0, 1)|(0, 0, 0)) + qzS13((0, 0, 1)|(1, 1, 1)) + qS13((0, 1, 1)|(0, 0, 1)) ++ (1 − z)S13((1, 0, 0)|(0, 0, 0)) − qz(1 − z)S13((1, 0, 0)|(1, 1, 1)) +− q(1 − z)S13((1, 0, 1)|(0, 0, 1)). +Then, by the q-difference equation for H(5,2,3)(z, q), +H(5,2,3)(z, q) − H(6,2,2)(qz, q) − H(5,1,4)(qz, q) − H(4,3,3)(qz, q) + (1 − qz)H(6,1,3)(q2z, q) +(5.10) ++ (1 − qz)H(5,3,2)(q2z, q) + (1 − qz)H(4,4,2)(q2z, q) − (1 − qz)(1 − q2z)H(5,2,3)(q3z, q) = 0, +together with (5.1) and (5.9) we claim that +H(5,1,4)(z, q) = S13((−1, 0, 1)|(0, 0, 0)) − qS13((−1, 1, 1)|(0, 0, 1)) − S13((0, 0, 0)|(0, 0, 0)) +(5.11) ++ (1 − z)S13((0, 0, 0)|(0, 0, 1)) − (1 − q)S13((0, 0, 1)|(0, 0, 1)) +− q(1 − z)S13((0, 0, 1)|(0, 1, 1)) + qS13((0, 1, 1)|(0, 1, 1)) ++ (1 − z)S13((1, 0, 0)|(0, 0, 1)) − q(1 − z)zS13((1, 0, 0)|(0, 1, 1)) +− (1 − z)S13((1, 0, 1)|(0, 0, 0)) − q(1 − z)S13((1, 0, 1)|(0, 1, 1)) ++ q2(1 − z)zS13((1, 0, 1)|(1, 1, 1)) + (1 − z)S13((1, 1, 1)|(0, 0, 0)) ++ q(1 − z)S13((1, 1, 1)|(0, 0, 1)) + (1 − z)(1 − qz)S13((2, 0, 0)|(0, 0, 0)) +− q2z(1 − z)(1 − qz)S13((2, 0, 0)|(1, 1, 1)) − (1 − z)(1 − qz)S13((2, 0, 1)|(0, 0, 0)) +− q(1 − z)(1 − qz)S13((2, 0, 1)|(0, 0, 1)) − q2z(1 − z)S13((2, 1, 1)|(0, 0, 1)). + +14 +ALI KEMAL UNCU +Finally, by replacing the formulas in (5.1) and (5.11) in +(5.12) +H(6,0,4)(z, q) − H(7,0,3)(qz, q) − H(5,1,4)(qz, q) + (1 − qz)H(6,1,3)(q2z, q) = 0 +we conjecture that +H(6,0,4)(z, q) = S13((0, 0, 1)|(0, 0, 0)) − qS13((0, 1, 1)|(0, 0, 1)) − S13((1, 0, 0)|(0, 0, 0)) +(5.13) ++ (1 − qz)S13((1, 0, 0)|(0, 0, 1)) − (1 − q)S13((1, 0, 1)|(0, 0, 1)) +− q(1 − qz)S13((1, 0, 1)|(0, 1, 1)) + qS13((1, 1, 1)|(0, 1, 1)) ++ (1 − qz)S13((2, 0, 0)|(0, 0, 1)) − q2z(1 − qz)S13((2, 0, 0)|(0, 1, 1)) +− (1 − qz)S13((2, 0, 1)|(0, 0, 0)) − q(1 − qz)S13((2, 0, 1)|(0, 1, 1)) ++ q3z(1 − qz)S13((2, 0, 1)|(1, 1, 1)) + S13((2, 1, 1)|(0, 0, 0)) ++ q(1 − qz)S13((2, 1, 1)|(0, 0, 1)) + (1 − qz)(1 − q2z)S13((3, 0, 0)|(0, 0, 0)) +− q3z(1 − qz)(1 − q2z)S13((3, 0, 0)|(1, 1, 1)) − (1 − qz)(1 − q2z)S13((3, 0, 1)|(0, 0, 0)) +− q(1 − qz)(1 − q2z)S13((3, 0, 1)|(0, 0, 1)) − q3z(1 − qz)S13((3, 1, 1)|(0, 0, 1)). +We can prove that the later claimed H(6,4,0)(z, q), H(5,5,0)(z, q), H(5,4,1)(z, q), H(5,3,2)(z, q), H(5,2,3)(z, q), +and H(6,0,4)(z, q) the initial conditions Hc(0, q) = 1/(q; q)∞ and Hc(z, 0) = 1 in the succession from (5.2), +(5.4), (5.6), (5.8), (5.10), and (5.12), respectively. To prove the Hc(0, q) = 1/(q; q)∞ initial condition we need +to first shift z �→ z/q in all but the last of the functional equations. +The q-difference equations for H(10,0,0)(z, q), H(9,0,1)(z, q), H(8,0,2)(z, q), and H(7,0,3)(z, q) becomes tau- +tologies once translated into S13 form using (5.1) and (2.12). The q-difference equations for H(7,3,0)(z, q), +H(6,4,0)(z, q), H(6,3,1)(z, q), H(5,3,2)(z, q), H(5,2,3)(z, q), and H(6,0,4)(z, q) are the recurrences used to define the +missing Hc(z, q) functions in the modulo 13 family (see (5.2), (5.4), (5.6), (5.8), (5.10), and (5.12), resp.). +Hence, these equations also trivializes once the relevant functions are written in their claimed S13 forms using +(5.1), (5.3), (5.5), (5.7), (5.9), (5.11), and (5.13) together with (2.12). +After the considerations above, we end up with 10 non-trivial coupled q-difference equations to prove. +Showing that the q-difference equations’ in the claimed S13((a1, a2, a3)|(b1, b2, b3)) belong to the ideal I13, +which is generated by the relations of S13((a1, a2, a3)|(b1, b2, b3))s (see Lemma 2.5), is done by the method +outlined in Section 3. Explicit linear combination of (R(i) +1 (ρ|σ))-(R4(ρ|σ)) equivalents of these 12 functional +equations in S13 form can, once again, be found in the ancillary files portion of ArXiv and on the author’s +website [36] under the file names M13RecHXYZ Explicit.txt. Here XYZ is to be replaced by the relevant +profile’s digits such as 910 for the profile (9, 1, 0). One can check that the elements of I13 given in these +text files are equivalent to the q-difference equations (2.6) satisfied by H(X,Y,Z)(z, q) after they are translated +to S11(ρ|σ) form using (5.1), (5.3), (5.5), (5.7), (5.9), (5.11), and (5.13) and (2.12). The recurrence names +are reflected in the text as RX[{Y},{{a1,a2,a3},{b1,b2,b3}}] for X and Y to be replaced by 1 or 2 to +denote R(Y ) +X ((a1, a2, a3)|(b1, b2, b3)), or RZ[{{a1,a2,a3},{b1,b2,b3}}] for Z to be replaced by 3 or 4 to +denote R3((a1, a2, a3)|(b1, b2, b3)) and R4((a1, a2, a3)|(b1, b2, b3)), respectively. Finally, a guide document that +explicitly lists each R functional relation for modulo 10 is given in M13R text file. +This tedious, error prone and impossible by hand calculation proves the following theorem and its corollary. +Theorem 5.1. Conjecture 3.2 is correct for m = 13 and N = 6. +Corollary 5.2. Conjecture 3.1 is correct for m = 13. +Corollary 5.2 is equivalent to the following theorem: +Theorem 5.3. The claimed expressions of (5.1), (5.3), (5.5), (5.7), (5.9), (5.11), and (5.13) hold. +As before, Theorem 5.3 adds another new witness to Corollary 2.6, and increases our confidence in it. +Now that the main conjectures are proven for the modulus 13 cases, we can set z = 1 and see the 22 +sum-product identities coming from the cylindric partitions paradigm. + +MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES +15 +Theorem 5.4. The following identities hold +� +r1≥r2≥r3≥0 +s1≥s2≥s3≥0 +qr2 +1−r1s1+s2 +1+r2 +2−r2s2+s2 +2+r2 +3−r3s3+s2 +3 pc(r1, r2, r3, s1, s2, s3, q) +(q; q)r1−r2(q; q)r2−r3(q; q)r3(q; q)s1−s2(q; q)s2−s3(q; q)s3(q; q)r3+s3+1 +(5.14) += +1 +(q; q)∞ +1 +θ(qi1, qi2, qi3, qi4, qi5, qi6, qi7, qi8, qi9; q13), +where the polynomials pc(r1, r2, r3, s1, s2, s3, q) and the 9-tuples (i1, i2, i3, i4, i5, i6, i7, i8, i9) for each profile is +given in the following table: +Profile c +pc(r1, r2, r3, s1, s2, s3, q) +(i1, i2, i3, i4, i5, i6, i7, i8, i9) +(10, 0, 0) +qr1+r2+r3+s1+s2+s3 +(2, 3, 3, 4, 4, 5, 5, 6, 6) +(9, 1, 0) +qr2+r3+s1+s2+s3 +(1, 2, 3, 4, 4, 5, 5, 6, 6) +(9, 0, 1) +qr1+r2+r3+s2+s3 +(1, 2, 3, 4, 4, 5, 5, 6, 6) +(8, 2, 0) +qr3+s1+s2+s3 +(1, 2, 2, 3, 4, 5, 5, 6, 6) +(8, 1, 1) +qr2+r3+s2+s3(1 − qr1+s1+1) +(1, 1, 3, 3, 4, 5, 5, 6, 6) +(8, 0, 2) +qr1+r2+r3+s3 +(1, 2, 2, 3, 4, 5, 5, 6, 6) +(7, 3, 0) +qs1+s2+s3 +(1, 2, 2, 3, 3, 4, 5, 6, 6) +(7, 2, 1) +qr3+s2+s3(1 − qr2+s1+1) +(1, 1, 2, 3, 4, 4, 5, 6, 6) +(7, 1, 2) +qr2+r3+s3(1 − qr1+s2+1) +(1, 1, 2, 3, 4, 4, 5, 6, 6) +(7, 0, 3) +qr1+r2+r3 +(1, 2, 2, 3, 3, 4, 5, 6, 6) +(6, 3, 1) +qs2+s3(1 − qr3+s1+1) +(1, 1, 2, 3, 3, 4, 5, 5, 6) +(6, 2, 2) +qr3+s3(1 − qr2+s2+1) +(1, 1, 2, 2, 4, 4, 5, 5, 6) +(6, 1, 3) +qr2+r3(1 − qr1+s3+1) +(1, 1, 2, 3, 3, 4, 5, 5, 6) +(5, 3, 2) +qs3(1 − qr3+s2+1) +(1, 1, 2, 2, 3, 4, 5, 5, 6) +(5, 2, 3) +qs3(1 − qr3+s2+1) +(1, 1, 2, 2, 3, 4, 5, 5, 6) +(4, 3, 3) +(1 − qr3+s3+1) +(1, 1, 2, 2, 3, 3, 5, 6, 6) +(6, 4, 0) +qs2+s3(q−r1+s1 − qr3 + qr2+r3+s1+1) +(1, 2, 2, 3, 3, 4, 4, 5, 6) +(6, 0, 4) +qr3 − qr2+r3+s3+1 − qr1 + q2r1+r2+r3 + qr1+r2+r3+s2+s3+1 ++(1 − q)qr1+s3(1 − qr3 − qr3+s3+1) +−(1 − q)q2r1(qr3 − qs3 − qr2+r3+s3+1 + qr1+r2+r3+s3+3 ++qs2+s3+2 + qr3+s2+s3+1 − qr3+s1+s2+s3+3) ++(1 − q)(1 − q2)qr3(1 − qr3 − qr3+s3+1 + qs1+s2+s3+3) +(1, 2, 2, 3, 3, 4, 4, 5, 6) +(5, 5, 0) +q−2r1+s1+s2+s3 − q−r1+r3+s2+s3 − qs2+s3−1 + qr3+s1+s2+s3 ++q−r1+r2+r3+s1+s2+s3+1 +(1, 2, 2, 3, 3, 4, 4, 5, 5) +(5, 4, 1) +q−r1+s2+s3 − q−r1+r3+s1+s2+s3+1 − qs1+s2+s3 − qr3+s3 ++qr2+r3+s2+s3+1 +(1, 1, 2, 3, 3, 4, 4, 5, 6) +(5, 1, 4) +q−r1+r3 − q−r1+r2+r3+s3+1 + qr2+r3+s2+s3+1 − (1 − q)qr3+s3 − 1 +(1, 1, 2, 3, 3, 4, 4, 5, 6) +(4, 4, 2) +q−r1+s3 − q−r1+r3+s2+s3+1 − qs2+s3 − qr3 + qr3+s1+s2+s3+1 ++qr2+r3+s3+1 +(1, 1, 2, 2, 3, 4, 4, 6, 6) +Once we ignore the symmetries between variables r and s, Theorem 5.4 proves 16 essentially unique sum- +product identities. It can easily be seen that within the under-the-line identities, we do not see these sym- +metries. The product related to the first profile, (10, 0, 0), on the table is presented in the introduction as +Theorem 1.9. +We also note that among these identities the ones related to profiles (10, 0, 0), (8, 1, 1), (6, 2, 2), (4, 3, 3) are +the i = 1, . . . , 4 cases of (5.22), respectively, and (7, 3, 0) and (8, 2, 0) are the σ = 0 and 1 cases of (5.23), +respectively, of [7, Theorem 5.1]. + +16 +ALI KEMAL UNCU +6. Future Directions +There are many mathematical questions that arose from the recent studies on cylindric partitions. It is +relevant to mention some of the future directions we plan to pursue. +The approach outlined in [29] and in this paper attempts to prove sum-representations for all the normalized +generating function Hc(z, q) in one stroke for any fixed |c| where #(c) = 3. The proof requires hefty calculations +after the under-the-line sums are recovered. Then by setting z = 1 and using (2.1), we prove sum-product +identities for all profiles within a cylindric partition system for a fixed modulus, again in one stroke. Therefore, +to prove A2 Rogers–Ramanujan identities we first prove a more general and more complicated combinatorial +connection with a free variable z. The success of this method depends on the completion of these calculations, +which is virtually impossible by hand. +Warnaar [43] mentioned that he build the necessary theory of the Bailey machinery for profiles with 3 +parts. This machinery will allow us to prove one sum-product identity at a time. This is wonderful to hear +and a great advancement in mathematics. Sadly, it comes with its own short-comings. Warnaar acknowledged +that this Bailey machinery can not prove any under-the-line identity at the moment. It can only find the +sum-product relation related to the z = 1 specializations of Conjecture 2.13. This is similar to the situation of +the original Andrews–Schilling–Warnaar paper, where for example at the modulo 7 case the Bailey machinery +there couldn’t reach the under-the-line identity related to the profile (2, 2, 0), which was later proven in [18]. +Be that as it may, we plan to investigate ways to simplify calculations necessary to prove the identities as +a whole in one stroke for the free z case by adding the extra information we gather from Warnaar’s results. +At the very least, for the z = 1 specialization, we should pursue ways to prove under-the-line identities using +the Bailey-machinery-proven over-the-line identities. +There are other sum-product identities that are not visible through the cylindric partitions paradigm.These +identities do not have a related cylindric partition profiles attached to them either. Similar to the under-the- +line identities, we discover and prove these sum representations using the proven relations in the cylindric +partitions system. For example, there are the following two modulo 10 examples similar to (1.4): +� +r1≥r2≥0 +s1≥s2≥0 +qr2 +1−r1s1+s2 +1+r2 +2−r2s2+s2 +2 qs1+s2(1 + qr1+r2+1) +(q; q)r1−r2(q; q)s1−s2(q; q)r2(q; q)s2(q; q)r2+s2+1 += +1 +(q; q)∞ +1 +θ(q, q, q3, q4, q4, q4; q10), +(6.1) +� +r1≥r2≥0 +s1≥s2≥0 +qr2 +1−r1s1+s2 +1+r2 +2−r2s2+s2 +2 qs1+s2(1 − qr1+r2+1) +(q; q)r1−r2(q; q)s1−s2(q; q)r2(q; q)s2(q; q)r2+s2+1 += +1 +(q; q)∞ +1 +θ(q2, q2, q2, q3, q3, q3; q10). +(6.2) +All the products associated to principal characters of modulo 10 A2 Rogers–Ramanujan identities are covered +by the products that appear in (2.1). The identities (6.1) and (6.2) are outside of this system and appear, so +to say, on the dark-side of the cylinder. We hope to find a cylindric partition interpretation of these identities +in the future. Nevertheless, we plan to present the proofs of these theorems using q-theoretic means in an +upcoming paper. +It is still highly relevant to find manifestly positive sum representations for any one of the identities men- +tioned here. We are looking for ways to see the positivity of the series coefficients. In [7], Andrews–Schilling– +Warnaar suggests applying hypergeometric transformations to eliminate the (q; q)∞ factor that appear in the +identities (such as (1.4)) to get a manifestly positive representation. That suggestion is limited and might not +be widely applicable, especially for the under-the-line identities. +In the study of symmetric cylindric partitions [12] another two fundamental modulo 8 partition theoretic +identity families, namely G¨ollnitz–Gordon and little G¨ollnitz identities, showed up. +The G¨ollnitz–Gordon +identities are known to be related to the level 2 modules of affine Lie algebra A(2) +5 +[27]. This raises new +questions of whether, similar to the symmetric partitions paradigm, we can also relate symmetric cylindric +partitions to character formulas of some affine Lie algebras. The product formula analogous to (2.1) for the +count of symmetric cylindric partitions’ is present in [12]. At the moment, the relation of these products’ to +affine Lie algebra character formulas are fuzzy, and there are no general conjectural series representations for +symmetric cylindric partitions either. We plan to study these objects further. + +MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES +17 +Finally, we plan to pursue sum representations of any generating functions for cylindric partitions with +profiles of more than 3 parts. +The product representation (2.1) and the functional equations (2.3) apply +regardless of the size and length of the profiles. +So far, we are only able to prove and conjecture sum +representations for the profiles with up to 3 parts. +7. Comments on Computations +In the computerized proofs of [19], we make extensive use of [1] and [30]. Those proofs had three main +steps. Finding a recurrence relation (over the exponent of z) for claimed sum formulas of the (normalized) +generating functions of cylindric partitions, uncoupling the q-difference equation system laid out by the (2.3) +to get a recurrence satisfied by the coefficient of the z’s in the true generating functions of cylindric partitions, +comparing recurrences (taking greatest common divisors of recurrences as operators if needed) and showing +that both sequences satisfy the same recurrences with the same initial conditions. Once the critical mass +of proved identities were reached the rest of the identities were shown by series manipulations guided by +(2.3). That way we showed that all the claimed sum and the true combinatorial generating function were the +same. This proof required two hefty algorithms, namely Creative Telescoping algorithm and Gr¨obner bases +calculations, to find the recurrence of a given hypergeometric sum dependent of a discrete variable and to +uncouple a coupled system of recurrences, respectively. +We tried using the same method to prove some claims Warnaar [42] made for cylindric partitions with 3 +part profiles where the modulus is not divisible by 3. Then we quickly saw that the Creative Telescoping +calculations were not terminating (in any definition of reasonable time). This is due to the increasing number +of nested summations in these conjectures. However, uncoupling of recurrences could still be performed. +Kanade–Russell’s approach [29] to prove that the claimed series representations for the bivariate generating +functions of cylindric partitions are the true generating functions is a fresh take on things. It is somehow +backwards compared to the proofs of [19], in the sense that we first extend our conjectural identities using the +explicit conjectures of Conjecture 2.13 and series manipulations, then prove all these conjectural identities by +showing that the coupled relations are satisfied and that we still satisfy the initial conditions. The key idea +of reducing coupled q-difference equation with the functional relations of the claimed hypergeometric sums +was also used in [16] in a different context. Moreover, this approach replaces (the old bottle-neck) Creative +Telescoping with the contiguous relations of Lemma 2.5. However, rewriting the coupled relations of (2.3) in +the new language as a linear combination of terms in the ideal Im (see Section 3) with coefficients in Z((q, z)) is +highly non-trivial. Kanade [26] mentioned that they found these linear combinations by first making an ansatz +for a single case at a time and then solving for undetermined coefficients. The identification of the minimal +necessary ansatz is impossible. They also mentioned that each hard-case proof of modulo 10 calculations took +about 8 hours to terminate on a home computer. With the matrix reduction approach of this paper, we are +order of 2 faster in the modulo 10 cases. This is basically because once we reduce a matrix, we can use it +repeadetly for all the functional relations, whereas the previous approach needs to make a single ansatz and +solve if for all cases individually. It is with this speed upgrade that we could prove the new modulo 11 and +modulo 13 cases. On the other hand, modulo 9 and modulo 12 cases are still open. This is likely due to the +extra degree of complication the q-binomial coefficients in (2.9)’s introduce. As the order of the recurrences +the Sm(ρ|σ) satisfy increases, the systems we need to reduce also become larger. +Mathematica’s Gaussian elimination function RowReduce is adamant in calculating the reduced row echelon +form of matrices. +This is not only not necessary, it also overcomplicates the calculations by introducing +large rational function expressions for upper triangular coefficients. This forced us to implementing our own +Gaussian elimination algorithm within the Mathematica computer algebra system. This basic implementation +sorts, performs row elimination of a matrix with entries in a polynomial ring with integer coefficients, such as +Z[q, z], while not introducing rational functions, and it terminates when a row echelon matrix (a triangular +system of equations) is reached. This function will be made a part of the impending next version release of +qFunctions package. As a side note, we implemented a naive parallelization of this elimination but we have +not seen any benefits of splitting calculations yet. +We should also acknowledge that there are at least two crucial optimizations waiting to be implemented to +aid proos of families in cylindric partitions scheme and other similar schemes. First task that should be done + +18 +ALI KEMAL UNCU +is to keep track of nullified relations and to remove the contributions of the nullspace in later calculations. To +put it in concrete terms, at the moment we do not know if N = 6 is the minimal number to prove Theorems 4.1 +and/or 5.1. We know that it is a sufficient number. By removing any and all nullified relations we would only +see a minimal representation (dependent on the choice of N) of these recurrences as elements in the ideals Im, +and that can give us an idea of what the optimal bound for N is supposed to be in general. The second pending +addition is dynamic extension of the matrix to be reduced. At the moment, we fix an N experimentally hoping +that it is enough to show that the relations of interest are in the nullspace of this matrix. This is in the same +spirit of making a fixed ansatz. Row reduction as a preprocessing step helps for the repeated calculations. +Having an echelon system boosts the speed of later calculations immensely. If the chosen N is not enough, +then we need to pick a larger N and start all over. This requires performing row reduction of the matrix for +N once more as a subproblem. This should be changed by extending the already triangularized matrix for N +to N + 1 and doing the row reduction again for only the added relations. The incrementality of the matrix +would also carry us to the minimal necessary N for any given m (assuming that Conjecture 3.2 is correct) +naturally. +References +[1] J. Ablinger and A. K. Uncu. qFunctions - a Mathematica package for q-series and partition theory applications. Submitted. +arXiv:1910.12410, 2019. +[2] A. Agrawal, G. E. Andrews, and D. Bressoud. The Bailey lattice. J. Indian Math. Soc., 51:57–73, 1987. +[3] G. E. Andrews. An analytic generalization of the Rogers-Ramanujan identities for odd moduli. Proc. Nat. Acad. Sci. USA, +71:4082–4085, 1974. +[4] G. E. Andrews. q-series: +their development and application in analysis, number theory, combina- torics, physics, and +computer algebra. Vol. 66. CBMS Regional Conference Series in Math- ematics. Published for the Conference Board of the +Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986, pp. xii+130. +[5] G. E. Andrews. The Theory of Partitions. Cambridge University Press, 1984. +[6] G. E. Andrews. On the proofs of the Rogers-Ramanujan identities. In q-Series and Partitions, pages 1–14. Springer-Verlag, +New York, 1989. +[7] G. E. Andrews, A. Schilling, and S. O. Warnaar. An A2 Bailey lemma and Rogers-Ramanujan-type identities. J. Amer. +Math. Soc., 12(3):677–702, 1999. +[8] C. Armond and O. T. Dasbach. Rogers-Ramanujan type identities and the head and tail of the colored Jones polynomial. +arXiv:1106.3948 [math.GT]. +[9] W. N. Bailey. Identities of the Rogers-Ramanujan type. Proc. London Math. Soc., 50(2):1–10, 1949. +[10] R. J. Baxter. Rogers-Ramanujan identities in the hard hexagon model. J. Stat. Phys., 26:427–452, 1981. +[11] A. Borodin. Periodic Schur process and cylindric partitions. Duke Math. J., 140(3):391–468, 2007. +[12] W. Bridges, and A. K. Uncu. Weighted cylindric partitions. J. Algebraic Combin., 56 (2022), no. 4, 1309—1337. +[13] D. M. Bressoud. A generalization of the Rogers-Ramanujan identities for all moduli. J. Comb. Th. A, 27:64–68, 1979. +[14] D. M. Bressoud. An easy proof of the Rogers-Ramanujan identities. J. Number Th., 16:335–241, 1983. +[15] C. Bruschek, H. Mourtada, and J. Schepers. Arc spaces and Rogers-Ramanujan identities. Ramanujan J., 30:9–38, 2013. +[16] S. Chern Linked partition ideals, directed graphs and q-multi-summations. Electron. J, Combin. 27(3): Paper No. 3.33, 29 +pp. +[17] S. Corteel. Rogers-Ramanujan identities and the Robinson-Schensted-Knuth correspondence. Proc. Amer. Math. Soc., +145(5):2011–2022, 2017. +[18] S. Corteel and T. A. Welsh. The A2 Rogers–Ramanujan identities revisited. Annals of Combinatorics, 23(3):683–694, 2019. +[19] S. Corteel, J. Dousse and A. K. Uncu. Cylindric partitions and some new A2 Rogers–Ramanujan identities. Proc. Amer.Math. +Soc., 150(2):481—497, 2021. +[20] B. Feigin, O. Foda, and T. A. Welsh. Andrews–Gordon type identities from combinations of Virasoro characters. Ramanujan +J., 17(1):33–52, 2008. +[21] O. Foda and T. A. Welsh. Cylindric partitions, Wr characters and the Andrews-Gordon-Bressoud identities. J. Phys. A, +49(16):164004, 37, 2016. +[22] A. M. Garsia and S. C. Milne. A Rogers-Ramanujan bijection. J. Combin. Theory Ser. A, 31:289–339, 1981. +[23] I. M. Gessel and C. Krattenthaler. Cylindric partitions. Trans. Amer. Math. Soc., 349(2):429–479, 1997. +[24] B. Gordon. A combinatorial generalisation of the Rogers-Ramanujan identities. Amer. J. Math., 83:393–399, 1961. +[25] M. J. Griffin, K. Ono, and S. O. Warnaar. A framework of Rogers–Ramanujan identities and their arithmetic properties. +Duke Math. J., 8:1475–1527, 2016. +[26] S. Kanade. Private communications. +[27] S. Kanade. Structure of certain level 2 standard modules for A(2) +5 +and G¨ollnitz–Gordon identities. Ramanujan J., 45(3):873– +893, 2018. +[28] S. Kanade. On the A2 Andrews—Schilling—Warnaar identities. preprint. + +MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES +19 +[29] S. Kanade, and M. C. Russell. Completing the A2 Andrews–Schilling–Warnaar identities. arXiv:2203.05690 [math.CO]. +[30] C. Koutschan. Advanced applications of the holonomic systems approach. PhD thesis, RISC, Johannes Kepler University, +Linz, 2009. +[31] J. Lepowsky and R. L. Wilson. The structure of standard modules, I: Universal algebras and the Rogers-Ramanujan identities. +Invent. Math., 77:199–290, 1984. +[32] J. Lepowsky and R. L. Wilson. The structure of standard modules, II: The case A(1) +1 , principal gradation. Invent. Math., +79:417–442, 1985. +[33] P. A. MacMahon. Combinatory Analysis, volume 2. Cambridge University Press, New York, NY, USA, 1916. +[34] S. C. Milne and G. M. Lilly. The Aℓ and Cℓ Bailey transform and lemma. Bull. Amer. Math.Soc., 26:258ˆa€“263, 1992. +[35] S. C. Milne and G. M. Lilly. Consequences of the Aℓ and Cℓ Bailey transform and lemma. Discrete Math., 139:319–346, +1995. +[36] A.K. Uncu Last accessed January 3, +2023. +[37] A. Pascadi. Several new product identities in relation to two-variable Rogers–Ramanujan type sums and mock theta functions. +arXiv:2009.05878, 2020. +[38] L. J. Rogers and S. Ramanujan. Proof of certain identities in combinatory analysis. Cambr. Phil. Soc. Proc., 19:211–216, +1919. +[39] I. Schur. Ein Beitrag zur Additiven Zahlentheorie und zur Theorie der Kettenbr¨uche. S.-B. Preuss. Akad. Wiss. Phys. Math. +Klasse, pages 302–321, 1917. +[40] A. V. Sills. An invitation to the Rogers-Ramanujan identities. CRC Press, 2017. +[41] S. Tsuchioka. An example of A2 Rogers-Ramanujan bipartition identities of level 3. arXiv:2205.04811 [math.RT]. +[42] S. O. Warnaar. The A2 Andrews-Gordon identities and cylindric partitions. arXiv:2111.07550 [math.CO]. +[43] S. O. Warnaar. Private communications. +Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Science, Altenberg- +erstraße 69, A-4040 Linz, Austria +Email address: akuncu@ricam.oeaw.ac.at +University of Bath, Faculty of Science, Department of Computer Science, Bath, BA2 7AY, UK +Email address: aku21@bath.ac.uk + diff --git a/ctAzT4oBgHgl3EQfZ_wC/content/tmp_files/load_file.txt b/ctAzT4oBgHgl3EQfZ_wC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c16349f9c8c9b8c575eb8beedc6ca764aac17b39 --- /dev/null +++ b/ctAzT4oBgHgl3EQfZ_wC/content/tmp_files/load_file.txt @@ -0,0 +1,2051 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf,len=2050 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='01359v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='NT] 3 Jan 2023 PROOFS OF MODULO 11 AND 13 CYLINDRIC KANADE-RUSSELL CONJECTURES FOR A2 ROGERS–RAMANUJAN TYPE IDENTITIES ALI KEMAL UNCU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We present proofs of two new families of sum-product identities arising from the cylindric parti- tions paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Most of the presented expressions, the related sum-product identities, and the ingredients for the proofs were first conjectured by Kanade–Russell in the spirit of Andrews–Schilling–Warnaar identities of the A2 Rogers–Ramanujan type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We follow the footsteps of Kanade–Russell while we alter the computations heavily to accomplish our goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Introduction There is an ever-growing synergy between number theory, combinatorics, q-series, and affine Lie algebras that led to groundbreaking techniques and beautiful mathematical discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Among these are the Rogers– Ramanujan type identities where an infinite q-series is equal to a infinite product with a modular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' First appeared at the intersection of number theory and combinatorics, the Rogers–Ramanujan identities have been of great interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' These sum-product identities have been studied, proved and generalized in many different ways over the years [3, 6, 13, 14, 17, 22, 24, 25, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' These identities also naturally arose in many other fields including mathematical physics [10], representation theory of affine Lie algebras and vector operator algebras [31, 32], knot theory in relation to the colored Jones polynomials [8], and algebraic geometry [15] over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' For some non-negative integer L and formal variables a and q, let q-Pochhammer symbol be (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)L := (1 − a)(1 − aq) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' (1 − aqL−1), and (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ := limL→∞(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)L, θ(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) := (a, q/a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞, and for a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' , ak some formal variables, define the shorthand notation θ(a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' , ak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) := θ(a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)θ(a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' θ(ak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The Rogers–Ramanujan identities are as follows [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1 (Rogers–Ramanujan identities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) � n≥0 qn2 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)n = 1 θ(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q5) and � n≥0 qn2+n (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)n = 1 θ(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The reciprocal q-Pochhammer products on the right-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) has the ±1 and ±2 residue classes modulo 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We call these modulo 5 identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' A composition c of n is a finite list of non-negative integers that sum up to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' A partition is a composition where no element of the list (called parts) are zero and the list elements are ordered in a non-increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We define the size of a composition π as the sum of all its parts and denote this by |π|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We denote the number of parts in a composition π by #(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' A composition (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' partition) with size n is called “a composition (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' partition) of n.” The empty list is considered as the unique composition/partition of 0 with 0 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' For example, (2, 0, 2) is a composition with 3 parts and (1, 1, 1, 1), (4, 3, 1), and (2, 2) are partitions of 4, 8, and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' MacMahon [33] and Schur [39] gave combinatorial interpretations to Rogers–Ramanujan identities inde- pendently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Date: January 5, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Primary 05A15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Secondary 05A17, 05A19, 11B65, 11P84, 17B65, 68R05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Cylindric partitions, Partition identities, Rogers–Ramanujan identities, Andrews–Schilling–Warnaar identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Research of the author is partly supported by EPSRC grant number EP/T015713/1 and partly by FWF grant P-34501N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1 2 ALI KEMAL UNCU Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2 (Combinatorial interpretaton of Rogers–Ramanujan identities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let i = 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' For every natural number n, the number of partitions of n such that the difference between two consecutive parts is at least 2 and the the smallest part is strictly greater than i is equal to the number of partitions of n into parts congruent to ±(1 + i) mod 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Gordon [24] presented a wide generalization of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2 to all odd modulus ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3 (Gordon’s identities, 1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let r and i be integers such that r ≥ 2 and 1 ≤ i ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The number of partitions π = (π1, π2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' , πs) of n such that πj − πj+r−1 ≥ 2 for all j with at most i− 1 1s appears as parts in π are equal to the number of partitions of n whose parts are not congruent to 0, ±i mod 2r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The Rogers–Ramanujan identities correspond to the cases r = i = 2 and r = 2, i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Andrews found the q-series counterpart to Gordon’s identities [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4 (Andrews–Gordon identities, 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let r ≥ 2 and 1 ≤ i ≤ r be two integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We have (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2) � n1≥···≥nr−1≥0 qn2 1+···+n2 r−1+ni+···+nr−1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)n1 � n1 n1 − n2 � q · · � nr−2 nr−2 − nr−1 � q = θ(qi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q2r+1)(q2r+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q2r+1)∞ (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ , where for two integers n and m, �m + n m � q := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)m+n (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)m(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)n for m, n ≥ 0, 0 otherwise, is the classical q-binomial coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Note that the Rogers–Ramanujan identities are the particular case of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2) where r = i = 2, and r = 2 and i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Interested readers can get a great overview of the history of the Rogers–Ramanujan identities, their significance, and some generalizations in the recent book of Sills [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The identities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2) can be proven by the Bailey machinery coming from the world of q-series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This powerful mechanism starts with a pair of q-expressions, called a Bailey pair, that satisfies a pre-defined relation and modifies this pair iteratively (using Bailey lemma or one of its generalizations) to make a new Bailey pair (see [2, 4, 9, 40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' That way, by starting with the pair related to Rogers–Ramanujan identities, a whole infinite chain of identities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2) can be acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The identities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2) are certain characters related to affine Lie algebra A(1) 1 , and we thus refer to them as A1 Rogers–Ramanujan identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The original Bailey mechanism was later extended to An−1 for general n [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' However,these works did not yield An−1 Rogers–Ramanujan identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In their influential paper, Andrews, Schilling and Warnaar [7] were able to describe an A2 Bailey lemma and the associated Bailey machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' They found several infinite families of identities, One of their modulo 7 identities is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5 (Andrews–Schilling–Warnaar, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3) � r1,s1≥0 qr2 1−r1s1+s2 1+r1+s1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r1 �2r1 s1 � q = 1 θ(q2, q3, q3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Andrews–Schilling–Warnaar found several very general families of sum-product identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Of particular interest to representation theory, the product-sides of these identities are character formulas of the W3 algebra multiplied by an extra factor (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)−1 ∞ [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' These formulas do not yield manifestly positive sum-sides for the character formulas because of this extra factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' For example, one of Andrews–Schilling–Warnaar’s modulo 10 identities after clearing the extra factor (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)−1 ∞ is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6 (Andrews–Schilling–Warnaar, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) (q, q)∞ � r1≥r2≥0 s1≥s2≥0 qr2 1−r1s1+s2 1+r2 2−r2s2+s2 2+r1+r2+s1+s2 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r1−r2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s1−r2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r2+s2+1 = 1 θ(q2, q3, q3, q4, q4, q5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q10) MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES 3 Recall the Euler’s Pentagonal Number Theorem [5] (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5) (q, q)∞ = ∞ � i=−∞ (−1)iqi(3i+1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Although it is easy to see that the right-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) has positive coefficients, in light of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5) this is not directly visible on the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In contrast, both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3) are manifestly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The manifestly positive sum representations give insight to the structure of certain modules for the affine Lie algebra A(1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' These mentioned character of standard modules for the affine Lie algebra A(1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Interested readers can find more on this connection in [7, 29, 28, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Recently, the discovery of manifestly positive identities of these character formulas through a scheme with combinatorial roots attracted the attention and led to many new Rogers–Ramanujan type identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In 1997, Gessel and Krattenthaler [23] defined cylindric partitions in context of non-intersecting lattice paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Borodin [11] gave univariate product formulas for the generating functions of the number of cylindric partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Foda and Welsh [21] proved the A1 Rogers–Ramanujan identities using the combinatorics of cylin- dric partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This led to Corteel’s combinatorial proof of the Rogers–Ramanujan identities using cylindric partitions [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In 2019, Corteel and Welsh [18] derived functional equations for the bivariate generating func- tions for the number the number of cylindric partitions using the largest part statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' While doing so, they also gave a new proof of Andrews–Schilling–Warnaar’s modulo 7 A2 Rogers–Ramanujan identities (including (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3)) and a fifth missing identity which was originally conjectured by Feigin–Foda–Welsh [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' All these modulo 7 identities have manifestly positive sum sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' [18] has been the catalyst for the recent developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Ablinger and the author [1] implemented the Corteel–Welsh’s cylindric partitions related functional equations in their symbolic computation implementation qFunctions to be able to exploit this combinatorial idea using formal manipulation and computer algebra techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Corteel, Dousse and the author [19] later proved the modulo 8 identities that arise from the cylindric partitions paradigm with the help of this implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' One of such identities is as follows (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6 in [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='7 (Corteel–Dousse–U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6) � r1≥s1≥r2≥0 r1≥s2≥0 qr2 1−r1s1+s2 1+r2 2+s2 2+s1s2+r1+r2+s1+s2 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r1 �r1 s1 � q �r1 s2 � q �s1 r2 � q = 1 θ(q2, q3, q3, q4, q4, q5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q10) Unlike (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6) has a manifestly positive sum-side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Shortly after [19], in late 2021, Warnaar [42] come up with many beautiful conjectures for manifestly positive sum-sides related to higher moduli (not divisible by 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In 2022, Tsuchioka [41] proved manifestly positive sum-sides for modulus 6 using finite-automata and automated proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' He was also able to analyze the structure of relevant level 3 standard modules for the affine Lie algebra A(1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In a different vein, Bridges and the author studied weighted versions of cylindric partitions as well as cylindric partitions into distinct parts in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Earlier in 2022, Kanade and Russell [29] aimed (and succeeded) at conjecturing A2 Rogers–Ramanujan type identities in the form of Andrews–Schilling–Warnaar instead of aiming for manifestly positive sum-sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' They were able to make explicit claims for each modulus ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' They proved the cases for moduli 5, 6, 7, 8 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Their exploration came to an end due to the increasing computational difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In this paper, we approach the conjectures of Kanade–Russell by changing the computational techniques used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We prove all modulo 11 and 13 A2 Rogers–Ramanujan identities coming from the cylindric partitions paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Two such identities are as follows: 4 ALI KEMAL UNCU Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' � r1≥r2≥r3≥0 s1≥s2≥s3≥0 qr2 1−r1s1+s2 1+r2 2−r2s2+s2 2+r2 3+r3s3+s2 3+r1+r2+r3+s1+s2+s3 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r1−r2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r2−r3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s1−s2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s2−s3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r3+s3+1 = 1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ 1 θ(q2, q3, q3, q4, q4, q5, q5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' � r1≥r2≥r3≥0 s1≥s2≥s3≥0 qr2 1−r1s1+s2 1+r2 2−r2s2+s2 2+r2 3−r3s3+s2 3+r1+r2+r3+s1+s2+s3 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r1−r2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r2−r3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s1−s2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s2−s3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r3+s3+1 = 1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ 1 θ(q2, q3, q3, q4, q4, q5, q5, q6, q6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The organization of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In Section 2, we introduce cylindric partitions, the relevant results and the conjectures of Kanade–Russell of which we prove some cases of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Section 3 is dedicated to rewording the conjectures and the description ot the proof methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In Sections 4 and 5 we present the proofs of the modulo 11 and 13 A2 Rogers–Ramanujan identities in Andrews–Schilling–Warnaar form, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We outline some natural questions and mathematical challenges that arise from this work in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Section 7 is reserved for a discussion on how the computerized proofs have been carried in earlier work [19, 29] and this paper and what future improvements can be done to take us further mathematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Acknowledgement The author would like to thank the workshop on cylindric partitions group that came together in November 2022 in Linz for all the stimulating discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In particular, the author would like to thank Shashank Kanade for suggesting that the researchers working on cylindric partitions should come together and join forces in the first place, and for all his comments on this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The author would also like to thank Christian Koutschan his encouragement of the author in the necessary implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Research of the author is partly supported by EPSRC grant number EP/T015713/1 and partly by FWF grant P-34501N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Necessary definitions We shall start with the definition of a cylindric partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' A cylindric partition is made up of a composition c = (c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' , cr) called profile with r parts, and a vector π = (π(1), π(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' , π(r)) consisting of r partitions π(i) = (π(i) 1 , π(i) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ), that satisfy the inequalities π(i) j ≥ π(i+1) j+ci+1 and π(r) j ≥ π(1) j+c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' For example, the vector partition π = {(1, 1, 1, 1), (4, 3, 1), (2, 2)} together with the profile (2, 0, 2) is a cylindric partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Note that the same vector partition can also satisfy the cylindric partition inequalities with different profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' For example, π is also a cylindric partition for profiles (2, 0, 0), (2, 0, 1), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We can define the total size of a cylindric partition π as the sum of all the sizes of the partitions included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We denote the total size, once again, by |π|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let c be a composition and let Pc be the set of all vector partitions that are cylindric partitions with profile c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' For a given profile c, let Pc be the set of all cylindric partitions with profile c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let Fc(z, q) := � π∈Pc zmax(π)q|π|, the bivariate generating function for the number of cylindric partitions where the exponents of z and q are keeping record of the largest parts size and the total of the parts in π, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Borodin [11] showed that when z = 1, Fc(z, q) generating functions have product formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES 5 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2 (Borodin, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let r and l be positive integers, and let c = (c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' , cr) be a composition of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Define m := r + l and s(i, j) := ci + ci+1 + · · · + cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Then, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) Fc(1, q) = 1 (qm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' qm)∞ r � i=1 r � j=i ci � k=1 1 (qk+j−i+s(i+1,j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' qm)∞ r � i=2 i� j=2 ci � k=1 1 (qm−k+j−i−s(j,i−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' qm)∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Focusing on replacing the largest part in a given cylindric partition, Corteel–Welsh [18] defined a q-difference equation for Fc(z, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This functional equation relates Fc(z, q) with other generating functions Fc∗(z, q) where #(c) = #(c∗) and |c| = |c∗|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let c = (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' , cr) (with the convention that c0 = cr) be a given composition and define Ic to be the set of indices for the non-zero entries in c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Given a non-empty subset J ⊆ Ic, the composition c(J) = (c1(J), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' , cr(J)) is defined by: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2) ci(J) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ci − 1 if i ∈ J and i − 1 /∈ J, ci + 1 if i /∈ J and i − 1 ∈ J, ci otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Then the explicit q-difference equation Fc(z, q) satisfies is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3 (Corteel–Welsh, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' For any profile c, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3) Fc(z, q) = � ∅⊂J⊆Ic (−1)|J|−1 Fc(J)(zq|J|, q) (1 − zq|J|) , with the initial conditions Fc(0, q) = Fc(z, 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let c be a profile and c′ be a cyclic shift of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' There is a clear one-to-one correspondence between cylindric partitions in Pc and Pc′ by cyclically shifting the vector of partitions counted in Pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This is enough to see that the generating functions for these sets of cylindric partitions are equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Fc(z, q) = Fc′(z, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Therefore, we can cyclically shift the profiles and lower the number of (seemingly different) generating functions that appear in the coupled system of q-difference equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We can also normalize (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3) and get an equivalent q-difference equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' For example, let Gc(z, q) := (zq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞Fc(z, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3) is equivalent to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) Gc(z, q) = � ∅⊂J⊆Ic (−1)|J|−1(zq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)|J|−1Gc(J)(zq|J|, q), with the initial conditions Gc(0, q) = Gc(z, 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This q-difference equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) with polynomial coeffi- cients, in practice, played a central role in the proofs of modulo 7 and modulo 8 cylindric partition with 3-part profile identities in [18] and [19], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Weighted versions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) are later presented in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In [29], Kanade–Russell decided to change the initial conditions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' While this does not change the q-difference equations, this lead to the conjectural discovery of explicit formulas for most of these 3-part profile cylindric partition generating functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5) Hc(z, q) := (zq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ Fc(z, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Then Hc(z, q) satisfies the same q-difference equation as Gc(z, q), namely (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6) Hc(z, q) = � ∅⊂J⊆Ic (−1)|J|−1(zq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)|J|−1Hc(J)(zq|J|, q), with the initial conditions Hc(0, q) = 1/(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ and Hc(z, 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' From this point forward we only focus on cylindric partition profiles with 3-parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let k ≥ 2, let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='7) ρ = (ρ1, ρ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' , ρk−1) ∈ Zk−1, and σ = (σ1, σ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' , σk−1) ∈ Zk−1 6 ALI KEMAL UNCU and define S3k−1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ρ|σ) = � r,s∈Zk−1 ≥0 zr1 q �k−1 i=1 r2 i −risi+s2 i +ρiri+σisi �k−2 i=1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)ri−ri+1(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)si−si+1 q2rk−1sk−1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)rk−1(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)sk−1(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)rk−1+sk−1+1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='8) S3k(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ρ|σ) = � r,s∈Zk−1 ≥0 zr1 q �k−1 i=1 r2 i −risi+s2 i +ρiri+σisi �k−2 i=1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)ri−ri+1(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)si−si+1 1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)rk−1+sk−1(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)rk−1+sk−1+1 �rk−1 + sk−1 rk−1 � q3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='9) S3k+1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ρ|σ) = � r,s∈Zk−1 ≥0 zr1 q �k−1 i=1 r2 i −risi+s2 i +ρiri+σisi �k−2 i=1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)ri−ri+1(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)si−si+1 1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)rk−1(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)sk−1(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)rk−1+sk−1+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='10) Let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='11) ei = (0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' , 0 � �� � i , 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=', 1) ∈ Zk−1 and δi := (δij)1≤j≤k−1 ∈ Zk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' It is easy to see that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12) Sm(zqn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ρ|σ) = Sm(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ρ + nδ1|σ) for any n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Kanade–Russell conjectured that for any fixed k ≥ 3, the H(c1,c2,c3)(z, q) can be expressed as linear combi- nations of the Sm(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ρ|σ) functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Precisely they claimed the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4 (Kanade–Russell, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let k ≥ 3 and |c|+3 = m = 3k+{−1, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Using cyclis symmetries asusme that c1 ≥ c2, c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' If c2, c3 ≤ k − 1, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='13) H(c1,c2,c3)(z, q) = \uf8f1 \uf8f2 \uf8f3 Sm(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ec2|ec3) − qSm(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ec2−1|ec3−1), c2, c3 > 0, Sm(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ec2|e0), c3 = 0, Sm(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' e0|ec3) − q(1 − z)Sm(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' e0 + δ0|ec3−1), c2 = 0, c3 ̸= 0, where ei and δi be defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The explicit claims that Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4 provide do not cover all the functions Hc(z, q) with |c| + 3 = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' It does provide enough claims to recover explicit expression claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' How to find the conjectural Sm(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ρ|σ) equivalents of the other Hc(z, q) that appear in the coupled q-difference equation system is explained in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The profiles related to the functions to be recovered are called ”under-the-line” profiles by Kanade–Russell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We will also call these profiles as such while we ignore to explain anything about the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' These under-the-line profile related functions can have shifts of z in the Sm(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ρ|σ) language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' These shifts are inherited from the q-difference equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' One can use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12) to clear all the shifts in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Therefore, from now on in all our expressions we will translate any z shift of Sm(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ρ|σ) using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12) and this way ignore any and all shifts in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' To further emphasize this moving forward on we suppress the variable z from our notation and write Sm(ρ|σ) := Sm(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ρ|σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Proof of Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4 (and its extension to all 3-part profiles with total m − 3) requires one to show that the expressions in Sm(ρ|σ) are the correct expressions for the respective Hc(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This can be done by showing that the expressions in Sm(ρ|σ) satisfies the same recurrence relation specified by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6) and the initial conditions of the expression holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In [29], it is already proven that for c2, c3 ≤ k − 1, the conjectural formulas of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='13) all satisfy the necessary initial conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='14) Hc(z, 0) = 1 and Hc(0, z) = 1/(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' It was noted in the [29, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2] that Sm(ρ|σ) functions satisfy the following list of recur- rences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES 7 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5 (Kanade–Russell, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let k ≥ 3, let m = 3k + {−1, 0, 1} and let δi := (δij)1≤j≤k−1 ∈ Zk−1, where δij is the Kronecker delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The following recurrence relations follow for all 1 ≤ i ≤ k − 2, Sm(ρ|σ) − Sm(ρ + δi − δi+1|σ) − zqi+�i j=1 ρjSm(ρ + 2 i � j=1 δj|σ − i � j=1 δj) = 0, (R(i) 1 (ρ|σ)) Sm(ρ|σ) − Sm(ρ|σ + δi − δi+1) − zqi+�i j=1 σjSm(ρ − i � j=1 δj|σ + 2 i � j=1 δj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' (R(i) 2 (ρ|σ)) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' If m ≡ −1 (mod 3), a) and if σk−1 = 0, then (R3(ρ|σ)) Sm(ρ|σ) − Sm(ρ|σ + δk−1) − qSm(ρ + δk−1|σ + δk−1) + qSm(ρ + δk−1|σ + δk−2 + δk−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' b) and if ρk−1 = 0, then (R4(ρ|σ)) Sm(ρ|σ) − Sm(ρ + δk−1|σ) − qSm(ρ + δk−1|σ + δk−1) + qSm(ρ + δk−2 + δk−1|σ + δk−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' If m ≡ 0 (mod 3), then Sm(ρ|σ) − (1 + q)Sm(ρ + δk−1|σ + δk−1) + qSm(ρ + 2δk−1|σ + 2δk−1) (R3(ρ|σ)) − zqk−1+�k−1 j=1 ρjSm(ρ + 2 k−1 � j=1 δj|σ − k−1 � j=1 δj) − qk−1+�k−1 j=1 σjSm(ρ − k−2 � j=1 δj + 2δk−1|σ + 2 k−1 � j=1 δj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Sm(ρ|σ) − (1 + q)Sm(ρ + δk−1|σ + δk−1) + qSm(ρ + 2δk−1|σ + 2δk−1) (R4(ρ|σ)) − zqk−1+�k−1 j=1 ρjSm(ρ + 2 k−1 � j=1 δj|σ − k−2 � j=1 δj + 2δk−1) − qk−1+�k−1 j=1 σjSm(ρ − k−1 � j=1 δj|σ + 2 k−1 � j=1 δj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' If m ≡ 1 (mod 3), then Sm(ρ|σ) − Sm(ρ|σ + δk−1) − qSm(ρ + δk−1|σ + 2δk−1) (R3(ρ|σ)) + qSm(ρ + δk−1|σ + 2δk−1) − qk−1+�k−1 j=1 σjSm(ρ − k−1 � j=1 δj|σ + 2 k−1 � j=1 δj) = 0 Sm(ρ|σ) − Sm(ρ + δk−1|σ) − qSm(ρ + δk−1|σ + δk−1) (R4(ρ|σ)) + qSm(ρ + 2δk−1|σ + δk−1) − zqk−1+�k−1 j=1 ρjSm(ρ + 2 k−1 � j=1 δj|σ − k−1 � j=1 δj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Then they made the following claim (see [29, Conjecture 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6 (Kanade–Russell, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In each modulus m ≥ 5, the relations (R(i) 1 (ρ|σ))-(R4(ρ|σ)) are enough to prove recurrences necessary for the proof of Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We find this conjecture highly sensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' For all m ≥ 5, the explicit Sm’s are 2⌊m/3⌋-fold sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Same is true for the number of distinct functional equations (R(i) 1 (ρ|σ))-(R4(ρ|σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' One can easily check that these relations are distinct by comparing the first two terms in each left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Each second term corresponds to a canonical shift in one of the summation variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' One would expect to see every relation that the Sm(ρ|σ) functions satisfy to be translated and recovered as a combination the relations (R(i) 1 (ρ|σ))-(R4(ρ|σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Hence, 8 ALI KEMAL UNCU if the claims of Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4 are correct, for any fixed profile c the coupled q-difference equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6) written using the explicit claims of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='13) (together with the “under-the-line” expressions) can be recovered as a combination of the relations (R(i) 1 (ρ|σ))-(R4(ρ|σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Proof Methodology Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6 can be rephrased as a set inclusion question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let m = 3k + {−1, 0, 1} with k ≥ 3, ρ and σ as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='7) and 1 ≤ i ≤ k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Define (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) Im := ⟨R(i) 1 (ρ|σ), R(i) 2 (ρ|σ), R3(ρ|σ), R4(ρ|σ)⟩, the ideal generated by the left-hand sides of the recurrences (R(i) 1 (ρ|σ))-(R4(ρ|σ)) as polynomials in the ring Z((q, z))[Sm(ρ|σ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Here which R3(ρ|σ) and R4(ρ|σ) to be included in Im is to be understood by the residue class of m modulo 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Recall that ρ and σ are integer vectors with k −1 entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Therefore the ring Z((q, z))[Sm(ρ|σ)] is a formal polynomial ring defined on a countable set of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' For any given fixed m = 3k + {−1, 0, 1} with k ≥ 3, let the set of all the coupled system of q-difference equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6) for the profiles c with |c| + 3 = m be Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Any relation in Hm can be written in Sm(ρ|σ) functions using the Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4 (and the paragraph below it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let Sm be the set of all relations in Hm written in the conjectural Sm(ρ|σ) form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Now we can write Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6 in its equivalent form: Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let m = 3k + {−1, 0, 1} with k ≥ 3 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ∀h ∈ Sm, we have h ∈ Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The infinite set {R(i) 1 (ρ|σ), R(i) 2 (ρ|σ), R3(ρ|σ), R4(ρ|σ) : 1 ≤ i ≤ k − 2, ρ, σ ∈ Zk−1} that spans Im has non-trivial relations within itself and not all the elements of this set are generators of Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' However, we do not know an exact pattern of which elements are related at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Nevertheless, it is easy to understand that Im is generated by infinitely many elements since ρ and σ ∈ Zk−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' On the other hand, for any fixed m, the Sm(ρ|σ) functions that appear within the formulas from Sm make up a finite list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' One can easily find explicit bounds for the entries of vectors ρ and σ such that every Sm(ρ|σ) that appear in Sm is within the bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This observation suggests that instead of attempting to prove Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1, we can instead go after a stronger conjecture that is more suitable for computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' To that end, let [N] := {−N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' , −1, 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=', N} and we define Im,N := ⟨{R(i) 1 (ρ|σ), R(i) 2 (ρ|σ), R3(ρ|σ), R4(ρ|σ) : ρ, σ ∈ [N]k−1}⟩ ⊂ Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' With this definition we form the stronger conjecture: Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let m = 3k + {−1, 0, 1} with k ≥ 3 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' There is some N ∈ N such that ∀h ∈ Sm, we have h ∈ Im,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Since Im,N ⊂ Im, it is clear that Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2 implies Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Finally we transferred the open problems into a linear algebra setting, and we can approach it as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let m and N be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' we can order all the Sm(ρ|σ) that appears in the spanning set of Im,N and write in a column vector ⃗s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Then the matrix A is uniquely defined by A⃗s = ⃗0A, where ⃗0A is the colum vector with the same number of rows as A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Every row of A, corresponds to a functional relation Rj(σ|ρ) ∈ {R(i) 1 (ρ|σ), R(i) 2 (ρ|σ), R3(ρ|σ), R4(ρ|σ) : ρ, σ ∈ [N]k−1} and every column of A corresponds to the coefficients of Sm(ρ|σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Also observe that A is a finite dimensional matrix with entries in Z[q, z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' One can use Gaussian elimination on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Any non-trivial relation within the functional relations (R(i) 1 (ρ|σ))- (R4(ρ|σ)) within the defining bounds of A would yield 0 rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Let B be the matrix consisting of non-zero rows of A after the Gaussian elimination is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' It should still be clear that B⃗s = ⃗0B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES 9 Moreover, the ideal Im,N is generated by the equations that appear in B⃗s = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Therefore, for any element of h ∈ Sm one can check whether that element is in Im,N by simply writing that relation as a row vector ⃗h (with respect to the vector ⃗s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' vech is defined by h := [⃗h⃗s = 0]), add the row vector ⃗h to B and perform Gaussian elimination to this new matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' If the Gaussian elimination yields a zero row, this means that ⃗h is a linear combination of rows in B, or equivalently this means h ∈ Im,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' If no zero row appears, then h ̸∈ Im,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This approach is clearly algorithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Furthermore, termination of the algorithm and a definitive answer among the termination are both guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Top it all up, the explicit combination of (R(i) 1 (ρ|σ))-(R4(ρ|σ)) functional equations that is equivalent to a given h ∈ Sm is also easy to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' One only needs to use an augmented version of A where one more column is added to keep track of the name of the relations (R(i) 1 (ρ|σ))- (R4(ρ|σ)) while doing the row reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' After these considerations, proof of Conjectures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2 (and consequently Conjectures 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) comes down to experimentally identifying an N and being able to perform the Gaussian elimination calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Modulo 11 Identities Let m = 11 (= 3k − 1 with k = 4), for this family of identities ρ and σ ∈ Z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' There are a total of 15 essentially unique 3 part compositions of 8 that appear in the coupled q-difference system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4 suggests that the following sum representations for Hc(z, q) hold for all but one of these: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) H(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S11((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S11((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S11((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − q(1 − z)S11((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S11((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S11((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − qS11((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S11((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − q(1 − z)S11((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S11((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0) | (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S11((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − qS11((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S11((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − qS11((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S11((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)) − q(1 − z)S11((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S11((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − qS11((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S11((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − qS11((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S11((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)) − qS11((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S11((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − qS11((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) | (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Only H(4,4,0)(z, q) misses a claimed formula and that can be recovered by the q-difference equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We know that H(4,4,0)(z, q) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2) H(4,4,0)(z, q) + (1 − qz)H(4,1,3)(q2z, q) − H(4,3,1)(qz, q) − H(5,0,3)(qz, q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Using the conjectured series equivalents (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) of H(4,1,3)(z, q), H(4,3,1)(z, q) and H(5,0,3)(z, q), we see that H(4,4,0)(z, q) = −(S11(qz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' (0, 0, 0)|(0, 1, 1)) − qS11(qz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' (0, 0, 1)|(1, 1, 1))) + (1 − qz)(S11(q2z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' (0, 1, 1)|(0, 0, 0)) − qS11(q2z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' (1, 1, 1)|(0, 0, 1))) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3) − (S11(qz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' (1, 1, 1)|(0, 0, 0)) − q(1 − z)S11(qz(2, 1, 1)|(0, 0, 1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Notice that we used the shifts in the variable z in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We clear these shifts by employing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This yields an explicit claim for H(4,4,0)(z, q): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) H(4,4,0)(z, q) = S11((1, 0, 0) | (0, 1, 1)) − qS11((1, 0, 1) | (1, 1, 1)) + qzS11(2, 1, 1) | (0, 0, 0)), with no shifts in z, where the S11(ρ|σ) functions fit the forms in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We can also see that H(4,4,0)(z, q) satisfies the necessary initial conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The initial condition H(4,4,0)(z, 0) = 1 is immediate by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We can also see that H(4,4,0)(0, q) = 1/(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ by plugging 10 ALI KEMAL UNCU in z = 0 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2) and using the initial conditions of the other proven initial conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='14) for the functions in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Our proof routine explained in Section 3 can start once all the normalized generating functions Hc(z, q)’s are (conjecturally) translated in the S11((a1, a2, a3)|(b1, b2, b3)) language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' It is easy to see that the following four recurrences, H(8,0,0)(z, q) − H(7,1,0)(qz, q) = 0, H(7,0,1)(z, q) − H(6,1,1)(qz, q) + (1 − qz)H(7,1,0)(q2z, q) − H(8,0,0)(qz, q) = 0, H(6,0,2)(z, q) − H(5,1,2)(qz, q) + (1 − qz)H(6,1,1)(q2z, q) − H(7,0,1)(qz, q) = 0, H(5,0,3)(z, q) − H(4,1,3)(qz, q) + (1 − qz)H(5,1,2)(q2z, q) − H(6,0,2)(qz, q) = 0, trivializes to 0 = 0 once the terms on the left-hand sides are written in S11((a1, a2, a3)|(b1, b2, b3)) using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Therefore, these relations are trivially in I11, the ideal generated by the functional relations of the S11((a1, a2, a3)|(b1, b2, b3)) series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Recall that we used the coupled q-difference equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2) to make an explicit claim for H(4,4,0)(z, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Hence, the functional relation of H(4,4,0)(z, q) also trivializes to 0 = 0 once written in the claimed S11(ρ|σ) forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The very claim (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) is instrumental in proving that the q-difference equations satisfied by H(5,3,0)(z, q), H(4,3,1)(z, q), and H(4,1,3)(z, q) in S11((a1, a2, a3)|(b1, b2, b3)) language are elements of I11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Next, we look at the q-difference equation satisfied by H(7,1,0)(z, q) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6): H(7,1,0)(z, q) − H(7,0,1)(qz, q) − H(6,2,0)(qz, q) + (1 − qz)H(6,1,1)(q2z, q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' After the use of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12), we see that this q-difference equation is equivalent to the following conjectural form S11((0, 1, 1)|(1, 1, 1)) − S11((1, 0, 1)|(1, 1, 1)) − qzS11((2, 1, 1)|(0, 1, 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This is nothing but the relation R(1) 1 (0, 1, 1)|(1, 1, 1) of (R(i) 1 (ρ|σ)) given in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Hence, this relation is also within I11 and covered by the relations of S11((a1, a2, a3)|(b1, b2, b3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' As a second explicit example, consider the q-difference equation satisfied by H(6,1,1)(z, q), H(6,1,1)(z, q) − H(7,1,0)(qz, q) − H(6,0,2)(qz, q) − H(5,2,1)(qz, q) + (1 − qz)H(7,0,1)(q2z, q) + (1 − qz)H(6,2,0)(q2z, q) + (1 − qz)H(5,1,2)(q2z, q) − (1 − qz)(1 − q2z)H(6,1,1)(q3z, q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Using employing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12), we see get the (conjecturally) equivalent form S11((0, 1, 1)|(0, 1, 1)) − S11((1, 0, 1)|(0, 1, 1)) − S11((1, 1, 1)|(1, 1, 1)) + (1 − qz)S11((2, 0, 1)|(1, 1, 1)) − qzS11((2, 1, 1)|(0, 0, 1)) + q2z(1 − qz)S11((3, 1, 1)|(0, 1, 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This relation can be checked to be the side-by-side additions of R(1) 1 ((0, 1, 1)|(0, 1, 1)) − (1 − qz)R(1) 1 ((1, 1, 1)|(1, 1, 1)) + qzR(1) 2 ((2, 1, 1)|(−1, 1, 1)) ∈ I11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We can one-by-one write down the remaining 8 recurrences, their S11(ρ|σ) equivalents, and what combina- tion of (R(i) 1 (ρ|σ))-(R4(ρ|σ)) is equivalent to the functional equations in the S11(ρ|σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This way we prove that these relations are all included in the ideal I11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We need to say that these relations gets messier, pages long and not hand-verifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Printing these would be a waste of page/paper and instead we keep these in the digital realm for interested readers to check it easily, or print on paper as they wish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' To that end, similar to how it was handled in [29], we include text files M11RecHXYZ Explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='txt in the ancillary files portion of ArXiv and on the author’s website [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Here XYZ is to be replaced by the relevant profile’s digits such as 620 for the profile (6, 2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' One can check that the elements of I11 given in these text files are equivalent to the q-difference equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6) satisfied by H(X,Y,Z)(z, q) after they are translated to S11(ρ|σ) form using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The functional equation names are reflected in the text as RX[{Y},{{a1,a2,a3},{b1,b2,b3}}] for X and Y to be replaced by 1 or 2 to denote R(Y ) X ((a1, a2, a3)|(b1, b2, b3)), or RZ[{{a1,a2,a3},{b1,b2,b3}}] for Z to be replaced by 3 or 4 to denote R3((a1, a2, a3)|(b1, b2, b3)) and R4((a1, a2, a3)|(b1, b2, b3)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES 11 The definitions of these functional equations can be found in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5 for m = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' A guide document that explicitly lists each R functional relation for modulo 10 is given in M11R text file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' One also can see that the largest entry within ρ = (a1, a2, a3) and σ = (b1, b2, b3) of the relations (R(i) 1 (ρ|σ))-(R4(ρ|σ)) for the modulo 11 case given in the additional documents is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This proves the following theorem and its corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2 is correct for m = 11 and N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1 is correct for m = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2 is equivalent to the following theorem: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The claimed expressions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Observe that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3 adds a new supporting case to Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Now that the main conjectures are proven for the modulus 11 cases, we can specialize z = 1 and see the 15 sum-product identities coming from the cylindric partitions paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The following identities hold � r1≥r2≥r3≥0 s1≥s2≥s3≥0 qr2 1−r1s1+s2 1+r2 2−r2s2+s2 2+r2 3+r3s3+s2 3 pc(r1, r2, r3, s1, s2, s3, q) (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r1−r2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r2−r3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s1−s2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s2−s3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r3+s3+1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5) = 1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ 1 θ(qi1, qi2, qi3, qi4, qi5, qi6, qi7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' where the polynomials pc(r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) and the 7-tuples (i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i7) for each profile is given in the following table: Profile c pc(r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) (i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i7) (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0) qr1+r2+r3+s1+s2+s3 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0) qr2+r3+s1+s2+s3 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) qr1+r2+r3+s2+s3 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0) qr3+s1+s2+s3 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) qr2+r3+s2+s3(1 − qr1+s1+1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2) qr1+r2+r3+s3 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0) qs1+s2+s3 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) qr3+s2+s3(1 − qr2+s1+1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2) qr2+r3+s3(1 − qr1+s2+1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3) qr1+r2+r3 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) qs2+s3(1 − qr3+s1+1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2) qr3+s3(1 − qr2+s2+1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3) qr2+r3(1 − qr1+s3+1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2) qs3(1 − qr3+s2+1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0) qr1(qs2+s3 − qr3+s1+s2+s3+1 + qr1+r2+r3+1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4) In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4, we chose to put the profile (4, 4, 0) related sum-product identity under a line in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This is to indicate that this identity is not a direct claim made by combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) with z = 1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We first recovered a formula for H(4,4,0)(z, q) as a combination of S11((a1, a2, a3)|(b1, b2, b3)) series and then made this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This line also has the added benefit that it aligns us with Kanade–Russell’s language as this is the sum-product identity related to the under-the-line Hc(z, q) function, which we chose not to directly define.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The sum sides are the expressions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) with z = 1 written explicitly using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='8) with k = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The product sides follow from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5) with z = 1 followed by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The product related to the first profile, (8, 0, 0), on the table is presented in the introduction as Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Observe that the products that appear on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5) related to the profiles (c1, c2, c3) and (c1, c3, c2) are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The symmetry for the generating functions have been observed and noted before, for example in [19, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This symmetry is visible on the sum side of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' One can get 12 ALI KEMAL UNCU the “other” sum by merely replacing the variable ‘r’s and ‘s’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Note that this is a byproduct of setting z = 1 and this similarity does not exist on the sum side for generic z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In that light, this theorem consisting of 15 sum-product identities actually provide a total of 10 essentially unique sum-product identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We also note that among these identities the ones related to profiles (8, 0, 0), (6, 1, 1), (4, 2, 2), (3, 3, 2) are the i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' , 4 cases of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='28), respectively, and (5, 3, 0) and (6, 2, 0) are the σ = 0 and 1 cases of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='29), respectively, of [7, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Modulo 13 Identities Similar to Section 4, we start by listing the explicit claims of Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4 for the modulus m = 13 family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) H(10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − q(1 − z)S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − qS13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − q(1 − z)S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)|(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − qS13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − qS13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)) − q(1 − z)S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − qS13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − qS13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)) − qS13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − qS13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)) − qS13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' H(4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3)(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) = S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)) − qS13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' There are six profiles that are not covered by Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Once again, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6) explicit claims for the normalized generating functions related to the number of cylindric partitions with these profiles can be recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We make the claims in the following succession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' First we look at the q-difference equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6) that H(7,3,0): (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2) H(7,3,0)(z, q) − H(7,2,1)(qz, q) − H(6,4,0)(qz, q) + (1 − qz)H(6,3,1)(q2z, q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' By writing the S13((a1, a2, a3)|(b1, b2, b3)) equivalents for the functions in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12), we get H(6,4,0)(z, q) = S13((−1, 0, 0)|(1, 1, 1)) − S13((0, 0, 1)|(0, 1, 1)) + qS13((0, 1, 1)|(1, 1, 1)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3) + (1 − z)S13((1, 0, 0)|(0, 1, 1)) − q(1 − z)S13((1, 0, 1)|(1, 1, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Note that we did not use the q-difference equation of H(6,4,0)(z, q) to make a claim for its formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In Section 4, there was only a single missing formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' That allowed us to use the q-difference equation for that very function and get a formula in S11((a1, a2, a3)|(b1, b2, b3))’s with no backwards shifts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' z �→ z/q, which also reflects as negative indices in the first variable a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This may not be possible in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The q-difference equation H(6,4,0)(z, q) satisfies is (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) H(6,4,0)(z, q) − H(6,3,1)(qz, q) − H(5,5,0)(qz, q) + (1 − qz)H(5,4,1)(q2z, q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The conjectural formulas (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) does not cover H(5,5,0)(z, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Hence, we cannot fully translate H(6,4,0)(z, q) to a formula made up of S13((a1, a2, a3)|(b1, b2, b3)) series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Nevertheless, as also noted in [29], we can recover formulas for all the missing functions using other recurrences and backwards shifts in a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES 13 In fact, the recurrence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3) can be put together to claim a formula for H(5,5,0)(z, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' After the similar considerations we claim H(5,5,0)(z, q) = S13((−2, 0, 0)|(1, 1, 1)) − S13((−1, 0, 1)|(0, 1, 1)) + qS13((−1, 1, 1)|(1, 1, 1)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5) + (1 − z/q − z)S13((0, 0, 0)|(0, 1, 1)) − (q − z − qz)S13((0, 0, 1)|(1, 1, 1)) − q(1 − z)zS13((1, 0, 0)|(1, 1, 1)) − (1 − z)S13((1, 0, 1)|(0, 0, 1)) + q(1 − z)S13((1, 1, 1)|(0, 1, 1)) + (1 − z)(1 − qz)S13((2, 0, 0)|(0, 0, 1)) − q(1 − z)(1 − qz)S13((2, 0, 1)|(0, 1, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We point out that the coefficients of the claimed H(5,5,0)(z, q) formula now can be seen to have a Laurent polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This is a byproduct of the backwards shifts in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Using the q-difference equation for H(6,3,1)(z, q), H(6,3,1)(z, q) − H(7,3,0)(qz, q) − H(6,2,2)(qz, q) − H(5,4,1)(qz, q) + (1 − qz)H(7,2,1)(q2z, q) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6) + (1 − qz)H(6,4,0)(q2z, q) + (1 − qz)H(5,3,2)(q2z, q) − (1 − qz)(1 − q2z)H(6,3,1)(q3z, q) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3) we claim that H(5,4,1)(z, q) = S13((−1, 0, 0)|(0, 1, 1)) − qS13((−1, 0, 1)|(1, 1, 1)) − zS13((0, 0, 0)|(1, 1, 1)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='7) − S13((0, 0, 1)|(0, 0, 1)) + qS13((0, 1, 1)|(0, 1, 1)) + (1 − z)S13((1, 0, 0)|(0, 0, 1)) − q(1 − z)S13((1, 0, 1)|(0, 1, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Using the q-difference equation for H(5,3,2)(z, q), H(5,3,2)(z, q) − H(6,3,1)(qz, q) − H(5,2,3)(qz, q) − H(4,4,2)(qz, q) + (1 − qz)H(6,2,2)(q2z, q) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='8) + (1 − qz)H(5,4,1)(q2z, q) + (1 − qz)H(4,3,3)(q2z, q) − (1 − qz)(1 − q2z)H(5,3,2)(q3z, q) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='7) we claim that H(4,4,2)(z, q) = S13((−1, 0, 0)|(0, 0, 1)) − qS13((−1, 0, 1)|(0, 1, 1)) − zS13((0, 0, 0)|(0, 1, 1)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='9) − S13((0, 0, 1)|(0, 0, 0)) + qzS13((0, 0, 1)|(1, 1, 1)) + qS13((0, 1, 1)|(0, 0, 1)) + (1 − z)S13((1, 0, 0)|(0, 0, 0)) − qz(1 − z)S13((1, 0, 0)|(1, 1, 1)) − q(1 − z)S13((1, 0, 1)|(0, 0, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Then, by the q-difference equation for H(5,2,3)(z, q), H(5,2,3)(z, q) − H(6,2,2)(qz, q) − H(5,1,4)(qz, q) − H(4,3,3)(qz, q) + (1 − qz)H(6,1,3)(q2z, q) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='10) + (1 − qz)H(5,3,2)(q2z, q) + (1 − qz)H(4,4,2)(q2z, q) − (1 − qz)(1 − q2z)H(5,2,3)(q3z, q) = 0, together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='9) we claim that H(5,1,4)(z, q) = S13((−1, 0, 1)|(0, 0, 0)) − qS13((−1, 1, 1)|(0, 0, 1)) − S13((0, 0, 0)|(0, 0, 0)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='11) + (1 − z)S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − (1 − q)S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − q(1 − z)S13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) + qS13((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) + (1 − z)S13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − q(1 − z)zS13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − (1 − z)S13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)) − q(1 − z)S13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) + q2(1 − z)zS13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) + (1 − z)S13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)) + q(1 − z)S13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) + (1 − z)(1 − qz)S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)) − q2z(1 − z)(1 − qz)S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)|(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − (1 − z)(1 − qz)S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)) − q(1 − z)(1 − qz)S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − q2z(1 − z)S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 14 ALI KEMAL UNCU Finally, by replacing the formulas in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='11) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12) H(6,0,4)(z, q) − H(7,0,3)(qz, q) − H(5,1,4)(qz, q) + (1 − qz)H(6,1,3)(q2z, q) = 0 we conjecture that H(6,0,4)(z, q) = S13((0, 0, 1)|(0, 0, 0)) − qS13((0, 1, 1)|(0, 0, 1)) − S13((1, 0, 0)|(0, 0, 0)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='13) + (1 − qz)S13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − (1 − q)S13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − q(1 − qz)S13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) + qS13((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) + (1 − qz)S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − q2z(1 − qz)S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − (1 − qz)S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)) − q(1 − qz)S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) + q3z(1 − qz)S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) + S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)) + q(1 − qz)S13((2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) + (1 − qz)(1 − q2z)S13((3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)) − q3z(1 − qz)(1 − q2z)S13((3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)|(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − (1 − qz)(1 − q2z)S13((3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0)) − q(1 − qz)(1 − q2z)S13((3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)) − q3z(1 − qz)S13((3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)|(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We can prove that the later claimed H(6,4,0)(z, q), H(5,5,0)(z, q), H(5,4,1)(z, q), H(5,3,2)(z, q), H(5,2,3)(z, q), and H(6,0,4)(z, q) the initial conditions Hc(0, q) = 1/(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ and Hc(z, 0) = 1 in the succession from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='8), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='10), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' To prove the Hc(0, q) = 1/(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ initial condition we need to first shift z �→ z/q in all but the last of the functional equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The q-difference equations for H(10,0,0)(z, q), H(9,0,1)(z, q), H(8,0,2)(z, q), and H(7,0,3)(z, q) becomes tau- tologies once translated into S13 form using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The q-difference equations for H(7,3,0)(z, q), H(6,4,0)(z, q), H(6,3,1)(z, q), H(5,3,2)(z, q), H(5,2,3)(z, q), and H(6,0,4)(z, q) are the recurrences used to define the missing Hc(z, q) functions in the modulo 13 family (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='8), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='10), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Hence, these equations also trivializes once the relevant functions are written in their claimed S13 forms using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='7), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='9), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='11), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='13) together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' After the considerations above, we end up with 10 non-trivial coupled q-difference equations to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Showing that the q-difference equations’ in the claimed S13((a1, a2, a3)|(b1, b2, b3)) belong to the ideal I13, which is generated by the relations of S13((a1, a2, a3)|(b1, b2, b3))s (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5), is done by the method outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Explicit linear combination of (R(i) 1 (ρ|σ))-(R4(ρ|σ)) equivalents of these 12 functional equations in S13 form can, once again, be found in the ancillary files portion of ArXiv and on the author’s website [36] under the file names M13RecHXYZ Explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='txt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Here XYZ is to be replaced by the relevant profile’s digits such as 910 for the profile (9, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' One can check that the elements of I13 given in these text files are equivalent to the q-difference equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6) satisfied by H(X,Y,Z)(z, q) after they are translated to S11(ρ|σ) form using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='7), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='9), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='11), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='13) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The recurrence names are reflected in the text as RX[{Y},{{a1,a2,a3},{b1,b2,b3}}] for X and Y to be replaced by 1 or 2 to denote R(Y ) X ((a1, a2, a3)|(b1, b2, b3)), or RZ[{{a1,a2,a3},{b1,b2,b3}}] for Z to be replaced by 3 or 4 to denote R3((a1, a2, a3)|(b1, b2, b3)) and R4((a1, a2, a3)|(b1, b2, b3)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Finally, a guide document that explicitly lists each R functional relation for modulo 10 is given in M13R text file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This tedious, error prone and impossible by hand calculation proves the following theorem and its corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2 is correct for m = 13 and N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1 is correct for m = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2 is equivalent to the following theorem: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The claimed expressions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='7), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='9), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='11), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='13) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' As before, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3 adds another new witness to Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='6, and increases our confidence in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Now that the main conjectures are proven for the modulus 13 cases, we can set z = 1 and see the 22 sum-product identities coming from the cylindric partitions paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES 15 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The following identities hold � r1≥r2≥r3≥0 s1≥s2≥s3≥0 qr2 1−r1s1+s2 1+r2 2−r2s2+s2 2+r2 3−r3s3+s2 3 pc(r1, r2, r3, s1, s2, s3, q) (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r1−r2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r2−r3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s1−s2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s2−s3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s3(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r3+s3+1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='14) = 1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ 1 θ(qi1, qi2, qi3, qi4, qi5, qi6, qi7, qi8, qi9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q13),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' where the polynomials pc(r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) and the 9-tuples (i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i9) for each profile is given in the following table: Profile c pc(r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q) (i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' i9) (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0) qr1+r2+r3+s1+s2+s3 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 6) (9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0) qr2+r3+s1+s2+s3 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 6) (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0) qr3+s1+s2+s3 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 6,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3) qr2+r3(1 − qr1+s3+1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 6) (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2) qs3(1 − qr3+s2+1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3) qs3(1 − qr3+s2+1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 6) (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3) (1 − qr3+s3+1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 6) (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0) qs2+s3(q−r1+s1 − qr3 + qr2+r3+s1+1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 6) (6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4) qr3 − qr2+r3+s3+1 − qr1 + q2r1+r2+r3 + qr1+r2+r3+s2+s3+1 +(1 − q)qr1+s3(1 − qr3 − qr3+s3+1) −(1 − q)q2r1(qr3 − qs3 − qr2+r3+s3+1 + qr1+r2+r3+s3+3 +qs2+s3+2 + qr3+s2+s3+1 − qr3+s1+s2+s3+3) +(1 − q)(1 − q2)qr3(1 − qr3 − qr3+s3+1 + qs1+s2+s3+3) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 6) (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 0) q−2r1+s1+s2+s3 − q−r1+r3+s2+s3 − qs2+s3−1 + qr3+s1+s2+s3 +q−r1+r2+r3+s1+s2+s3+1 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5) (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1) q−r1+s2+s3 − q−r1+r3+s1+s2+s3+1 − qs1+s2+s3 − qr3+s3 +qr2+r3+s2+s3+1 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 6) (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4) q−r1+r3 − q−r1+r2+r3+s3+1 + qr2+r3+s2+s3+1 − (1 − q)qr3+s3 − 1 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 6) (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2) q−r1+s3 − q−r1+r3+s2+s3+1 − qs2+s3 − qr3 + qr3+s1+s2+s3+1 +qr2+r3+s3+1 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 6) Once we ignore the symmetries between variables r and s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4 proves 16 essentially unique sum- product identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' It can easily be seen that within the under-the-line identities, we do not see these sym- metries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The product related to the first profile, (10, 0, 0), on the table is presented in the introduction as Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We also note that among these identities the ones related to profiles (10, 0, 0), (8, 1, 1), (6, 2, 2), (4, 3, 3) are the i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' , 4 cases of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='22), respectively, and (7, 3, 0) and (8, 2, 0) are the σ = 0 and 1 cases of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='23), respectively, of [7, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 16 ALI KEMAL UNCU 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Future Directions There are many mathematical questions that arose from the recent studies on cylindric partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' It is relevant to mention some of the future directions we plan to pursue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The approach outlined in [29] and in this paper attempts to prove sum-representations for all the normalized generating function Hc(z, q) in one stroke for any fixed |c| where #(c) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The proof requires hefty calculations after the under-the-line sums are recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Then by setting z = 1 and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1), we prove sum-product identities for all profiles within a cylindric partition system for a fixed modulus, again in one stroke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Therefore, to prove A2 Rogers–Ramanujan identities we first prove a more general and more complicated combinatorial connection with a free variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The success of this method depends on the completion of these calculations, which is virtually impossible by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Warnaar [43] mentioned that he build the necessary theory of the Bailey machinery for profiles with 3 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This machinery will allow us to prove one sum-product identity at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This is wonderful to hear and a great advancement in mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Sadly, it comes with its own short-comings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Warnaar acknowledged that this Bailey machinery can not prove any under-the-line identity at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' It can only find the sum-product relation related to the z = 1 specializations of Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This is similar to the situation of the original Andrews–Schilling–Warnaar paper, where for example at the modulo 7 case the Bailey machinery there couldn’t reach the under-the-line identity related to the profile (2, 2, 0), which was later proven in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Be that as it may, we plan to investigate ways to simplify calculations necessary to prove the identities as a whole in one stroke for the free z case by adding the extra information we gather from Warnaar’s results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' At the very least, for the z = 1 specialization, we should pursue ways to prove under-the-line identities using the Bailey-machinery-proven over-the-line identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' There are other sum-product identities that are not visible through the cylindric partitions paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='These identities do not have a related cylindric partition profiles attached to them either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Similar to the under-the- line identities, we discover and prove these sum representations using the proven relations in the cylindric partitions system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' For example, there are the following two modulo 10 examples similar to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4): � r1≥r2≥0 s1≥s2≥0 qr2 1−r1s1+s2 1+r2 2−r2s2+s2 2 qs1+s2(1 + qr1+r2+1) (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r1−r2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s1−s2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r2+s2+1 = 1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ 1 θ(q, q, q3, q4, q4, q4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q10), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) � r1≥r2≥0 s1≥s2≥0 qr2 1−r1s1+s2 1+r2 2−r2s2+s2 2 qs1+s2(1 − qr1+r2+1) (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r1−r2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s1−s2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)s2(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)r2+s2+1 = 1 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ 1 θ(q2, q2, q2, q3, q3, q3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2) All the products associated to principal characters of modulo 10 A2 Rogers–Ramanujan identities are covered by the products that appear in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The identities (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2) are outside of this system and appear, so to say, on the dark-side of the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We hope to find a cylindric partition interpretation of these identities in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Nevertheless, we plan to present the proofs of these theorems using q-theoretic means in an upcoming paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' It is still highly relevant to find manifestly positive sum representations for any one of the identities men- tioned here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We are looking for ways to see the positivity of the series coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In [7], Andrews–Schilling– Warnaar suggests applying hypergeometric transformations to eliminate the (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' q)∞ factor that appear in the identities (such as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='4)) to get a manifestly positive representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' That suggestion is limited and might not be widely applicable, especially for the under-the-line identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' In the study of symmetric cylindric partitions [12] another two fundamental modulo 8 partition theoretic identity families, namely G¨ollnitz–Gordon and little G¨ollnitz identities, showed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The G¨ollnitz–Gordon identities are known to be related to the level 2 modules of affine Lie algebra A(2) 5 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This raises new questions of whether, similar to the symmetric partitions paradigm, we can also relate symmetric cylindric partitions to character formulas of some affine Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The product formula analogous to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) for the count of symmetric cylindric partitions’ is present in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' At the moment, the relation of these products’ to affine Lie algebra character formulas are fuzzy, and there are no general conjectural series representations for symmetric cylindric partitions either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We plan to study these objects further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES 17 Finally, we plan to pursue sum representations of any generating functions for cylindric partitions with profiles of more than 3 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The product representation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1) and the functional equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3) apply regardless of the size and length of the profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' So far, we are only able to prove and conjecture sum representations for the profiles with up to 3 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Comments on Computations In the computerized proofs of [19], we make extensive use of [1] and [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Those proofs had three main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Finding a recurrence relation (over the exponent of z) for claimed sum formulas of the (normalized) generating functions of cylindric partitions, uncoupling the q-difference equation system laid out by the (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3) to get a recurrence satisfied by the coefficient of the z’s in the true generating functions of cylindric partitions, comparing recurrences (taking greatest common divisors of recurrences as operators if needed) and showing that both sequences satisfy the same recurrences with the same initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Once the critical mass of proved identities were reached the rest of the identities were shown by series manipulations guided by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' That way we showed that all the claimed sum and the true combinatorial generating function were the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This proof required two hefty algorithms, namely Creative Telescoping algorithm and Gr¨obner bases calculations, to find the recurrence of a given hypergeometric sum dependent of a discrete variable and to uncouple a coupled system of recurrences, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We tried using the same method to prove some claims Warnaar [42] made for cylindric partitions with 3 part profiles where the modulus is not divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Then we quickly saw that the Creative Telescoping calculations were not terminating (in any definition of reasonable time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This is due to the increasing number of nested summations in these conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' However, uncoupling of recurrences could still be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Kanade–Russell’s approach [29] to prove that the claimed series representations for the bivariate generating functions of cylindric partitions are the true generating functions is a fresh take on things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' It is somehow backwards compared to the proofs of [19], in the sense that we first extend our conjectural identities using the explicit conjectures of Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='13 and series manipulations, then prove all these conjectural identities by showing that the coupled relations are satisfied and that we still satisfy the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The key idea of reducing coupled q-difference equation with the functional relations of the claimed hypergeometric sums was also used in [16] in a different context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Moreover, this approach replaces (the old bottle-neck) Creative Telescoping with the contiguous relations of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' However, rewriting the coupled relations of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='3) in the new language as a linear combination of terms in the ideal Im (see Section 3) with coefficients in Z((q, z)) is highly non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Kanade [26] mentioned that they found these linear combinations by first making an ansatz for a single case at a time and then solving for undetermined coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The identification of the minimal necessary ansatz is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' They also mentioned that each hard-case proof of modulo 10 calculations took about 8 hours to terminate on a home computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' With the matrix reduction approach of this paper, we are order of 2 faster in the modulo 10 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This is basically because once we reduce a matrix, we can use it repeadetly for all the functional relations, whereas the previous approach needs to make a single ansatz and solve if for all cases individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' It is with this speed upgrade that we could prove the new modulo 11 and modulo 13 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' On the other hand, modulo 9 and modulo 12 cases are still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This is likely due to the extra degree of complication the q-binomial coefficients in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='9)’s introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' As the order of the recurrences the Sm(ρ|σ) satisfy increases, the systems we need to reduce also become larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Mathematica’s Gaussian elimination function RowReduce is adamant in calculating the reduced row echelon form of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This is not only not necessary, it also overcomplicates the calculations by introducing large rational function expressions for upper triangular coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This forced us to implementing our own Gaussian elimination algorithm within the Mathematica computer algebra system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This basic implementation sorts, performs row elimination of a matrix with entries in a polynomial ring with integer coefficients, such as Z[q, z], while not introducing rational functions, and it terminates when a row echelon matrix (a triangular system of equations) is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This function will be made a part of the impending next version release of qFunctions package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' As a side note, we implemented a naive parallelization of this elimination but we have not seen any benefits of splitting calculations yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We should also acknowledge that there are at least two crucial optimizations waiting to be implemented to aid proos of families in cylindric partitions scheme and other similar schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' First task that should be done 18 ALI KEMAL UNCU is to keep track of nullified relations and to remove the contributions of the nullspace in later calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' To put it in concrete terms, at the moment we do not know if N = 6 is the minimal number to prove Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1 and/or 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' We know that it is a sufficient number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' By removing any and all nullified relations we would only see a minimal representation (dependent on the choice of N) of these recurrences as elements in the ideals Im, and that can give us an idea of what the optimal bound for N is supposed to be in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The second pending addition is dynamic extension of the matrix to be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' At the moment, we fix an N experimentally hoping that it is enough to show that the relations of interest are in the nullspace of this matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This is in the same spirit of making a fixed ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Row reduction as a preprocessing step helps for the repeated calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Having an echelon system boosts the speed of later calculations immensely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' If the chosen N is not enough, then we need to pick a larger N and start all over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This requires performing row reduction of the matrix for N once more as a subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' This should be changed by extending the already triangularized matrix for N to N + 1 and doing the row reduction again for only the added relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' The incrementality of the matrix would also carry us to the minimal necessary N for any given m (assuming that Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='2 is correct) naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Combinatory Analysis, volume 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Cambridge University Press, New York, NY, USA, 1916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' [34] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Milne and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' M.' metadata={'source': 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26:258ˆa€“263, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' [35] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Milne and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Lilly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Consequences of the Aℓ and Cℓ Bailey transform and lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=', 139:319–346, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' [36] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAzT4oBgHgl3EQfZ_wC/content/2301.01359v1.pdf'} +page_content=' Uncu 9 and BLS +peak significance > 5 (for stars with T < 12 mag) or +> 9 (for stars with T > 12 mag) are labelled threshold- +crossing events (TCEs). These filters give slightly dif- +ferent perspectives on transit significance: (1) signal-to- +pink-noise compares the transit depth to pink noise in +the light curve (Pont et al. 2006), while (2) BLS peak +significance compares the BLS spectrum’s peak height +2 QLP data can be found at doi:10.17909/t9-r086-e880 +to its noise. In combination, these checks help filter out +events that are clearly not transit-like. +In addition, we filter out instances where the planet +would orbit “inside the star.” For each signal we com- +pute the expected semi-major axis to stellar radius ratio +assuming a Keplerian orbit.3 If the ratio < 1, the signal +is labeled as inside the star. Typically, these signals sig- +nify stellar variability or blended signals from a smaller +nearby star. +2.2. Assembling a set of signals to label +Even with filters described in the previous subsection, +manually labeling every TCE would take an enormous +amount of time, so we select a subset of TCEs for train- +ing / testing. +Over time, we gradually accumulated +three batches of labeled TCEs from the first two years of +TESS Primary Mission (observed with 30 min cadence) +and the first year of the TESS 1st Extended Mission +(observed with 10 min cadence). +The year 1 (Y1) TESS observations for the southern +hemisphere went through significant changes in noise +property due to the spacecraft pointing strategy change +in Sector 4,4 and the subsequent tweaking of the momen- +tum dump frequency. We selected 8992 TCEs detected +in Sector 13 (the last sector of Y1) for the labeling. This +was not an intentional choice, but after spending hun- +dreds of person-hours labeling these TCEs, we opted to +make use of them regardless. Fortunately, despite the +fact that our Y1 TCEs came only from Sector 13, the +observations that led to these detections still included +a diversity of spacecraft pointing control strategies and +data artifacts (for example detector warmups following +instrument anomaly events5). In particular, stars ob- +served in Sector 13 have been observed by TESS in Y1 +between one to thirteen sectors and cover a variety of +prior sectors. +For the year 2 (Y2) TESS observations in the northern +hemisphere, the data has more uniform characteristics +including a consistent momentum dump frequency of +every 4.4 days starting in Sector 146. We sorted TCEs +by their target’s TESS magnitude, and then took the +13372 brightest TCEs detected from Sectors 14–26. +3 When computing the semi-major axis we use two times the +detected BLS period in case the detected period is half the true +period, which often happens for eclipsing binaries. If the star has +an estimate for its mass in the TIC, we use that value; if not, we +assume a mass of 1 M⊙. We also assume a circular orbit. +4 +https://archive.stsci.edu/missions/tess/doc/tess drn/ +tess sector 04 drn05 v04.pdf +5 +https://archive.stsci.edu/missions/tess/doc/tess drn/ +tess sector 08 drn10 v02.pdf +6 +https://archive.stsci.edu/missions/tess/doc/tess drn/ +tess sector 14 drn19 v02.pdf + +4 +Tey/Moldovan et al. +In year 3 (Y3), TESS returned to observe the south- +ern hemisphere, with faster cadence and a further im- +proved momentum dump strategy (only once each or- +bit)7. We added an additional 2588 TCEs from Sectors +27-39, which increased the sky coverage and brightness +range for our southern hemisphere labels. +We note that TCEs around stars only observed in one +of the CCDs in Sector 13 Camera 1, and Camera 1 and +2 for Sector 24 and 25 are not included in our sample +due to temporary unavailability of the data at the time +of vetting. +Altogether, these TCEs create a broad sample of +transit-like events detected in the first three years of +TESS observation. The final TCE distribution across +the sky is shown in Figure 1, and across TESS magni- +tude (Tmag) in Figure 2. Due to the different selection +criteria of the TCEs from three different years, they have +somewhat different data characteristics. As discussed in +Section §5.2, these differences do not significantly im- +pact our results. +2.3. Labels and their definitions +For each TCE we assigned one of the following five +labels: +• E denotes a periodic eclipsing signal. This includes +both planetary transits and non-contact eclipsing +binaries. +In the triage process, we do not take +into account information that would distinguish +an eclipsing signal from background stars from an +eclipsing signal on the target star. +Both cases +would be labeled as E if they satisfy all the other +criteria. +• S denotes events containing only a single transit +or events where an incorrect period or period alias +is assessed to be reported from BLS. +• B denotes contact eclipsing binaries. +They are +distinguishable from non-contact binaries through +their continuous ingress/egress slope. +• J denotes junk. This includes other astrophysical +phenomena like stellar variability as well as instru- +mental phenomena like scattered light (due to the +Earth or the Moon approaching the field of view +and reflecting light into the camera) or artifacts +introduced at the times of spacecraft momentum +dumps (when the spacecraft’s reaction wheels cor- +rect for the spacecraft’s speed). +7 +https://archive.stsci.edu/missions/tess/doc/tess drn/ +tess sector 27 drn38 v02.pdf +• N denotes not sure. No conclusive label decision +could be made for these TCEs. Often an N label +was given when a weak signal bordered on being +an E or J. +These labels are not necessarily mutually exclusive. +We detail the rules we use in labeling when resolving +marginal/ambiguous cases: +• E vs S: If there is ambiguity in the period (e.g. both +the reported period and the double period are con- +sistent with the data) or the period is only slightly +off, we default to an E label. +Only if the pe- +riod is explicitly incorrect (e.g there are flat light +curve segments during expected transits, or there +are multiple regular transits outside of expected +transit times) do we choose an S label. If there +is only one regular transit outside the expected +transit time, i.e. it might represent a secondary +eclipse, we use an E label, and if the reported pe- +riod potentially includes the secondary eclipse, we +also use an E label. +• B vs S: If we have a contact binary with the incor- +rect period, we default to a B label. +We choose these labels first because they mirror astro- +physical phenomena. This means the labeled TCEs pro- +vide good targets for follow-up (e.g. Es will be good can- +didates for exoplanet and binary star detection). Sec- +ond, we expect similarities in light curve morphology +within a label. This should help our model learn labels +more accurately. +For the purposes of finding exoplanets, we are particu- +larly interested in high precision and recall metrics for E +labels. S and N labels may also be important candidates +for further investigation. +2.4. Labeling process +All TCEs were manually assigned labels based on +human-visual representations (see Figure 3) similar to +the model input representations described in Section 3. +On a weekly basis, batches of targets were independently +vetted by 3 – 7 of the authors. At the end of the week, +targets with conflicting labels where at least one human +chose an E or S were discussed in order to reach a con- +sensus on the target’s final label. If a target had only B, +J, or N votes, we assigned weights to each label based on +the number of votes. Altogether, this process took over +2 years. We expect the multiplicity of vetters to reduce +the number of label errors, giving us a very high-quality +dataset. +Table A contains examples of signal data along with +individually-assigned labels and their consensus dispo- + +Improved TESS Triage with Neural Networks +5 +Figure 1. Sky map showing the locations of the 24926 TCEs presented here (black starred data points) compared to the +coverage of each TESS Prime Mission sector (colored data points). The black and red labels are the Prime Mission sector +numbers in the southern and northern ecliptic hemispheres, respectively. Note that we also include 2588 TCEs from the 1st +Extended Mission, for which sector coverage is not shown here. The under- and over-densities of TCEs are due to the selection +criteria as described in the text. +Figure 2. Distribution of Tmag across our dataset. Both +Y1 and Y2 portions of the dataset focused on the brightest +TCEs, while Y3 added TCEs more uniformly across magni- +tudes. More details on TCE selection can be found in Section +2.1. +sitions. The full table (and accompanying light curve +data) can be found online in Tey et al. (2022). +Following common practice in ML, we randomly sep- +arate the dataset into a training, validation, and test +set. The model is initially fit on the training set, a set +of examples used to fit the parameters of the model. +Next, the validation set provides a measure of predic- +tive accuracy and model fit. The validation set consists +of examples that the model has not seen in the training +set and allows for optimization of the architecture and +hyperparameters. Lastly, after the model architecture +and hyperparameters are finalized, the test set is used +as one last objective test of the model accuracy and fit. +1. Training set (19919 targets): +used for model +training. +(15414 J + 2102 E + 1681 B + 224 +S + 498 N) +2. Validation set (2491 targets): used to calculate +precision, recall, detection threshold for binary +classification, and model debugging. +(1945 J + +261 E + 198 B + 17 S + 70 N) + +80 +60 +40 + [deg] +195.18 4 +20 +F16 15 +8 21＇20 +24 +173 +14 +26 25 +922 +E2 +0 +1023 +Dec +1 +11 +13 +12 +-20 +-40 +-60 +-80 +20 +15 +10 +5 +0 +RA [hr]5000 - +三 +Y1 +Y2 +4000- +Y3 +Counts +3000 - +2000 - +1000 - +0: +0 +5 +10 +Tmag6 +Tey/Moldovan et al. + +EBImproved TESS Triage with Neural Networks +7 + +S +H +TTTIT +H +TS8 +Tey/Moldovan et al. +Figure 3. Six example visual representations used for human labeling with labels in red. The different figures within each +representation were made to mirror the information described in Section 3. Each image was individually labeled by at least 3 +individual vetters. Conflicting labels were discussed and resolved each week. + +:ww +HHH +HHH +H +HH +HHFN +亚王 +【 +亚 +TII +HITImproved TESS Triage with Neural Networks +9 +Figure 4. Distribution of labels across our dataset (see Sec- +tion 2.3 for descriptions of each type). As described in Sec- +tion 2.4, some TCEs were assigned fractional B and J labels +so these counts have been rounded to the nearest integer. +Figure 5. Scatterplot of transit depth vs. orbital period for +our dataset. TCEs with E labels are shown in blue. Red +lines mark 13.7 and 27.4, the orbital period and twice the +orbital period of TESS. +3. Test set (2516 targets): hold-out set used for fi- +nal evaluation; this set was never used for training +or debugging, or any other evaluation. (1970 J + +250 E + 200 B + 34 S + 62 N) +2.5. Distribution of the labels +Figure 6. Scatterplot of planet radii vs. orbital period for +our dataset. TCEs with E labels are shown in blue. Red +lines mark 13.7 and 27.4, the orbital period and twice the +orbital period of TESS. +Figure 7. Scatterplot of transit duration vs. orbital period +for our dataset. TCEs with E labels are shown in blue. Red +lines mark 13.7 and 27.4, the orbital period and twice the +orbital period of TESS. +Figure 4 shows the distribution of labels in our train- +ing set. Out of the total 24926 labels, the majority are +J labels (19329). The amount of signals identified as +eclipsing objects (E, 2613) is comparable to that iden- +tified by contact binaries (B, 2079). + +Counts +1000 +0 +107 +106 +Transit Depth [ppm] +105 +104 +103 +102 +101 +10-1 +100 +101 +102 0 +5000 +Period [days] +CountsCounts +1000 +0 +103 +102 +101 +100 +10-1 +100 +101 +102 0 +2500 +Period [days] +CountsCounts +1000 +0 +102 +Transit Duration [hrs] +101 +100 +10- +10-1 +100 +101 +102 0 +5000 +Period [days] +Counts20000 +19329 +15000 : +Counts +10000 +5000 - +2613 +2079 +630 +275 +0 +E +B +N +s +Disposition10 +Tey/Moldovan et al. +We examine the distribution of the fundamental tran- +sit parameters (i.e., orbital period, transit depth, esti- +mated planet radius, and transit duration) of the la- +bels in Figure 5, 6, and 7. Specifically, we compare the +parameter spaces resided by the E labels to the other +labels. +The comparison reveals the following charac- +teristics: (1) a majority number of the TCEs with pe- +riod smaller than ∼ 0.5 days are not caused by eclipses; +(2) a majority of the shallow events with period longer +than 10 days are not caused by eclipses; (3) there is +clear pile-up of TCEs at the TESS orbital period and +its alias, which are not caused by eclipses; (4) a major- +ity of TCEs with extremely short/long transit duration +are not caused by eclipses. +3. MODEL INPUT REPRESENTATIONS +For each TCE, we pass the raw flux time series leading +to the detection and all the relevant information describ- +ing the detected periodic signal and target star to the +neural network. +3.1. Time series data +We preprocess the raw flux time series into dif- +ferent input representations before passing them to +Astronet-Triage-v2. +We use the same basis spline +techniques used in QLP, however, the transit signals +are masked out based on the BLS-detected period, +epoch and duration before the optimal spline is com- +puted. This approach will often prevent over-fitting of +the transit signals during the detrending process. +To +account for different time scales of the stellar variabil- +ity, we adopt multiple detrending settings to provide +Astronet-Triage-v2 a more complete view of the light +curve noise characteristics. Unlike in QLP, which only +uses one set of splines with spacing between 0.3 and +1.5 days to create the final detrended light curves, we +use three different settings (0.3, 5.0, and a value which +minimizes the Bayesian Information Criterion, Schwarz +1978) to create three different sets of detrended light +curves. The light curves detrended with larger spacing +are also less likely to over-fit the transit signals with +long transit duration. +For each detrended light curve we generate seven dif- +ferent plots or views (see Figure 8). Each view is binned +using a robust binning technique to de-weight outliers. +During this binning, we also account for the change in +exposure time between the Primary and 1st Extended +Mission by weighing points according to their exposure +time in a given bin. After this, we normalize the binned +data so that the minimum value is -1 and the median +value is 0. The complete list of views can be found in +the source code 8. A detailed description of each view +type is below: +• Global View: The global view uses the full light +curve folded on the reported period with 201 bins. +In addition to the median values, the view also +includes the standard deviations for each bin, a +mask indicating whether the bin was empty, and +a mask indicating whether the bin falls inside the +detected transit. +• Local View: The local view uses points within two +transit durations of the transit center (for a full +timespan of four transit durations), again folded +on the reported period. +The local view uses 61 +bins, and includes standard deviation and mask +values like the global view. In addition, we also +record the scale factor used in normalization, as a +scalar feature. +• Secondary View: The secondary view is similar to +the local view, but is centered around the most +significant secondary transit, determined by per- +forming a grid search9 on the out-of-transit por- +tion of the phase folded view, for a duration equal +to the primary transit duration, and selecting the +region with the highest signal/noise ratio. +This +view is accompanied by two scalar features: the +normalization scale factor, and the phase of the +secondary transit’s center. +• Local Half-Period View: Similar to the local view, +but folded at half the detected period. This view +only contains the standard deviation value, since +the median value can appear very noisy when fold- +ing a transit over a non-transit. +• Global Double Period View: Similar to the global +view, but folded at twice the period of the global +view. +• Sample Global Segments: This view contains the +entire period (similar to the global view), but +showing up to 7 of the folds that contain the most +points (ties are broken at random). Each fold is ac- +companied by a mask indicating whether the bin +contains any points. +If the light curve contains +8 +https://github.com/mdanatg/Astronet-Triage/blob/ +e4ec517b175b2a3dfb74cf6c6e3f5273dd8749c7/astronet/ +astro cnn model/configurations.py#L2254 +9 +https://github.com/mdanatg/Astronet-Triage/blob/ +e4ec517b175b2a3dfb74cf6c6e3f5273dd8749c7/light curve util/ +find secondary.py#L62 + +Improved TESS Triage with Neural Networks +11 +fewer transits, the extra views remain empty. Each +fold is independently binned with 201 bins. +• Sample Local Segments: +Similar to the sample +global segments, this view contains the transit cen- +ter of up to 4 of the folds that contain the most +points (ties are broken at random), for a total of 8 +folds. Each fold is independently binned with 61 +bins. +3.2. Scalar data +We also use scalar values that describe characteristics +of the transit, host star and the light curve itself. Transit +features include period in days (P), transit duration in +days (Tdur), transit depth (δ), and the number of full +periods observed in the flux-time series (nfolds), while +host star features include TESS magnitude (Tmag), mass +in M⊙, and radius in R⊙. The host star features are +directly extracted from the TESS Input Catalog v8.2 +(Paegert et al. 2021). +For TCEs without stellar radii in the catalog, we per- +form a rough estimate using a Bayesian estimate of the +distance (Bailer-Jones et al. 2021), apparent magnitude +(either Gaia G, Bp, and Rp, or Gaia G and 2MASS K if +Bp and Rp are unavailable), and color/temperature and +color/bolometric corrections from MIST models (Choi +et al. 2016). In brief, we estimate the temperature and +bolometric correction from either the target’s Bp-Rp or +G-K colors, use the bolometric correction to estimate +the target’s apparent bolometric magnitude, use the es- +timated distance to the target to convert to an absolute +magnitude, convert to bolometric luminosity, and solve +for the stellar radius from the temperature and lumi- +nosity via the Stefan Boltzmann Law. In our testing, +we were able to determine radii within about 10% of +the TIC values when present, and provided radius esti- +mates for ∼ 2400 from the ∼ 2800 TCEs missing stellar +radii in our dataset. +Light curve features include the total number of +points. +Each scalar value is normalized to be zero +mean and unit variance across the dataset, except for +nfolds which is truncated to a maximum value of 100 +and a log-scaled to fit between 0 and 1. In addition, +we also include as scalar inputs the detected phase of +the secondary eclipse, as well as the calculated scaling +factor when normalizing the global, local and secondary +views. +4. NEURAL NETWORK ARCHITECTURE +Our model uses a convolutional neural network archi- +tecture derived from Astronet. The high level architec- +ture is shown in Figure 8. +Each time series feature is grouped together with +similar features and then passed through a separate +convolutional tower. +For example, the global view +flux is grouped together with the standard deviation +of the global view, so that they form a 2-channel, 1- +dimensional image. +The structure of a convolutional +tower is shown in Figure 9. Each tower consists of con- +volutional layers with Rectified Linear Unit (ReLU) ac- +tivation, alternating with pooling layers. The pooling +layers aggregate neighboring pixels, in effect increasing +the field of view of the subsequent convolutional layer. +The output of each convolutional tower is flattened +into a vector shape. The flattened outputs from all tow- +ers are concatenated together with the auxiliary inputs +to form the input into the next section of the network, +the fully-connected tower, whose structure is shown in +Figure 10. +The fully-connected tower is composed of +several fully-connected neural network layers, alternat- +ing with dropout layers. The dropout layers randomly +set inputs to zero, and serve a role of regularization, to +mitigate over-fitting. The dropout layers are only ac- +tive during training. The final layer has five outputs, +and uses a sigmoid activation function, so that its out- +put is in the interval [0..1]. +Each of the five outputs +corresponds to one of the five labels. +The various hyper-parameters of each network can be +found in the configuration file included with the source +code.10 +The hyper-parameters are tuned using Vizier +(Golovin et al. 2017a; Song et al. 2022) by minimizing +the loss on the validation set. +4.1. Training +We train the model using the Adam, a popular variant +of stochastic gradient descent optimization (Kingma & +Ba 2014), for 20,000 steps. The complete set of training +parameters can be found in the code 11. +For the loss function we use binary cross-entropy +loss12. Notably, this means that the model is not trained +to choose between the five labels exclusively. Instead, it +produces independent scores for each label, so a model +is free to assign high scores for both “E” and “J” la- +bels, for instance. This loss function enables us to assign +weighted labels to uncertain examples (e.g. 50 percent +10 +https://github.com/mdanatg/Astronet-Triage/blob/ +e4ec517b175b2a3dfb74cf6c6e3f5273dd8749c7/astronet/ +astro cnn model/configurations.py +11 +https://github.com/mdanatg/Astronet-Triage/blob/ +e4ec517b175b2a3dfb74cf6c6e3f5273dd8749c7/astronet/ +astro cnn model/configurations.py#L2254 +12 See https://www.tensorflow.org/api docs/python/tf/keras/ +losses/BinaryCrossentropy for the implementation and Good +(1952) and Shallue & Vanderburg (2018) for more information + +12 +Tey/Moldovan et al. +Figure 8. Astronet-Triage-v2 neural network architecture. +Figure 9. +Structure of a CNN tower. +Each convolution +tower has 1 to 4 blocks. Each block has 1 to 4 layers. +Figure 10. Structure of the fully-connected tower. +“B”, 50 percent “J”). The weight is determined as fol- +lows: if a target had a single label (as resulting from +the group resolution, or if the vote was unanimous), the +weight is 1.0; if the target had multiple votes, the weight +is the maximum number of votes for any label divided by +the total number of votes. This means targets for which +a label didn’t receive a majority of votes are weighted +less. +We don’t apply data augmentation, although that is +something we intend to do in future work (see Section +6.4.2). +4.2. Prediction and ensembling +As a multi-class classifier, our model outputs a predic- +tion score for each label. Predictions where the “E” label +score exceeds a threshold chosen beforehand are consid- +ered to predict the label “E”. Otherwise, the model is +considered to predict the label with the highest predic- +tion score. +We then construct an ensemble of 10 models trained +separately (hence with different initial weight values, +and different shuffling of the input data). The compound +prediction of the ensemble is constructed as follows: +1. If any of the models predicts “E”, then the ensem- +ble prediction is also “E”. +2. Otherwise, the ensemble prediction is the label +predicted by a majority of models, with ties bro- +ken at random. + +Global +Local +Secondary +Half-Period +2x Period +Ind. Transits +(1 x 200 x 6) +(1 x 60 x 7) +(1 x 60 x 7) +(1 x 200 x 1) +(1 × 200 x 1) +(1 x 200 x 14) +Aux. Inputs +period +duration +depth +Convolution +Convolution +Convolution +Convolution +Convolution +Convolution +T mag +Network +Network +Network +Network +Network +Network +star mass +star radius +3 layers +3 layers +3 layers +3 layers +3 layers +3 layers +#folds +#points +Fully-connected Network +5 layers +E +s +B +J +N + score +score +score + score +scoreConv1D +Conv block +ReLU +Conv block +MaxPool1D +Conv block +FlattenConv +aux features +Fully-connected +Fully-connected block +ReLU +Fully-connected block +Dropout +Fully-connected +Logits block +Sigmoid +Class scoresImproved TESS Triage with Neural Networks +13 +Although the model predicts five different labels, we +are primarily interested in the “E” label. +The other +labels are mainly used at training, to encourage the net- +work to learn natural representations. We found that +the extra labels greatly help understand a model’s pre- +dictions, as well as validate whether the model does in- +deed create correct internal representations. +5. RESULTS +Here we report the results of our ML activity predic- +tions. First we discuss the metrics we used to evaluate +the performance and then we summarize how the differ- +ent models performed on each dataset. +The two primary metrics we use to evaluate our per- +formance are precision and recall. The precision, or re- +liability, of a model on a labelled dataset is the number +of true positives divided by the number of true posi- +tives and false positives. Recall, or completeness, is the +number of true positives divided by the number of true +positives and false negatives. As we are interested in +“E” labels as potential planet candidates, they gener- +ally are used as the “positive” class. In this context, a +high precision means our model outputs fewer false posi- +tives, meanwhile a high recall means successful recovery +of more planet candidates (fewer potential planets lost +by Astronet-Triage-v2). Since labels are determined +by comparing output prediction scores against a chosen +threshold, each specific threshold yields its own precision +and recall. When plotted over many different thresh- +olds, we can form a precision-recall curve (see Figure +11). By taking the area under the precision-recall curve +(AUC-PR), also known as the average precision, we can +characterize our model’s overall performance and com- +pare against other models with the highest achievable +value being a 1. +5.1. Performance on validation and test sets +On the validation dataset we obtained an AUC-PR +value of 0.977. The model achieves 100% recall at 41% +precision, at a prediction threshold of 0.0105. If we in- +crease the threshold to 0.215, we obtain 96.9% recall at +79.8% precision. +On the test set, we obtained an AUC-PR value of +0.965. The model achieves 100% recall at 15% preci- +sion, at a prediction threshold of 0.0005. This suggests +the test set contains more difficult examples (possibly +incorrect ones). With the thresholds suggested by the +validation set, we obtain 99.6% recall at 39.7% precision +for the 0.0105 threshold, and respectively 97.2% recall +at 75.7% precision for the 0.215 threshold. +5.2. Generalizing to TESS 1st Extended Mission data +We explore the adaptability of our network, and the +generalization of training on non-uniform datasets in +this section. In practice, models like Astronet-Triage-v2 +are trained on previously observed sectors with a goal +of classifying new observations taken by TESS in the +future. Since noise characteristics and TESS observa- +tion strategy can change sector-to-sector, it is important +that our models generalize well to new data. +Nearly 90% of our total training dataset comes from +the TESS Primary Mission, so we use QLP data from +TESS 1st Extended Mission (Sector 33, observed during +Year 3 from UT 2020 December 17 – UT 2021 January +13) to test how our model generalizes to unseen or out- +of-distribution data. +Following the QLP convention, we ran a BLS search +and Astronet-Triage-v2 on the full multi-sector light +curves (including both Primary Mission and 1st Ex- +tended Mission data) for each star. +Of the discov- +ered TCEs, we selected a random sample of 759 targets +with Tmag < 11 from camera 1 and 590 targets with +11 < Tmag < 13.5 from camera 2. Due to the TESS +pointing strategy, we focus on these cameras because +their light curves have roughly equal amounts of Pri- +mary vs. 1st Extended Mission observations. The mag- +nitude ranges also allow us to compare performance on +stars in different brightness bins. +One of our vetters (CH) independently labeled all 1349 +TCEs before evaluation, among which, 255 TCEs were +assigned an E label. +To better understand our ability to generalize, we +apply the following models to the Sector 33 dataset: +Astronet-Triage, the fully trained Astronet-Triage-v2, +and three independent instances of the Astronet-Triage-v2 +architecture trained on different subsets of our original +TCE dataset (Section 2). +These three separate training sets were formed by +splitting our original training set on observation year, +meaning roughly 40% went into training the Y1 model, +50% into the Y2 model, and 10% into the Y3 model. +The differences between these datasets are described in +Section 2.1, but briefly: Both the Y1 and Y2 datasets +feature brighter stars, but the Y1 dataset were only +taken from Sector 13, so they cover a small region of +the Southern ecliptic hemisphere. The Y2 dataset, on +the other hand, were selected more uniformly and cover +most of the Northern ecliptic hemisphere. Neither has +much overlap in sky coverage with the evaluation set +(the 1349 Sector 33 TCEs) – Y1 having little overlap +and Y2 having none. +Both datasets also have much +shorter observation baselines than the evaluation set, +and finally, due to the change in TESS momentum dump +strategy, the Y1 dataset also differs from the evaluation + +14 +Tey/Moldovan et al. +Table 1. Performance on previously unseen S33 data +Model +Cam +Threshold +Precision +Recall +Astronet-Triage-v2 +1 +0.0105 +0.64 +0.98 +Astronet-Triage-v2 +2 +0.0105 +0.53 +1.00 +Astronet-Triage-v2 +1 +0.215 +0.89 +0.91 +Astronet-Triage-v2 +2 +0.215 +0.84 +0.99 +Astronet-Triage +1 +0.08 +0.89 +0.85 +Astronet-Triage +2 +0.08 +0.82 +0.90 +set in noise characteristics. The Y3 dataset bears the +most similarity to the evaluation set in terms of data +characteristic. +It is, however, much smaller than the +other datasets. Altogether, these different datasets and +models provide useful views at our ability to general- +ize to data that can be fairly different from the training +data. +Since Astronet-Triage only distinguishes between +transit-like and non-transit-like, it’s trained to give +high scores TCEs we consider E- or S- labeled. +As +Astronet-Triage-v2 provides independent E and S +scores, we choose remove all S-labeled data from pre- +cision and recall calculations for a simple direct perfor- +mance comparison with Astronet-Triage. This leaves +us with 1315 TCEs. +Precision and recall numbers split across Astronet-Triage +and Astronet-Triage-v2 for each camera can be seen +in Table 1. +In both cameras we see that for similar +(or better) levels of precision, Astronet-Triage-v2 +provides better recall than Astronet-Triage, with a +slightly more pronounced effect in camera 2 (fainter tar- +gets). In other words, for the same amount of human +vetting time, Astronet-Triage-v2 would recover more +potential planets than Astronet-Triage. +The full precision-recall curves across all TCEs (ig- +noring S-labeled TCEs) are shown in Figure 11. Across +the board we see that Astronet-Triage-v2 (trained on +the full training set) improves on Astronet-Triage with +AUC-PR scores of 0.961 and 0.927. We also see that the +models trained only on Y1, Y2, and Y3 data perform +similarly to Astronet-Triage with AUC-PR scores of +0.954, 0.960, and 0.917 respectively. Even though the +Y1 and Y2 versions of the models don’t use any 1st Ex- +tended Mission training data, we see they’re still able to +perform highly in S33 (which occurred during Y3). This +supports Astronet-Triage-v2’s ability to generalize to +future sectors. +5.3. Performance on the TOI catalog +The TESS Objects of Interest (TOI) catalog (Guer- +rero et al. 2021), which lists the planetary candidates +detected by TESS, is a useful benchmark for high- +Figure 11. +Precision vs. recall for 1315 TCEs selected +from Sector 33 of the 1st Extended Mission. +Since +Astronet-Triage (Yu et al. 2019) only distinguishes between +transit-like and non-transit-like, it gives high scores to TCEs +we either consider to have E or S labels. For a more direct +comparison to Astronet-Triage-v2, we choose to ignore all +S-labeled TCEs when calculating precision and recall. We +see that across all levels of recall, Astronet-Triage-v2 pro- +vides higher precision even when trained only on Primary +Mission data taken during Y1 or Y2. +Although the Y3 +dataset bears the most resemblance to the S33 evaluation +set here, the size of the Y3 dataset is only ∼ 2500, so the +Y3-trained model doesn’t quite reach the performance of the +other models. +confidence E or S labels. +A good model should label +all TOI entries as E or S, since humans have inspected +each entry and considered them to be high-probability +planetary candidates (allowing for single-transit events). +On 2022 April 21 we downloaded the TOI catalog with +light curve data through Sector 47. We also use informa- +tion from TESS Follow-up Observing Program (TFOP) +Sub Groups 1 and 2 (SG1 & SG2), which use ground- +based photometry and reconnaissance spectroscopy to +follow-up on TOIs and help filter out false positives. Af- +ter keeping only planet candidates (PCs; meaning TOIs +that were not ruled out as false positives with follow-up +observations) and validated / confirmed / known planets +(Ps), we have a dataset of 4140 targets. +After evaluating all TOI signals with Astronet-Triage-v2, +Figure 12 shows the distribution of E scores. +Figure +13 shows the recall rate at different cutoff thresh- +old levels. +We see that 93% of the TOIs have E +scores > 0.0105 and as we increase the cutoff to 0.215, + +1.0 - +0.9 - +0.8 - +0.7 +Precision +0.6 - +0.5 +0.4 - +This work +This work (Y1) +0.3 +This work (Y2) +This work (Y3) +0.2 +Yu19 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +RecallImproved TESS Triage with Neural Networks +15 +Astronet-Triage-v2 passes 86% of the TOIs. +We +also see improved Astronet-Triage-v2 performance +on known, confirmed, or validated planets (Ps) com- +pared to the planet candidates (PCs) across the board. +For comparison, we also ran Astronet-Triage on all +TOI signals. Using a threshold of 0.09, as was originally +used in QLP, Astronet-Triage recovers 3349 TOIs. Us- +ing the dataset from Section 5.2, we find a precision- +matching threshold of 0.2 for Astronet-Triage-v2. By +finding the threshold of equal precision, we can com- +pare TOI recovery at a constant rate of human vet- +ter work. At this threshold, 3577 TOIs are recovered. +In other words, at least 200 TOIs are saved by us- +ing Astronet-Triage-v2 in place of Astronet-Triage +without introducing more false positives to human vet- +ters. +Some important caveats to note: +• The TOI catalog does include single-transit events. +Astronet-Triage-v2 is trained to give these S +rather than E labels. +Rather than keeping sep- +arate cutoffs for S and E scores, for simplicity +we choose to focus on E scores in reported re- +calls. This gives it a slight disadvantage in terms +of recovery numbers, though we leave them in the +dataset for fairer comparison to Astronet-Triage +which gives a score for transit-like (periodic or +single-transit) versus not transit-like. +• TOIs can also come from the SPOC pipeline, +which processes 2-minute cadence light curves. +For both Astronet-Triage-v2 and Astronet-Triage, +QLP light curves are binned down to 30 or 10 +minutes, so some signals may not be detectable +(e.g. due to low signal-to-noise in the binned light +curve) and should be assigned J labels. This con- +tributes partially to the lower recall numbers seen +at the cutoffs from Section 5.1. +• Only 130 TOI host stars appear in our dataset of +∼25,000, 100 of which were in the training set. +We also conducted this analysis with those TOIs +removed and saw similar results. +6. DISCUSSION +6.1. Use in producing the TOI catalog +A large piece of motivation for this work has been +improving on Astronet-Triage so fewer planet can- +didates are lost when searching for TOIs via QLP. +After signal detection via BLS, Astronet is one of +the finals triage steps before candidates are passed +along to human TOI vetters and potentially promoted +Figure 12. Top: Distribution of E score between this work +and Astronet-Triage (Yu et al. 2019) on the whole TOI +dataset. Bottom: Distribution of E scores from this work +when the dataset is separated into Planets (P, validated, +confirmed, and known planets) and Planet Candidates (PC, +TOIs that are not validated, confirmed, or known planets, +and were also not identified as false positives with follow-up +observations). +to TOIs (Guerrero et al. 2021). +Based on the re- +sults in Section 5 we expect Astronet-Triage-v2 to +save many planet candidates that would otherwise +be lost without adding false positives and increasing +the hours needed for human TOI vetting. Starting in +Sector 34, early versions of Astronet-Triage-v2 of- +ficially replaced Astronet-Triage within QLP. While +Astronet-Triage-v2 takes step towards a more au- +tomated process, it is still not developed enough for +population statistics (for a deeper discussion see Sec- +tion 6.4.1). +6.2. What is limiting our precision? +In our tests, we found a common source of false neg- +atives stemming from patterns with borderline label as- +sessments. The most common being eclipsing binaries +which are non-contact but still close enough to resem- +ble the pattern of a contact binary, due to, for example, +tidal distortion, hence it is unclear whether the label +should be “E” or “B” (Figure 14). Other instances of +ambiguous patterns are represented by very noisy tran- +sits, or transits on a background of high stellar variabil- + +This work +3000 +Yu19 +2500 +Counts +2000 : +1500 +1000 +500 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Score +8 +This work on Ps +This work on PCs +Normalized counts +6 +4 +2 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Score16 +Tey/Moldovan et al. +Figure 13. +Top: +Recall as a function of cutoff thresh- +old between this work and Astronet-Triage (Yu et al. +2019). For Astronet-Triage-v2 we choose to focus on just +E scores even though some TOIs are true S labels. Bottom: +Astronet-Triage-v2 recall as a function of cutoff thresh- +old when the dataset is separated into Planets (P, validated, +confirmed, and known planets) and Planet Candidates (PC, +TOIs that are not validated, confirmed, or known planets, +and were also not identified as false positives with follow-up +observations). +ity, where the distinction between “E” and “J” is more +subtle (Figure 15). +One particular element of sensitivity for the neural +network is on the correctness of the period and duration +values estimated by BLS. Errors in these values can lead +to de-trending distortions which can make phase-folded +views deviate from a transit-like light curve shape. Ex- +amples containing multi-year observations can be partic- +ularly sensitive, as even slight variations in the detected +period can lead to a blurring of the transit in the phase +folded view (Figure 16 and 17). +We also note that the phase folding and binning pro- +cesses are inherently lossy (similar to how compressing +an image is a lossy process). While we have not ascer- +tained the impact of such loss of information, it is to be +expected that it causes some loss of precision. +6.3. Comparison to other works +Our work is largely based on the original TESS +Astronet-Triage classifier described by Yu et al. +(2019), which was used for QLP planet candidate triage +from Sectors 6 to 33. +The following summarizes the +major differences in development and implementation +between classifiers: +1. Astronet-Triage was trained and tested on QLP +light curves from only TESS Sectors 1 – 5, while +Astronet-Triage-v2 was trained and tested on +Sectors 1 – 39. +2. Astronet-Triage was developed using 16,516 la- +beled TCEs (493 planet candidates, 2155 eclips- +ing binaries, and 13,868 noise/systematic signals), +which is roughly two-thirds the size of our labeled +set (24,926 TCEs). +3. Astronet-Triage used labels that were assigned +by only a single vetter who visually inspected all +TCEs, while 3 – 5 vetters independently inspected +each of the TCEs for Astronet-Triage-v2, and +group discussions resolved labeling disagreements. +As a result, our labels should be more reliable. +4. Astronet-Triage only labels signals as either +“planet” (for all eclipsing signals, including plan- +ets and eclipsing binaries) and “non-planet” (for +other false positives, including pulsating variables, +noise and systematics). The five-label model used +by Astronet-Triage-v2 (E, S, B, J, N) is more +flexible and informative. +5. Astronet-Triage takes the light curves already +detrended by QLP, and bins the data into two +views: a “global” view, showing the full light curve +phase diagram, and a “local” view, showing a +close-up of the transit in the phase diagram. As +described in Section 3, Astronet-Triage-v2 cre- +ates three sets of detrended light curves from the +raw QLP light curve, and generates seven views for +each one. In total, Astronet-Triage-v2 uses 21 +unique views to inform its classification compared +to the two used by Astronet-Triage. +These key differences result in improvements to our +ability to classify TESS signals in FFI data, as shown +Sections 5.2 and 5.3. +To our knowledge Yu et al. (2019) is the only truly +comparable work to ours, in that their source dataset +was the TESS Full Frame Images and not the pre- +selected targets processed by the SPOC pipeline, and, +their goal was to perform triage by identifying all eclips- +ing signals, rather than separating planet candidates +from eclipsing binaries and other false positives. Some +other groups have trained and tested neural networks on +TESS data from two-minute postage stamps processed + +0.8 +0.6 +Recall +0.4 +0.2 +This work +Yu19 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +Cutoff +0.9 +0.8 +Recall +0.7 +0.6 +This work on Ps +This work on PCs +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +CutoffImproved TESS Triage with Neural Networks +17 +Figure 14. Example of borderline pattern. The true label for this example is “E”, but the folded light curve appears very +similar to a “B”. +by the SPOC pipeline (Osborn et al. 2020; Rao et al. +2021; Valizadegan et al. 2021; Fiscale et al. 2021; Ofman +et al. 2022), and were successful in identifying planet +candidates. However, in general, these groups find that +the neural network performance is worse on TESS data +than a similar network on Kepler data, likely due to +TESS’s higher a priori TCE false positive fraction (due +to the larger TESS pixels resulting in more blending) +and shorter observational baseline. The false positive +rate for FFI targets is likely even higher because a) the +targets observed by QLP tend to be fainter than targets +observed in postage stamps and blending is more pro- +nounced, and b) the targets observed in the FFIs are +more often large, luminous stars like red giants, which +are difficult to find planets around, and are photomet- +rically noisy. +Therefore, TCEs detected by the QLP +likely have an even higher a priori false positive prob- +ability than TCEs detected by TESS in postage stamp +data. +6.4. Future work +6.4.1. Applications to exoplanet population statistics +Planet catalogs can be used to characterize exoplanet +population statistics through the estimation of occur- +rence rates. One of the key components of occurrence +rate methodologies is a characterization of catalog com- +pleteness, reflecting how many planets from the under- +lying population were missed. A second key component +is an understanding of catalog reliability (Bryson et al. +2020), reflecting how much of the catalog is polluted +with false positives. For these reasons, occurrence rate +studies require the ability to produce planet catalogs in a +fully automated, uniform, and reproducible way, rather +than relying on biased manual identification of planet +candidates. +NASA’s Kepler mission has dominated the past +decade of demographics work in large part thanks to +the fully automated Kepler Robovetter pipeline, which +enabled careful characterization of both completeness +and reliability across wide areas of exoplanet param- +eter space (Thompson et al. 2018; Christiansen et al. +2020). +However, there is not yet a fully automated +TESS planet vetting pipeline. Most previous work has +also focused on 2-minute cadence observations rather +than FFIs, which will be less suitable for demograph- +ics due to selection biases in 2-minute cadence target +lists. Astronet-Triage-v2 is an important step toward +uniformly vetted FFI planet catalogs, and it naturally +allows for a flexibility in balances between completeness +and reliability through the adjustment of prediction + +18 +Tey/Moldovan et al. +Figure 15. Example of borderline pattern. The very low signal to noise ratio of the transit signal is easily mistaken for a “J”. +thresholds for passing candidates. While the classifier is +not yet able to distinguish eclipsing binary false positives +from planets (labeling all such signals as “E”’s), it can be +used as a first round of automated and characterizable +triage. Future improvements to Astronet-Triage-v2 +(Section 6.4.2) are expected to improve the precision +and recall, and therefore the completeness and reliabil- +ity, of any resulting planet catalog. We have plans to +extend Astronet-Triage-v2 to be capable of all steps +of the vetting process in the future. +6.4.2. Further improvements to the neural network +In future work, we suggest a number of additions to +further improve the performance of our classifier. +Over the past few decades, the performance of deep +learning classifiers has seen unprecedented success. A +large part of this success has been attributed to the +increasing size of training datasets. In this work, the +number of training examples is relatively low, particu- +larly for the S-labelled class, with a large class-imbalance +(see Figure 4). +A common technique for increasing training datasets, +without obtaining new labelled data, is data augmen- +tation. This typically involve applying slight transfor- +mations to the training data to produce new data that +mimics real observation. Using a combination of a few +data augmentation techniques can magnify a training set +by several fold and helps reduce over-fitting. In future +work, we suggest applying data augmentation methods +such as randomly reversing or clipping light curves in +time and applying random Gaussian noise to the light +curves or scalar features. We note that these methods +were applied in Ansdell et al. (2018), where they showed +that the main benefit to data augmentation on exo- +planet classification was alleviating model over-fitting, +with only a small improvement to model performance. +More complex augmentation methods such as fitting a +model (e.g. Gaussian Process, see Boone 2019) to the mi- +nority class light curves and generating more synthetic +data may also help to improve the limited data for some +classes. +Since Astronet-Triage-v2 is used in production for +QLP’s monthly planet search, another way to increase +our training dataset is to use the existing human vetting +work that goes into producing the TOI catalog (Guer- +rero et al. 2021). +As this human vetting is the final +step in the TOI release process, there is a high level of +quality control in the labels and the signals being vet- +ted are often the most difficult to classify, making them +important examples for the model to learn. +7. CONCLUSION + +Improved TESS Triage with Neural Networks +19 +Figure 16. Example of incorrect BLS estimation. Although the phase and period are close, the transit duration is too small, +causing the transit to be clipped by the detrending process. +We have presented Astronet-Triage-v2, a convolu- +tional neural network designed to distinguish astrophys- +ical eclipsing candidates from other phenomena such as +stellar variability and instrumental systematics in TESS +FFI light curves. The network assigns input signals one +of five labels, namely “E” for eclipsing signals, “S” for +single transits or incorrect periods, “B” for contact bina- +ries, “J” for signals due to noise or systematics, and “N” +for inconclusive cases. We trained Astronet-Triage-v2 +using ∼ 25000 signals, which were detected by QLP from +TESS Sectors 1 – 39 and human-labeled through man- +ual review and group discussion. We make this training +set available to the community. +Astronet-Triage-v2 is the next in a line of Astronet +architectures, which were first used for Kepler (Shallue +et al. 2019) and later extended to K2 (Astronet-K2; +Dattilo et al. 2019) and TESS (Astronet-Triage; Yu +et al. 2019). This iteration features significant improve- +ments over Astronet-Triage, including a larger and +more robust training set, an expanded list of possi- +ble classifications, and more than ten times the num- +ber of unique views used to analyze each signal. +As +a result, we found Astronet-Triage-v2 is more suc- +cessful at correctly labeling known TOIs across al- +most all cutoff values, with 86% recall at a cutoff of +0.215 compared to 82% recall by Astronet-Triage. +When tested on a set of new signals from Sector 33, +Astronet-Triage-v2 provides better recall of E and S +labels than Astronet-Triage for similar (or better) lev- +els of precision, especially for fainter targets. Starting +in Sector 34, Astronet-Triage-v2 officially replaced +Astronet-Triage within QLP. +As both the TESS observing baseline and number of +observed stars continue to increase, automated TESS +planet vetting tools will become more important. This +is especially true of tools tuned for planet searches us- +ing FFIs, of which Astronet-Triage-v2 is one of the +few currently available. While Astronet-Triage-v2 is +not yet capable of distinguishing between eclipsing bina- +ries and transiting planets, it serves as an effective first +round of automated and characterizable triage. We plan +to continue to improve and extend the network into a +fully automated vetting tool in the future. +ACKNOWLEDGEMENTS +This paper includes data collected by the TESS mis- +sion. Funding for the TESS mission is provided by the +NASA’s Science Mission Directorate. + +HE20 +Tey/Moldovan et al. +Figure 17. Example of incorrect BLS estimation. The detected period is close, but when the light curve contains a large +number of folds, the error compounds and leads to a blurring of the transit view. This is due to QLP searching the light curve +with an undersampled BLS frequency grid (necessary due to the computational time needed to run BLS on a large number of +targets each sector), as discussed in Kunimoto et al. (2022, in prep.). +This work has made use of data from the Euro- +pean Space Agency (ESA) mission Gaia (https://www. +cosmos.esa.int/gaia), processed by the Gaia Data Pro- +cessing and Analysis Consortium (DPAC, https://www. +cosmos.esa.int/web/gaia/dpac/consortium). +Funding +for the DPAC has been provided by national institu- +tions, in particular the institutions participating in the +Gaia Multilateral Agreement. +This work was supported by an LSSTC Catalyst Fel- +lowship awarded by LSST Corporation to T.D. with +funding from the John Templeton Foundation grant ID +#62192. +The Astronet-Triage-v2 model was trained and +tuned on Google Compute Engine. +Facility: TESS, Gaia +Software: numpy (Oliphant 2006), matplotlib (Hunter +2007), pandas (pandas development team 2020; Wes +McKinney 2010), +statsmodels (Seabold & Perktold +2010), pydl, astropy (Astropy Collaboration et al. 2013; +Price-Whelan et al. 2018), TensorFlow (Abadi et al. +2016), Vizier (Golovin et al. 2017b), Jupyter (Kluyver +et al. 2016) +REFERENCES +Abadi, M., Agarwal, A., Barham, P., et al. 2016, +TensorFlow: Large-Scale Machine Learning on +Heterogeneous Distributed Systems, +doi:10.48550/ARXIV.1603.04467 +Ansdell, M., Ioannou, Y., Osborn, H. 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This table contains information about the signal detected from BLS (epoch, period, duration, +depth), information about the host star from TIC 8.2 (TIC ID, M∗, R∗, TMag). Est R∗ is described in Section 3.2, +and year describes the year the TCE was detected. MinT and MaxT specify the time range used from the light curve +for both detection and input to Astronet-Triage-v2, and Split specifies which dataset (train, val, test) the signal +was in. L1-L8 are labels assigned by individuals and Consensus Label is the label agreed upon by the group. + +24 +Tey/Moldovan et al. +TIC ID +Period +Epoch +Duration +Depth +TMag +M∗ +R∗ +Est R∗ +Year +MinT +MaxT +Split +Consensus +L1 +L2 +L3 +(days) +(BTJD) +(days) +(ppm) +M⊙ +R⊙ +R⊙ +BTJD +BTJD +Label +290603338 +13.6725 +1629.9162 +0.212 +380 +8.51 +15.62 +16.00 +1 +1624.9693 +1682.3443 +train +J +N +J +32092337 +1.3228 +1326.3441 +0.219 +110 +9.82 +1.41 +2.43 +2.43 +1 +1325.3226 +1652.8639 +train +J +J +J +J +278544052 +11.2664 +1600.6568 +0.206 +2350 +11.51 +1.36 +1.99 +2.06 +1 +1596.7819 +1682.3445 +train +J +J +J +J +380752037 +0.5639 +1657.5494 +0.069 +3970 +11.29 +1.96 +2.28 +2.18 +1 +1653.9262 +1682.3430 +train +J +J +J +J +259863095 +14.3676 +1338.8209 +0.315 +2660 +9.56 +0.82 +2.71 +2.50 +1 +1325.3233 +1682.3429 +train +J +J +J +J +272085506 +42.3842 +1328.3580 +0.367 +890 +11.39 +14.48 +15.56 +1 +1325.3218 +1652.8658 +train +J +J +J +J +306897664 +1.2464 +1325.8091 +0.175 +140 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+2.15 +1 +1381.7181 +1682.3452 +train +E +E +E +E +340060373 +12.9080 +1393.7434 +0.157 +1790 +11.07 +1.19 +1.20 +1.22 +1 +1381.7179 +1682.3454 +val +J +J +S +J +340929171 +1.0873 +1654.4122 +0.174 +2560 +11.25 +1.81 +1 +1653.9260 +1682.3428 +train +B +J +J +349647610 +36.1689 +1359.0938 +0.587 +300 +4.86 +41.18 +42.82 +1 +1325.3211 +1682.3450 +train +J +J +J +J +340065079 +16.5423 +1396.2365 +0.155 +570 +9.60 +10.50 +10.96 +1 +1381.7180 +1682.3453 +train +J +N +J +119088593 +0.3616 +1654.1666 +0.157 +18550 +10.82 +1.72 +1.85 +1.94 +1 +1653.9273 +1682.3439 +train +B +B +E +B +143769346 +0.8667 +1655.7849 +0.248 +3720 +8.99 +1.65 +1.65 +1.66 +1 +1653.9264 +1682.3432 +train +B +B +B +B +320004264 +1.0463 +1654.3056 +0.172 +570 +8.19 +1.34 +1.25 +1.25 +1 +1653.9262 +1682.3430 +train +J +J +J +J +63343395 +0.3958 +1657.3129 +0.039 +199070 +10.03 +1.35 +1.48 +1.54 +1 +1653.9275 +1682.3441 +train +B +E +B +B +261543672 +14.5671 +1333.9494 +0.359 +630 +10.56 +1.03 +1.75 +1.76 +1 +1325.3244 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Centre for Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' West Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Toowoomba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' QLD 4350 Australia 4Department of Astrophysical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 4 Ivy Lane,' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 77 Massachusetts Ave,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' USA 8Department of Earth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Atmospheric and Planetary Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 77 Massachusetts Ave,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' USA 9Department of Aeronautics and Astronautics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 77 Massachusetts Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' MA 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' USA ABSTRACT The TESS mission produces a large amount of time series data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' only a small fraction of which contain detectable exoplanetary transit signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Deep learning techniques such as neural networks have proved effective at differentiating promising astrophysical eclipsing candidates from other phenomena such as stellar variability and systematic instrumental effects in an efficient, unbiased and sustainable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This paper presents a high quality dataset containing light curves from the Primary Mission and 1st Extended Mission full frame images and periodic signals detected via Box Least Squares (Kov´acs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Hartman 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The dataset was curated using a thorough manual review process then used to train a neural network called Astronet-Triage-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' On our test set, for transiting/eclipsing events we achieve a 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='6% recall (true positives over all data with positive labels) at a precision of 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='7% (true positives over all predicted positives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Since 90% of our training data is from the Primary Mission, we also test our ability to generalize on held-out 1st Extended Mission data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Here, we find an area under the precision-recall curve of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='965, a 4% improvement over Astronet-Triage (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' On the TESS Object of Interest (TOI) Catalog through April 2022, a shortlist of planets and planet candidates, Astronet-Triage-v2 is able to recover 3577 out of 4140 TOIs, while Astronet-Triage only recovers 3349 targets at an equal level of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In other words, upgrading to Astronet-Triage-v2 helps save at least 200 planet candidates from being lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The new model is currently used for planet candidate triage in the Quick-Look Pipeline (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2020a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Kunimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Keywords: Neural networks, Transit photometry, Exoplanet detection methods, Exoplanet Catalogs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' INTRODUCTION For three decades, human judgement has played a crit- ical role in the exoplanet revolution that has yielded the discovery of more than 5000 planets outside of the Solar System1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Exoplanets are typically much cooler, smaller, and fainter than their host stars, so detecting them usu- ∗ These authors contributed equally to the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 1 NASA Exoplanet Archive: exoplanetarchive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='edu ally requires extremely precise observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' At the level of sensitivity required to detect exoplanets, numerous other systematic effects can be present in data that can mimic planetary signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Separating out these “false positive” signals from true exoplanets has been a major challenge (Jacob 1855;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' van de Kamp 1963;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Bailes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 1991) since before the discovery of the first exoplanets in the 1980s and 1990s (Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Latham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Wolszczan & Frail 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Mayor & Queloz 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Historically, classifying possible planet signals arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='01371v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='EP] 3 Jan 2023 2 Tey/Moldovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' as either false positives or viable planet candidates has most often been carried out by a human inspecting and making a judgement on each signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Humans are quite well suited for this type of work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' we can learn how to distinguish planet candidates and false positives with high accuracy, even after looking at a relatively small number of examples, and often without the benefit of a priori knowledge of the “ground truth” of any signal’s true classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' However, relying on human judgement to separate vi- able planet candidates from false positives has two main disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' First, humans are slow, both in terms of training time and actual classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' It often takes months or years of practice for a human to become adept at classifying planets and false positives, and once fully trained, it may take an experienced human several min- utes to review all of the information needed to make one classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' At these speeds, even classifying a mod- est number of possible planet signals (∼ 102 − 103) may take days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Given the rapid increase in the volume of astronomical data available for analysis, it will soon be impractical to rely on human classifications to identify viable planet candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Second, humans are incon- sistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Differences in external factors (mood, fatigue, hunger, etc) may cause a human to judge the same sig- nal differently on two different occasions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This makes characterizing and quantifying the biases introduced by human classification challenging and inexact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' An al- ternative system capable of quickly, accurately, and re- peatably identifying planet candidates would be highly attractive to planet hunters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In this paper, we focus on improving a deep neural net- work classifier used to identify viable planet candidates in data from the Transiting Exoplanet Survey Satellite (TESS) mission (Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' TESS identifies exoplanets by searching for “transits,” or slight peri- odic dimmings of the apparent brightness of a star as its planet passes between the star and our vantage point in the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Transit surveys like TESS produce copious numbers (≳ 106 so far) of false positive signals that must be separated from viable planet candidates to enable discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Machine learning has become a popular tool for iden- tifying promising planet candidates from transiting exo- planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Some work has focused on using machine learn- ing to perform the actual planet detection (Pearson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Zucker & Giryes 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021), but more often, efforts have focused on using machine learning to classify the large number of possible transit-like signals returned by existing planet detection pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' A push early in the Kepler mission (Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Borucki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2010) led to the development of two automated systems: a decision tree called the Robovetter (Coughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2018) and a random forest classifier called the Autovetter (McCauliff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In that initial work, the Robovetter proved more robust and easily extensible to new regimes and datasets, and therefore was used in the production of fully automated planet candidate catalogs from the Kepler mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' More recently, Shallue & Vanderburg (2018) intro- duced a convolutional neural network for vetting planet candidates from the Kepler mission called Astronet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Since then, Astronet and other similar architectures have been demonstrated on other datasets like K2 (Dat- tilo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2019), TESS (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Osborn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2020), WASP (Schanche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2019), and NGTS (Arm- strong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Chaushev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' New tweaks to the methdology including new input information and tweaks to the data representation (Ansdell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Jara-Maldonado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Valizadegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021) have yielded improvements in classification per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Our work is largely based upon the convolutional neu- ral network originally introduced by Shallue & Van- derburg (2018) and adapted to TESS by Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' (2019), known as Astronet-Triage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Starting in 2019, Astronet-Triage had been used in the TESS Quick- Look Pipeline (Guerrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021) to triage planet candidates and remove clear false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' However, our internal tests revealed that this step resulted in the loss of a fairly large number of viable planet candidates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=', “false negatives”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This paper describes our work to improve the performance of Astronet-Triage by in- troducing Astronet-Triage-v2 to reduce the number of lost planet candidates while throwing out a higher number of false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Our paper is organized as follows: In Section 2, we de- scribe the input transit signals and corresponding light curves which were used for training and testing our clas- sifier, and the labels assigned to each signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In Section 3, we describe how we processed the data before it is input to our neural network classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In Section 4, we describe the architecture of the neural network and the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We quantify and present the results of our classifier in Section 5, and we discuss the implica- tions of these results in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Finally, we conclude in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' DATA For training and testing our model, we use approxi- mately 25000 human vetted transit signals detected by Improved TESS Triage with Neural Networks 3 the Quick-Look Pipeline (QLP, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2020a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Kunimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021) across Sectors 1 – 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' TCEs from TESS FFIs During its Prime Mission (2018 July 25 – 2020 July 04), TESS collected full-frame images (FFIs) every 30 minutes for 2 years covering 70% of the entire sky (Guer- rero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The FFI cadence was updated to 10 minutes for the 1st Extended Mission (2020 July 04 – 2022 September 01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' QLP produces light curves from these images for all observed targets in the TESS In- put Catalog (TIC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Stassun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2018, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Paegert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021) with TESS-band magnitude (T) brighter than 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Flux time series (raw light curves) from five different sized circular apertures are extracted for each star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' These raw light curves are then filtered to remove low-frequency variability originating from stellar activ- ity or instrument noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Primarily, this is done by divid- ing the light curve from each separate orbit by a basis spline (following Vanderburg & Johnson 2014) fit using a break-point spacing between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3 days and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='5 days, selected as described by Shallue & Vanderburg (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Finally, these detrended light curves are merged with previous TESS sectors using a shared median value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' At this point, an optimal aperture is selected for target star based on its TESS magnitude – fainter stars get- ting smaller aperture sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' All subsequent processes use these multi-sector “best”-aperture detrended light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' QLP searches these light curves for transit signals us- ing the Box Least Squares (BLS) algorithm (Kov´acs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Hartman 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Because BLS spectra fea- ture a rising trend towards lower frequencies (longer pe- riods), QLP subtracts the low frequency baseline be- fore selecting the highest peak as the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' For each detected signal, the BLS implementation com- putes characteristic parameters (orbital period, tran- sit center, transit depth, the full transit duration) by performing a least square trapezoid fit for the transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' These parameters are used later in the input process for Astronet-Triage-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Transit signals with signal-to-pink-noise > 9 and BLS peak significance > 5 (for stars with T < 12 mag) or > 9 (for stars with T > 12 mag) are labelled threshold- crossing events (TCEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' These filters give slightly dif- ferent perspectives on transit significance: (1) signal-to- pink-noise compares the transit depth to pink noise in the light curve (Pont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2006), while (2) BLS peak significance compares the BLS spectrum’s peak height 2 QLP data can be found at doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='17909/t9-r086-e880 to its noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In combination, these checks help filter out events that are clearly not transit-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In addition, we filter out instances where the planet would orbit “inside the star.” For each signal we com- pute the expected semi-major axis to stellar radius ratio assuming a Keplerian orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3 If the ratio < 1, the signal is labeled as inside the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Typically, these signals sig- nify stellar variability or blended signals from a smaller nearby star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Assembling a set of signals to label Even with filters described in the previous subsection, manually labeling every TCE would take an enormous amount of time, so we select a subset of TCEs for train- ing / testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Over time, we gradually accumulated three batches of labeled TCEs from the first two years of TESS Primary Mission (observed with 30 min cadence) and the first year of the TESS 1st Extended Mission (observed with 10 min cadence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The year 1 (Y1) TESS observations for the southern hemisphere went through significant changes in noise property due to the spacecraft pointing strategy change in Sector 4,4 and the subsequent tweaking of the momen- tum dump frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We selected 8992 TCEs detected in Sector 13 (the last sector of Y1) for the labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This was not an intentional choice, but after spending hun- dreds of person-hours labeling these TCEs, we opted to make use of them regardless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Fortunately, despite the fact that our Y1 TCEs came only from Sector 13, the observations that led to these detections still included a diversity of spacecraft pointing control strategies and data artifacts (for example detector warmups following instrument anomaly events5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In particular, stars ob- served in Sector 13 have been observed by TESS in Y1 between one to thirteen sectors and cover a variety of prior sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' For the year 2 (Y2) TESS observations in the northern hemisphere, the data has more uniform characteristics including a consistent momentum dump frequency of every 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4 days starting in Sector 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We sorted TCEs by their target’s TESS magnitude, and then took the 13372 brightest TCEs detected from Sectors 14–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 3 When computing the semi-major axis we use two times the detected BLS period in case the detected period is half the true period, which often happens for eclipsing binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' If the star has an estimate for its mass in the TIC, we use that value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' if not, we assume a mass of 1 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We also assume a circular orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 4 https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='edu/missions/tess/doc/tess drn/ tess sector 04 drn05 v04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='pdf 5 https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='edu/missions/tess/doc/tess drn/ tess sector 08 drn10 v02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='pdf 6 https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='edu/missions/tess/doc/tess drn/ tess sector 14 drn19 v02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='pdf 4 Tey/Moldovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In year 3 (Y3), TESS returned to observe the south- ern hemisphere, with faster cadence and a further im- proved momentum dump strategy (only once each or- bit)7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We added an additional 2588 TCEs from Sectors 27-39, which increased the sky coverage and brightness range for our southern hemisphere labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We note that TCEs around stars only observed in one of the CCDs in Sector 13 Camera 1, and Camera 1 and 2 for Sector 24 and 25 are not included in our sample due to temporary unavailability of the data at the time of vetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Altogether, these TCEs create a broad sample of transit-like events detected in the first three years of TESS observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The final TCE distribution across the sky is shown in Figure 1, and across TESS magni- tude (Tmag) in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Due to the different selection criteria of the TCEs from three different years, they have somewhat different data characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' As discussed in Section §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2, these differences do not significantly im- pact our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Labels and their definitions For each TCE we assigned one of the following five labels: E denotes a periodic eclipsing signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This includes both planetary transits and non-contact eclipsing binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In the triage process, we do not take into account information that would distinguish an eclipsing signal from background stars from an eclipsing signal on the target star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Both cases would be labeled as E if they satisfy all the other criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' S denotes events containing only a single transit or events where an incorrect period or period alias is assessed to be reported from BLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' B denotes contact eclipsing binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' They are distinguishable from non-contact binaries through their continuous ingress/egress slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' J denotes junk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This includes other astrophysical phenomena like stellar variability as well as instru- mental phenomena like scattered light (due to the Earth or the Moon approaching the field of view and reflecting light into the camera) or artifacts introduced at the times of spacecraft momentum dumps (when the spacecraft’s reaction wheels cor- rect for the spacecraft’s speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 7 https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='edu/missions/tess/doc/tess drn/ tess sector 27 drn38 v02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='pdf N denotes not sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' No conclusive label decision could be made for these TCEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Often an N label was given when a weak signal bordered on being an E or J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' These labels are not necessarily mutually exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We detail the rules we use in labeling when resolving marginal/ambiguous cases: E vs S: If there is ambiguity in the period (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' both the reported period and the double period are con- sistent with the data) or the period is only slightly off, we default to an E label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Only if the pe- riod is explicitly incorrect (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='g there are flat light curve segments during expected transits, or there are multiple regular transits outside of expected transit times) do we choose an S label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' If there is only one regular transit outside the expected transit time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' it might represent a secondary eclipse, we use an E label, and if the reported pe- riod potentially includes the secondary eclipse, we also use an E label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' B vs S: If we have a contact binary with the incor- rect period, we default to a B label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We choose these labels first because they mirror astro- physical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This means the labeled TCEs pro- vide good targets for follow-up (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Es will be good can- didates for exoplanet and binary star detection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Sec- ond, we expect similarities in light curve morphology within a label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This should help our model learn labels more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' For the purposes of finding exoplanets, we are particu- larly interested in high precision and recall metrics for E labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' S and N labels may also be important candidates for further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Labeling process All TCEs were manually assigned labels based on human-visual representations (see Figure 3) similar to the model input representations described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' On a weekly basis, batches of targets were independently vetted by 3 – 7 of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' At the end of the week, targets with conflicting labels where at least one human chose an E or S were discussed in order to reach a con- sensus on the target’s final label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' If a target had only B, J, or N votes, we assigned weights to each label based on the number of votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Altogether, this process took over 2 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We expect the multiplicity of vetters to reduce the number of label errors, giving us a very high-quality dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Table A contains examples of signal data along with individually-assigned labels and their consensus dispo- Improved TESS Triage with Neural Networks 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Sky map showing the locations of the 24926 TCEs presented here (black starred data points) compared to the coverage of each TESS Prime Mission sector (colored data points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The black and red labels are the Prime Mission sector numbers in the southern and northern ecliptic hemispheres, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Note that we also include 2588 TCEs from the 1st Extended Mission, for which sector coverage is not shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The under- and over-densities of TCEs are due to the selection criteria as described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Distribution of Tmag across our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Both Y1 and Y2 portions of the dataset focused on the brightest TCEs, while Y3 added TCEs more uniformly across magni- tudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' More details on TCE selection can be found in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' sitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The full table (and accompanying light curve data) can be found online in Tey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Following common practice in ML, we randomly sep- arate the dataset into a training, validation, and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The model is initially fit on the training set, a set of examples used to fit the parameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Next, the validation set provides a measure of predic- tive accuracy and model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The validation set consists of examples that the model has not seen in the training set and allows for optimization of the architecture and hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Lastly, after the model architecture and hyperparameters are finalized, the test set is used as one last objective test of the model accuracy and fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Training set (19919 targets): used for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' (15414 J + 2102 E + 1681 B + 224 S + 498 N) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Validation set (2491 targets): used to calculate precision, recall, detection threshold for binary classification, and model debugging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' (1945 J + 261 E + 198 B + 17 S + 70 N) 80 60 40 [deg] 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='18 4 20 F16 15 8 21＇20 24 173 14 26 25 922 E2 0 1023 Dec 1 11 13 12 20 40 60 80 20 15 10 5 0 RA [hr]5000 - 三 Y1 Y2 4000- Y3 Counts 3000 - 2000 - 1000 - 0: 0 5 10 Tmag6 Tey/Moldovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' EBImproved TESS Triage with Neural Networks 7 S H TTTIT H TS8 Tey/Moldovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Six example visual representations used for human labeling with labels in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The different figures within each representation were made to mirror the information described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Each image was individually labeled by at least 3 individual vetters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Conflicting labels were discussed and resolved each week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' :ww HHH HHH H HH HHFN 亚王 【 亚 TII HITImproved TESS Triage with Neural Networks 9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Distribution of labels across our dataset (see Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3 for descriptions of each type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' As described in Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4, some TCEs were assigned fractional B and J labels so these counts have been rounded to the nearest integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Scatterplot of transit depth vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' orbital period for our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' TCEs with E labels are shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Red lines mark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='7 and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4, the orbital period and twice the orbital period of TESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Test set (2516 targets): hold-out set used for fi- nal evaluation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' this set was never used for training or debugging, or any other evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' (1970 J + 250 E + 200 B + 34 S + 62 N) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Distribution of the labels Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Scatterplot of planet radii vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' orbital period for our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' TCEs with E labels are shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Red lines mark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='7 and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4, the orbital period and twice the orbital period of TESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Scatterplot of transit duration vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' orbital period for our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' TCEs with E labels are shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Red lines mark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='7 and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4, the orbital period and twice the orbital period of TESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Figure 4 shows the distribution of labels in our train- ing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Out of the total 24926 labels, the majority are J labels (19329).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The amount of signals identified as eclipsing objects (E, 2613) is comparable to that iden- tified by contact binaries (B, 2079).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Counts 1000 0 107 106 Transit Depth [ppm] 105 104 103 102 101 10-1 100 101 102 0 5000 Period [days] CountsCounts 1000 0 103 102 101 100 10-1 100 101 102 0 2500 Period [days] CountsCounts 1000 0 102 Transit Duration [hrs] 101 100 10- 10-1 100 101 102 0 5000 Period [days] Counts20000 19329 15000 : Counts 10000 5000 - 2613 2079 630 275 0 E B N s Disposition10 Tey/Moldovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We examine the distribution of the fundamental tran- sit parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=', orbital period, transit depth, esti- mated planet radius, and transit duration) of the la- bels in Figure 5, 6, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Specifically, we compare the parameter spaces resided by the E labels to the other labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The comparison reveals the following charac- teristics: (1) a majority number of the TCEs with pe- riod smaller than ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='5 days are not caused by eclipses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' (2) a majority of the shallow events with period longer than 10 days are not caused by eclipses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' (3) there is clear pile-up of TCEs at the TESS orbital period and its alias, which are not caused by eclipses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' (4) a major- ity of TCEs with extremely short/long transit duration are not caused by eclipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' MODEL INPUT REPRESENTATIONS For each TCE, we pass the raw flux time series leading to the detection and all the relevant information describ- ing the detected periodic signal and target star to the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Time series data We preprocess the raw flux time series into dif- ferent input representations before passing them to Astronet-Triage-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We use the same basis spline techniques used in QLP, however, the transit signals are masked out based on the BLS-detected period, epoch and duration before the optimal spline is com- puted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This approach will often prevent over-fitting of the transit signals during the detrending process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' To account for different time scales of the stellar variabil- ity, we adopt multiple detrending settings to provide Astronet-Triage-v2 a more complete view of the light curve noise characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Unlike in QLP, which only uses one set of splines with spacing between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='5 days to create the final detrended light curves, we use three different settings (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0, and a value which minimizes the Bayesian Information Criterion, Schwarz 1978) to create three different sets of detrended light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The light curves detrended with larger spacing are also less likely to over-fit the transit signals with long transit duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' For each detrended light curve we generate seven dif- ferent plots or views (see Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Each view is binned using a robust binning technique to de-weight outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' During this binning, we also account for the change in exposure time between the Primary and 1st Extended Mission by weighing points according to their exposure time in a given bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' After this, we normalize the binned data so that the minimum value is -1 and the median value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The complete list of views can be found in the source code 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' A detailed description of each view type is below: Global View: The global view uses the full light curve folded on the reported period with 201 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In addition to the median values, the view also includes the standard deviations for each bin, a mask indicating whether the bin was empty, and a mask indicating whether the bin falls inside the detected transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Local View: The local view uses points within two transit durations of the transit center (for a full timespan of four transit durations), again folded on the reported period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The local view uses 61 bins, and includes standard deviation and mask values like the global view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In addition, we also record the scale factor used in normalization, as a scalar feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Secondary View: The secondary view is similar to the local view, but is centered around the most significant secondary transit, determined by per- forming a grid search9 on the out-of-transit por- tion of the phase folded view, for a duration equal to the primary transit duration, and selecting the region with the highest signal/noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This view is accompanied by two scalar features: the normalization scale factor, and the phase of the secondary transit’s center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Local Half-Period View: Similar to the local view, but folded at half the detected period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This view only contains the standard deviation value, since the median value can appear very noisy when fold- ing a transit over a non-transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Global Double Period View: Similar to the global view, but folded at twice the period of the global view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Sample Global Segments: This view contains the entire period (similar to the global view), but showing up to 7 of the folds that contain the most points (ties are broken at random).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Each fold is ac- companied by a mask indicating whether the bin contains any points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' If the light curve contains 8 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='com/mdanatg/Astronet-Triage/blob/ e4ec517b175b2a3dfb74cf6c6e3f5273dd8749c7/astronet/ astro cnn model/configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='py#L2254 9 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='com/mdanatg/Astronet-Triage/blob/ e4ec517b175b2a3dfb74cf6c6e3f5273dd8749c7/light curve util/ find secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='py#L62 Improved TESS Triage with Neural Networks 11 fewer transits, the extra views remain empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Each fold is independently binned with 201 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Sample Local Segments: Similar to the sample global segments, this view contains the transit cen- ter of up to 4 of the folds that contain the most points (ties are broken at random), for a total of 8 folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Each fold is independently binned with 61 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Scalar data We also use scalar values that describe characteristics of the transit, host star and the light curve itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Transit features include period in days (P), transit duration in days (Tdur), transit depth (δ), and the number of full periods observed in the flux-time series (nfolds), while host star features include TESS magnitude (Tmag), mass in M⊙, and radius in R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The host star features are directly extracted from the TESS Input Catalog v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2 (Paegert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' For TCEs without stellar radii in the catalog, we per- form a rough estimate using a Bayesian estimate of the distance (Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021), apparent magnitude (either Gaia G, Bp, and Rp, or Gaia G and 2MASS K if Bp and Rp are unavailable), and color/temperature and color/bolometric corrections from MIST models (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In brief, we estimate the temperature and bolometric correction from either the target’s Bp-Rp or G-K colors, use the bolometric correction to estimate the target’s apparent bolometric magnitude, use the es- timated distance to the target to convert to an absolute magnitude, convert to bolometric luminosity, and solve for the stellar radius from the temperature and lumi- nosity via the Stefan Boltzmann Law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In our testing, we were able to determine radii within about 10% of the TIC values when present, and provided radius esti- mates for ∼ 2400 from the ∼ 2800 TCEs missing stellar radii in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Light curve features include the total number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Each scalar value is normalized to be zero mean and unit variance across the dataset, except for nfolds which is truncated to a maximum value of 100 and a log-scaled to fit between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In addition, we also include as scalar inputs the detected phase of the secondary eclipse, as well as the calculated scaling factor when normalizing the global, local and secondary views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' NEURAL NETWORK ARCHITECTURE Our model uses a convolutional neural network archi- tecture derived from Astronet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The high level architec- ture is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Each time series feature is grouped together with similar features and then passed through a separate convolutional tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' For example, the global view flux is grouped together with the standard deviation of the global view, so that they form a 2-channel, 1- dimensional image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The structure of a convolutional tower is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Each tower consists of con- volutional layers with Rectified Linear Unit (ReLU) ac- tivation, alternating with pooling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The pooling layers aggregate neighboring pixels, in effect increasing the field of view of the subsequent convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The output of each convolutional tower is flattened into a vector shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The flattened outputs from all tow- ers are concatenated together with the auxiliary inputs to form the input into the next section of the network, the fully-connected tower, whose structure is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The fully-connected tower is composed of several fully-connected neural network layers, alternat- ing with dropout layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The dropout layers randomly set inputs to zero, and serve a role of regularization, to mitigate over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The dropout layers are only ac- tive during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The final layer has five outputs, and uses a sigmoid activation function, so that its out- put is in the interval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='.1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Each of the five outputs corresponds to one of the five labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The various hyper-parameters of each network can be found in the configuration file included with the source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='10 The hyper-parameters are tuned using Vizier (Golovin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2022) by minimizing the loss on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Training We train the model using the Adam, a popular variant of stochastic gradient descent optimization (Kingma & Ba 2014), for 20,000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The complete set of training parameters can be found in the code 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' For the loss function we use binary cross-entropy loss12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Notably, this means that the model is not trained to choose between the five labels exclusively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Instead, it produces independent scores for each label, so a model is free to assign high scores for both “E” and “J” la- bels, for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This loss function enables us to assign weighted labels to uncertain examples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 50 percent 10 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='com/mdanatg/Astronet-Triage/blob/ e4ec517b175b2a3dfb74cf6c6e3f5273dd8749c7/astronet/ astro cnn model/configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='py 11 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='com/mdanatg/Astronet-Triage/blob/ e4ec517b175b2a3dfb74cf6c6e3f5273dd8749c7/astronet/ astro cnn model/configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='py#L2254 12 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='org/api docs/python/tf/keras/ losses/BinaryCrossentropy for the implementation and Good (1952) and Shallue & Vanderburg (2018) for more information 12 Tey/Moldovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Astronet-Triage-v2 neural network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Structure of a CNN tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Each convolution tower has 1 to 4 blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Each block has 1 to 4 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Structure of the fully-connected tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' “B”, 50 percent “J”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The weight is determined as fol- lows: if a target had a single label (as resulting from the group resolution, or if the vote was unanimous), the weight is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' if the target had multiple votes, the weight is the maximum number of votes for any label divided by the total number of votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This means targets for which a label didn’t receive a majority of votes are weighted less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We don’t apply data augmentation, although that is something we intend to do in future work (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Prediction and ensembling As a multi-class classifier, our model outputs a predic- tion score for each label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Predictions where the “E” label score exceeds a threshold chosen beforehand are consid- ered to predict the label “E”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Otherwise, the model is considered to predict the label with the highest predic- tion score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We then construct an ensemble of 10 models trained separately (hence with different initial weight values, and different shuffling of the input data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The compound prediction of the ensemble is constructed as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' If any of the models predicts “E”, then the ensem- ble prediction is also “E”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Otherwise, the ensemble prediction is the label predicted by a majority of models, with ties bro- ken at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Global Local Secondary Half-Period 2x Period Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Transits (1 x 200 x 6) (1 x 60 x 7) (1 x 60 x 7) (1 x 200 x 1) (1 × 200 x 1) (1 x 200 x 14) Aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Inputs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='period ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='duration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='depth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='T mag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='star mass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='star radius ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3 layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3 layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3 layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3 layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3 layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3 layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='#folds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='#points ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Fully-connected Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='5 layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='scoreConv1D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Conv block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Conv block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='MaxPool1D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Conv block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='FlattenConv +aux features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Fully-connected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Fully-connected block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Fully-connected block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Dropout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Fully-connected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Logits block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Sigmoid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Class scoresImproved TESS Triage with Neural Networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='Although the model predicts five different labels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' we are primarily interested in the “E” label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The other labels are mainly used at training, to encourage the net- work to learn natural representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We found that the extra labels greatly help understand a model’s pre- dictions, as well as validate whether the model does in- deed create correct internal representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' RESULTS Here we report the results of our ML activity predic- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' First we discuss the metrics we used to evaluate the performance and then we summarize how the differ- ent models performed on each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The two primary metrics we use to evaluate our per- formance are precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The precision, or re- liability, of a model on a labelled dataset is the number of true positives divided by the number of true posi- tives and false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Recall, or completeness, is the number of true positives divided by the number of true positives and false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' As we are interested in “E” labels as potential planet candidates, they gener- ally are used as the “positive” class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In this context, a high precision means our model outputs fewer false posi- tives, meanwhile a high recall means successful recovery of more planet candidates (fewer potential planets lost by Astronet-Triage-v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Since labels are determined by comparing output prediction scores against a chosen threshold, each specific threshold yields its own precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' When plotted over many different thresh- olds, we can form a precision-recall curve (see Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' By taking the area under the precision-recall curve (AUC-PR), also known as the average precision, we can characterize our model’s overall performance and com- pare against other models with the highest achievable value being a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Performance on validation and test sets On the validation dataset we obtained an AUC-PR value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The model achieves 100% recall at 41% precision, at a prediction threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' If we in- crease the threshold to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='215, we obtain 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='9% recall at 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='8% precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' On the test set, we obtained an AUC-PR value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The model achieves 100% recall at 15% preci- sion, at a prediction threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This suggests the test set contains more difficult examples (possibly incorrect ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' With the thresholds suggested by the validation set, we obtain 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='6% recall at 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='7% precision for the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0105 threshold, and respectively 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2% recall at 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='7% precision for the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='215 threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Generalizing to TESS 1st Extended Mission data We explore the adaptability of our network, and the generalization of training on non-uniform datasets in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In practice, models like Astronet-Triage-v2 are trained on previously observed sectors with a goal of classifying new observations taken by TESS in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Since noise characteristics and TESS observa- tion strategy can change sector-to-sector, it is important that our models generalize well to new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Nearly 90% of our total training dataset comes from the TESS Primary Mission, so we use QLP data from TESS 1st Extended Mission (Sector 33, observed during Year 3 from UT 2020 December 17 – UT 2021 January 13) to test how our model generalizes to unseen or out- of-distribution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Following the QLP convention, we ran a BLS search and Astronet-Triage-v2 on the full multi-sector light curves (including both Primary Mission and 1st Ex- tended Mission data) for each star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Of the discov- ered TCEs, we selected a random sample of 759 targets with Tmag < 11 from camera 1 and 590 targets with 11 < Tmag < 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='5 from camera 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Due to the TESS pointing strategy, we focus on these cameras because their light curves have roughly equal amounts of Pri- mary vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 1st Extended Mission observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The mag- nitude ranges also allow us to compare performance on stars in different brightness bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' One of our vetters (CH) independently labeled all 1349 TCEs before evaluation, among which, 255 TCEs were assigned an E label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' To better understand our ability to generalize, we apply the following models to the Sector 33 dataset: Astronet-Triage, the fully trained Astronet-Triage-v2, and three independent instances of the Astronet-Triage-v2 architecture trained on different subsets of our original TCE dataset (Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' These three separate training sets were formed by splitting our original training set on observation year, meaning roughly 40% went into training the Y1 model, 50% into the Y2 model, and 10% into the Y3 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The differences between these datasets are described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='1, but briefly: Both the Y1 and Y2 datasets feature brighter stars, but the Y1 dataset were only taken from Sector 13, so they cover a small region of the Southern ecliptic hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The Y2 dataset, on the other hand, were selected more uniformly and cover most of the Northern ecliptic hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Neither has much overlap in sky coverage with the evaluation set (the 1349 Sector 33 TCEs) – Y1 having little overlap and Y2 having none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Both datasets also have much shorter observation baselines than the evaluation set, and finally, due to the change in TESS momentum dump strategy, the Y1 dataset also differs from the evaluation 14 Tey/Moldovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Performance on previously unseen S33 data Model Cam Threshold Precision Recall Astronet-Triage-v2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='98 Astronet-Triage-v2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='00 Astronet-Triage-v2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='91 Astronet-Triage-v2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='99 Astronet-Triage 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='85 Astronet-Triage 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='90 set in noise characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The Y3 dataset bears the most similarity to the evaluation set in terms of data characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' It is, however, much smaller than the other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Altogether, these different datasets and models provide useful views at our ability to general- ize to data that can be fairly different from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Since Astronet-Triage only distinguishes between transit-like and non-transit-like, it’s trained to give high scores TCEs we consider E- or S- labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' As Astronet-Triage-v2 provides independent E and S scores, we choose remove all S-labeled data from pre- cision and recall calculations for a simple direct perfor- mance comparison with Astronet-Triage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This leaves us with 1315 TCEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Precision and recall numbers split across Astronet-Triage and Astronet-Triage-v2 for each camera can be seen in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In both cameras we see that for similar (or better) levels of precision, Astronet-Triage-v2 provides better recall than Astronet-Triage, with a slightly more pronounced effect in camera 2 (fainter tar- gets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In other words, for the same amount of human vetting time, Astronet-Triage-v2 would recover more potential planets than Astronet-Triage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The full precision-recall curves across all TCEs (ig- noring S-labeled TCEs) are shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Across the board we see that Astronet-Triage-v2 (trained on the full training set) improves on Astronet-Triage with AUC-PR scores of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='961 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='927.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We also see that the models trained only on Y1, Y2, and Y3 data perform similarly to Astronet-Triage with AUC-PR scores of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='954, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='960, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='917 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Even though the Y1 and Y2 versions of the models don’t use any 1st Ex- tended Mission training data, we see they’re still able to perform highly in S33 (which occurred during Y3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This supports Astronet-Triage-v2’s ability to generalize to future sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Performance on the TOI catalog The TESS Objects of Interest (TOI) catalog (Guer- rero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021), which lists the planetary candidates detected by TESS, is a useful benchmark for high- Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Precision vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' recall for 1315 TCEs selected from Sector 33 of the 1st Extended Mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Since Astronet-Triage (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2019) only distinguishes between transit-like and non-transit-like, it gives high scores to TCEs we either consider to have E or S labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' For a more direct comparison to Astronet-Triage-v2, we choose to ignore all S-labeled TCEs when calculating precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We see that across all levels of recall, Astronet-Triage-v2 pro- vides higher precision even when trained only on Primary Mission data taken during Y1 or Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Although the Y3 dataset bears the most resemblance to the S33 evaluation set here, the size of the Y3 dataset is only ∼ 2500, so the Y3-trained model doesn’t quite reach the performance of the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' confidence E or S labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' A good model should label all TOI entries as E or S, since humans have inspected each entry and considered them to be high-probability planetary candidates (allowing for single-transit events).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' On 2022 April 21 we downloaded the TOI catalog with light curve data through Sector 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We also use informa- tion from TESS Follow-up Observing Program (TFOP) Sub Groups 1 and 2 (SG1 & SG2), which use ground- based photometry and reconnaissance spectroscopy to follow-up on TOIs and help filter out false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Af- ter keeping only planet candidates (PCs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' meaning TOIs that were not ruled out as false positives with follow-up observations) and validated / confirmed / known planets (Ps), we have a dataset of 4140 targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' After evaluating all TOI signals with Astronet-Triage-v2, Figure 12 shows the distribution of E scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Figure 13 shows the recall rate at different cutoff thresh- old levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We see that 93% of the TOIs have E scores > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0105 and as we increase the cutoff to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='215, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='9 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='7 Precision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4 - This work This work (Y1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3 This work (Y2) This work (Y3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2 Yu19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0 RecallImproved TESS Triage with Neural Networks 15 Astronet-Triage-v2 passes 86% of the TOIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We also see improved Astronet-Triage-v2 performance on known, confirmed, or validated planets (Ps) com- pared to the planet candidates (PCs) across the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' For comparison, we also ran Astronet-Triage on all TOI signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Using a threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='09, as was originally used in QLP, Astronet-Triage recovers 3349 TOIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Us- ing the dataset from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2, we find a precision- matching threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2 for Astronet-Triage-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' By finding the threshold of equal precision, we can com- pare TOI recovery at a constant rate of human vet- ter work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' At this threshold, 3577 TOIs are recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In other words, at least 200 TOIs are saved by us- ing Astronet-Triage-v2 in place of Astronet-Triage without introducing more false positives to human vet- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Some important caveats to note: The TOI catalog does include single-transit events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Astronet-Triage-v2 is trained to give these S rather than E labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Rather than keeping sep- arate cutoffs for S and E scores, for simplicity we choose to focus on E scores in reported re- calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This gives it a slight disadvantage in terms of recovery numbers, though we leave them in the dataset for fairer comparison to Astronet-Triage which gives a score for transit-like (periodic or single-transit) versus not transit-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' TOIs can also come from the SPOC pipeline, which processes 2-minute cadence light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' For both Astronet-Triage-v2 and Astronet-Triage, QLP light curves are binned down to 30 or 10 minutes, so some signals may not be detectable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' due to low signal-to-noise in the binned light curve) and should be assigned J labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This con- tributes partially to the lower recall numbers seen at the cutoffs from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Only 130 TOI host stars appear in our dataset of ∼25,000, 100 of which were in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We also conducted this analysis with those TOIs removed and saw similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' DISCUSSION 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Use in producing the TOI catalog A large piece of motivation for this work has been improving on Astronet-Triage so fewer planet can- didates are lost when searching for TOIs via QLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' After signal detection via BLS, Astronet is one of the finals triage steps before candidates are passed along to human TOI vetters and potentially promoted Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Top: Distribution of E score between this work and Astronet-Triage (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2019) on the whole TOI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Bottom: Distribution of E scores from this work when the dataset is separated into Planets (P, validated, confirmed, and known planets) and Planet Candidates (PC, TOIs that are not validated, confirmed, or known planets, and were also not identified as false positives with follow-up observations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' to TOIs (Guerrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Based on the re- sults in Section 5 we expect Astronet-Triage-v2 to save many planet candidates that would otherwise be lost without adding false positives and increasing the hours needed for human TOI vetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Starting in Sector 34, early versions of Astronet-Triage-v2 of- ficially replaced Astronet-Triage within QLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' While Astronet-Triage-v2 takes step towards a more au- tomated process, it is still not developed enough for population statistics (for a deeper discussion see Sec- tion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' What is limiting our precision?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In our tests, we found a common source of false neg- atives stemming from patterns with borderline label as- sessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The most common being eclipsing binaries which are non-contact but still close enough to resem- ble the pattern of a contact binary, due to, for example, tidal distortion, hence it is unclear whether the label should be “E” or “B” (Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Other instances of ambiguous patterns are represented by very noisy tran- sits, or transits on a background of high stellar variabil- This work 3000 Yu19 2500 Counts 2000 : 1500 1000 500 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0 Score 8 This work on Ps This work on PCs Normalized counts 6 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0 Score16 Tey/Moldovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Top: Recall as a function of cutoff thresh- old between this work and Astronet-Triage (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' For Astronet-Triage-v2 we choose to focus on just E scores even though some TOIs are true S labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Bottom: Astronet-Triage-v2 recall as a function of cutoff thresh- old when the dataset is separated into Planets (P, validated, confirmed, and known planets) and Planet Candidates (PC, TOIs that are not validated, confirmed, or known planets, and were also not identified as false positives with follow-up observations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' ity, where the distinction between “E” and “J” is more subtle (Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' One particular element of sensitivity for the neural network is on the correctness of the period and duration values estimated by BLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Errors in these values can lead to de-trending distortions which can make phase-folded views deviate from a transit-like light curve shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Ex- amples containing multi-year observations can be partic- ularly sensitive, as even slight variations in the detected period can lead to a blurring of the transit in the phase folded view (Figure 16 and 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We also note that the phase folding and binning pro- cesses are inherently lossy (similar to how compressing an image is a lossy process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' While we have not ascer- tained the impact of such loss of information, it is to be expected that it causes some loss of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Comparison to other works Our work is largely based on the original TESS Astronet-Triage classifier described by Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' (2019), which was used for QLP planet candidate triage from Sectors 6 to 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The following summarizes the major differences in development and implementation between classifiers: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Astronet-Triage was trained and tested on QLP light curves from only TESS Sectors 1 – 5, while Astronet-Triage-v2 was trained and tested on Sectors 1 – 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Astronet-Triage was developed using 16,516 la- beled TCEs (493 planet candidates, 2155 eclips- ing binaries, and 13,868 noise/systematic signals), which is roughly two-thirds the size of our labeled set (24,926 TCEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Astronet-Triage used labels that were assigned by only a single vetter who visually inspected all TCEs, while 3 – 5 vetters independently inspected each of the TCEs for Astronet-Triage-v2, and group discussions resolved labeling disagreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' As a result, our labels should be more reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Astronet-Triage only labels signals as either “planet” (for all eclipsing signals, including plan- ets and eclipsing binaries) and “non-planet” (for other false positives, including pulsating variables, noise and systematics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The five-label model used by Astronet-Triage-v2 (E, S, B, J, N) is more flexible and informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Astronet-Triage takes the light curves already detrended by QLP, and bins the data into two views: a “global” view, showing the full light curve phase diagram, and a “local” view, showing a close-up of the transit in the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' As described in Section 3, Astronet-Triage-v2 cre- ates three sets of detrended light curves from the raw QLP light curve, and generates seven views for each one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In total, Astronet-Triage-v2 uses 21 unique views to inform its classification compared to the two used by Astronet-Triage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' These key differences result in improvements to our ability to classify TESS signals in FFI data, as shown Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' To our knowledge Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' (2019) is the only truly comparable work to ours, in that their source dataset was the TESS Full Frame Images and not the pre- selected targets processed by the SPOC pipeline, and, their goal was to perform triage by identifying all eclips- ing signals, rather than separating planet candidates from eclipsing binaries and other false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Some other groups have trained and tested neural networks on TESS data from two-minute postage stamps processed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='6 Recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2 This work Yu19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0 Cutoff 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='8 Recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='6 This work on Ps This work on PCs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='0 CutoffImproved TESS Triage with Neural Networks 17 Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Example of borderline pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The true label for this example is “E”, but the folded light curve appears very similar to a “B”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' by the SPOC pipeline (Osborn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Valizadegan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Fiscale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Ofman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2022), and were successful in identifying planet candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' However, in general, these groups find that the neural network performance is worse on TESS data than a similar network on Kepler data, likely due to TESS’s higher a priori TCE false positive fraction (due to the larger TESS pixels resulting in more blending) and shorter observational baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The false positive rate for FFI targets is likely even higher because a) the targets observed by QLP tend to be fainter than targets observed in postage stamps and blending is more pro- nounced, and b) the targets observed in the FFIs are more often large, luminous stars like red giants, which are difficult to find planets around, and are photomet- rically noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Therefore, TCEs detected by the QLP likely have an even higher a priori false positive prob- ability than TCEs detected by TESS in postage stamp data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Future work 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Applications to exoplanet population statistics Planet catalogs can be used to characterize exoplanet population statistics through the estimation of occur- rence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' One of the key components of occurrence rate methodologies is a characterization of catalog com- pleteness, reflecting how many planets from the under- lying population were missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' A second key component is an understanding of catalog reliability (Bryson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2020), reflecting how much of the catalog is polluted with false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' For these reasons, occurrence rate studies require the ability to produce planet catalogs in a fully automated, uniform, and reproducible way, rather than relying on biased manual identification of planet candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' NASA’s Kepler mission has dominated the past decade of demographics work in large part thanks to the fully automated Kepler Robovetter pipeline, which enabled careful characterization of both completeness and reliability across wide areas of exoplanet param- eter space (Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Christiansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' However, there is not yet a fully automated TESS planet vetting pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Most previous work has also focused on 2-minute cadence observations rather than FFIs, which will be less suitable for demograph- ics due to selection biases in 2-minute cadence target lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Astronet-Triage-v2 is an important step toward uniformly vetted FFI planet catalogs, and it naturally allows for a flexibility in balances between completeness and reliability through the adjustment of prediction 18 Tey/Moldovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Example of borderline pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The very low signal to noise ratio of the transit signal is easily mistaken for a “J”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' thresholds for passing candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' While the classifier is not yet able to distinguish eclipsing binary false positives from planets (labeling all such signals as “E”’s), it can be used as a first round of automated and characterizable triage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Future improvements to Astronet-Triage-v2 (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2) are expected to improve the precision and recall, and therefore the completeness and reliabil- ity, of any resulting planet catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We have plans to extend Astronet-Triage-v2 to be capable of all steps of the vetting process in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Further improvements to the neural network In future work, we suggest a number of additions to further improve the performance of our classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Over the past few decades, the performance of deep learning classifiers has seen unprecedented success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' A large part of this success has been attributed to the increasing size of training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In this work, the number of training examples is relatively low, particu- larly for the S-labelled class, with a large class-imbalance (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' A common technique for increasing training datasets, without obtaining new labelled data, is data augmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This typically involve applying slight transfor- mations to the training data to produce new data that mimics real observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Using a combination of a few data augmentation techniques can magnify a training set by several fold and helps reduce over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' In future work, we suggest applying data augmentation methods such as randomly reversing or clipping light curves in time and applying random Gaussian noise to the light curves or scalar features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We note that these methods were applied in Ansdell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' (2018), where they showed that the main benefit to data augmentation on exo- planet classification was alleviating model over-fitting, with only a small improvement to model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' More complex augmentation methods such as fitting a model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Gaussian Process, see Boone 2019) to the mi- nority class light curves and generating more synthetic data may also help to improve the limited data for some classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Since Astronet-Triage-v2 is used in production for QLP’s monthly planet search, another way to increase our training dataset is to use the existing human vetting work that goes into producing the TOI catalog (Guer- rero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' As this human vetting is the final step in the TOI release process, there is a high level of quality control in the labels and the signals being vet- ted are often the most difficult to classify, making them important examples for the model to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' CONCLUSION Improved TESS Triage with Neural Networks 19 Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Example of incorrect BLS estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Although the phase and period are close, the transit duration is too small, causing the transit to be clipped by the detrending process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We have presented Astronet-Triage-v2, a convolu- tional neural network designed to distinguish astrophys- ical eclipsing candidates from other phenomena such as stellar variability and instrumental systematics in TESS FFI light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The network assigns input signals one of five labels, namely “E” for eclipsing signals, “S” for single transits or incorrect periods, “B” for contact bina- ries, “J” for signals due to noise or systematics, and “N” for inconclusive cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We trained Astronet-Triage-v2 using ∼ 25000 signals, which were detected by QLP from TESS Sectors 1 – 39 and human-labeled through man- ual review and group discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We make this training set available to the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Astronet-Triage-v2 is the next in a line of Astronet architectures, which were first used for Kepler (Shallue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2019) and later extended to K2 (Astronet-K2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Dattilo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2019) and TESS (Astronet-Triage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This iteration features significant improve- ments over Astronet-Triage, including a larger and more robust training set, an expanded list of possi- ble classifications, and more than ten times the num- ber of unique views used to analyze each signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' As a result, we found Astronet-Triage-v2 is more suc- cessful at correctly labeling known TOIs across al- most all cutoff values, with 86% recall at a cutoff of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='215 compared to 82% recall by Astronet-Triage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' When tested on a set of new signals from Sector 33, Astronet-Triage-v2 provides better recall of E and S labels than Astronet-Triage for similar (or better) lev- els of precision, especially for fainter targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Starting in Sector 34, Astronet-Triage-v2 officially replaced Astronet-Triage within QLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' As both the TESS observing baseline and number of observed stars continue to increase, automated TESS planet vetting tools will become more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This is especially true of tools tuned for planet searches us- ing FFIs, of which Astronet-Triage-v2 is one of the few currently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' While Astronet-Triage-v2 is not yet capable of distinguishing between eclipsing bina- ries and transiting planets, it serves as an effective first round of automated and characterizable triage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' We plan to continue to improve and extend the network into a fully automated vetting tool in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This paper includes data collected by the TESS mis- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Funding for the TESS mission is provided by the NASA’s Science Mission Directorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' HE20 Tey/Moldovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Example of incorrect BLS estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The detected period is close, but when the light curve contains a large number of folds, the error compounds and leads to a blurring of the transit view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This is due to QLP searching the light curve with an undersampled BLS frequency grid (necessary due to the computational time needed to run BLS on a large number of targets each sector), as discussed in Kunimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' (2022, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This work has made use of data from the Euro- pean Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='int/gaia), processed by the Gaia Data Pro- cessing and Analysis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='int/web/gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Funding for the DPAC has been provided by national institu- tions, in particular the institutions participating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This work was supported by an LSSTC Catalyst Fel- lowship awarded by LSST Corporation to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' with funding from the John Templeton Foundation grant ID #62192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' The Astronet-Triage-v2 model was trained and tuned on Google Compute Engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Facility: TESS, Gaia Software: numpy (Oliphant 2006), matplotlib (Hunter 2007), pandas (pandas development team 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Wes McKinney 2010), statsmodels (Seabold & Perktold 2010), pydl, astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=', & Frail, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 1992, Nature, 355, 145 Yu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=', Vanderburg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=', Huang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2019, AJ, 158, 25 Zucker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=', & Giryes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 2018, AJ, 155, 147 Improved TESS Triage with Neural Networks 23 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' EXAMPLE TCE TABLE Example TCE table that is passed into Astronet-Triage-v2 along-side raw light curve data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' All data is available in Tey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' This table contains information about the signal detected from BLS (epoch, period, duration, depth), information about the host star from TIC 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2 (TIC ID, M∗, R∗, TMag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' Est R∗ is described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content='2, and year describes the year the TCE was detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' MinT and MaxT specify the time range used from the light curve for both detection and input to Astronet-Triage-v2, and Split specifies which dataset (train, val, test) the signal was in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' L1-L8 are labels assigned by individuals and Consensus Label is the label agreed upon by the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' 24 Tey/Moldovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} +page_content=' TIC ID Period Epoch Duration Depth TMag M∗ R∗ Est R∗ Year MinT MaxT Split Consensus L1 L2 L3 (days) (BTJD) (days) (ppm) M⊙ R⊙ R⊙ BTJD BTJD Label 290603338 13.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftAzT4oBgHgl3EQfafzZ/content/2301.01371v1.pdf'} diff --git a/g9E0T4oBgHgl3EQf6gID/content/tmp_files/2301.02763v1.pdf.txt b/g9E0T4oBgHgl3EQf6gID/content/tmp_files/2301.02763v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..397542d09b94d90ab891777f71154a077f3bca46 --- /dev/null +++ b/g9E0T4oBgHgl3EQf6gID/content/tmp_files/2301.02763v1.pdf.txt @@ -0,0 +1,876 @@ +High-temperature thermoelectric properties with Th3−𝑥Te4 +Jizhu Hu,1, 2 Jinxin Zhong,1 and Jun Zhou3, ∗ +1Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China +2Max Plank Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany +3NNU-SULI Thermal Energy Research Center and Center for Quantum Transport and Thermal Energy Science, +School of Physics and Technology, Nanjing Normal University, Nanjing, Jiangsu 210023, China +(Dated: January 10, 2023) +Th3Te4 materials are potential candidates for commercial thermoelectric (TE) materials at high-temperature +due to their superior physical properties. We incorporate the multiband Boltzmann transport equations with +firstprinciples calculations to theoretically investigate the TE properties of Th3Te4 materials. As a demonstration +of our method, the TE properties of La3Te4 are similar with that of Ce3Te4 at low-temperature, which is consistent +with the experiment. Then we systematically calculate the electrical conductivity, the Seebeck coefficient, and +the power factor of the two materials above based on parameters obtained from first-principles calculations as +well as several other fitting parameters. Our results reveal that for the electron–optical-phonon scattering at high +temperatures, a linear dependence of optical phonon energy on temperature explains better the experimental +results than a constant optical phonon energy. Based on this, we predict that the TE properties of Ce3Te4 is +better than La3Te4 at high temperatures and the optimal carrier concentration corresponding to Ce3Te4 shifts +upward with increasing temperature. The optimal carrier concentration of Ce3Te4 is around 1.6 × 1021cm−3 +with the peak power factor 13.07 𝜇Wcm−1K−2 at 𝑇 = 1200K. +I. +INTRODUCTION +Thermoelectric (TE) materials have drawn great interest as +solid-state energy converters which can directly convert heat +to electricity and vice versa. +[1–3] The energy conversion +efficiency of TE materials is characterized by a dimension- +less figure of merit 𝑍𝑇 = 𝜎𝑆2𝑇/𝜅, where 𝜎 is the electrical +conductivity, 𝑆 is the Seebeck coefficient, 𝑇 is the absolute +temperature, and 𝜅 is the thermal conductivity consisting of +electronic 𝜅𝑐 and lattice thermal 𝜅𝑙 conductivity, 𝜅 = 𝜅𝑐 + 𝜅𝑙. +A higher cooling or power generation efficiency of TE devices +requires larger 𝑍𝑇 values. In the past decades, the 𝑍𝑇 of TE +materials has remained near 1 because 𝜎, 𝑆 and 𝜅𝑐 are cou- +pled to each other.[4] It is difficult to improve the TE properties +of materials by optimizing one of the parameters alone while +keeping the others constant. [5] A larger power factor, defined +as 𝜎𝑆2, is also required to gain larger output power. +Th3P4 (Th refers to rare earth metals, P refers to sulfur +group) has long been of interest due to its superior physical +properties, such as superconductivity, mixed valence, strong +electronic correlation, magnetic properties, optical proper- +ties, and TE properties.[6] Th3Te4 is a cubic crystal structure +with the space group 𝐼43𝑑. The Te atoms are hexa-aligned +with the rare earth metal lanthanide system through a twisted +octahedron.[7] It can be found by stoichiometry that the com- +pounds with Th3Te4 structure have good electrical properties +due to the presence of one extra electron. At the same time, +the presence of vacancies leads to disorder and distortion in +the lattice, which enhances phonon scattering and leads to a +lower lattice thermal conductivity.[8] +The properties of Th3Te4 have previously been investigated +by using solid-state diffusion and melt synthesis methods. [9] +However, the melt synthesis method leads to inhomogeneous +∗ zhoujunzhou@njnu.edu.cn +sample chemistry and carrier concentration caused by working +temperatures up to 2080 ∼ 2280K. May et al. in 2008 pro- +posed a mechanical alloying method to prepare La𝑥−3Te4.[7] +This method can effectively avoid the generation of inhomo- +geneous grains. +The authors estimated the lattice thermal +conductivity of La𝑥−3Te4 at 573K as 0.2 ∼ 0.4Wm−1K−1 +through the free electron Lorentz number. They also mea- +sured a power factor of 1.6Wm−1K−2 and a 𝑍𝑇 value of 1.1 +for La𝑥−3Te4 at 1273K. Recently, Pr2.74Te4 with a 𝑍𝑇 value as +high as 1.7 was prepared by Cheikh et al. using a mechanical +alloying method.[10] Ce𝑥−3Te4 and La𝑥−3Te4 have similar TE +properties in the low temperature region. [11] Using the first- +principle, Wang et al. found that the Ce3Te4 structure has a +𝛿-peak with 0.21eV in the density of states near the Fermi sur- +face. [12] Therefore, they predicted that Ce3Te4 has excellent +TE properties at high temperatures. Although the localized 𝑓 +electrons in Ce3Te4 make the density of states near the Fermi +surface sharp, the Seebeck coefficient is not increased by the +presence of 𝑓 electrons. [13] This is also confirmed by exper- +imental measurements. +In this paper, the multiband Boltzmann transport equations +(BTE) are used to explore and predict the TE transport prop- +erties under the relaxation time approximation (RTA). The pa- +rameters such as band gap and effective mass of each band are +calculated from first-principles calculations to solve the BTE. +The RTA based on the multiband carrier transport model is +also used. [14, 15] In order to demonstrate our method, we +study the TE properties of La3Te4 and Ce3Te4. Based on these +results, the optimal carrier concentrations for peak of power +factor are predicted for the Ce3Te4 materials at high tempera- +tures. The TE properties of other Th3−𝑥Te4 materials can be +studied similarly. +arXiv:2301.02763v1 [cond-mat.mtrl-sci] 7 Jan 2023 + +2 +II. +BAND STRUCTURE AND PHONON SPECTRUM +We employ the Vienna ab initio Simulation Package +(VASP), [16] which is based on the density function theory +(DFT) and generalized gradient approximation (GGA) with +the Perdew-Burke-Ernzerhof (PBE), [17, 18] to calculate the +band structure and phonon spectrum of Th3Te4 materials. The +structure of Th3Te4 were relaxed in cell shape, atom posi- +tions and volume. A plane-wave energy cutoff of 650 eV and +Monkhorst-Pack Γ-centered k-point meshes of 9×9×9 were +employed. For La3Te4, we consider the effect of spin-orbit +coupling (SOC).[19] Besides, due to the localization of f- +electrons in Ce3Te4, the on-site Coulomb interaction must be +considered to correct the self-interaction for f-electrons.[13] +The total energy is converged to less than 10−9 eV/unit. To de- +termine the phonon spectrum, a conventional cell is expanded +to a 2×2×2 supercell which contains 224 atoms, which further +undergoes a structure relaxation. Hellmann-Feynman forces +is calculated in relaxed supercell. Finally, The phonon spec- +trum of Th3Te4 are obtained via utilizing the phonopy package +combined with VASP. +A. +La3Te4 +The electronic band structure and phonon spectrum of +La3Te4 are shown in Figure 1. The main parameters for cal- +culating the TE properties are summarized in Table I. +-3 +-2 +-1 +0 +1 +P +G +N +H +0 +4 +8 +12 +16 +E (eV) +(a) +G +E (meV) +(b) +FIG. 1. (a) The band structure and (b) phonon spectrum of La3Te4. +The calculated band structure for La3Te4 is consistent with +that in Ref. 17, where a direct gap at Γ (0.99 eV) was obtained. +The energy minima of these bands which relative to E𝐹 are +E𝑚𝑖𝑛,1=-0.316 eV, E𝑚𝑖𝑛,2=-0.039 eV, and E𝑚𝑖𝑛,3=-0.018 eV, re- +spectively. The effective mass of each conduction and valance +bands near the Γ is obtained by fitting the band structure. The +density-of-state effective mass of each carriers pocket can be +expressed as +𝑚∗ +𝑖, 𝑗 = (𝑚∗2 +𝑖, 𝑗,∥𝑚∗ +𝑖, 𝑗,⊥) +1 +3 , +(1) +where 𝑚∗ +𝑖, 𝑗, ∥ (𝑚∗ +𝑖, 𝑗,⊥) is the parallel (perpendicular) effective +mass near the band edge and 𝑘𝑖, 𝑗, ∥ (𝑘𝑖, 𝑗,⊥) is the corresponding +wave vector. i represents the index of electron band. 𝑗 is the +type of carrier, 𝑗 = 𝑒 for electron and 𝑗 = ℎ for hole. Effective +mass and corresponding degeneracy for each conduction and +valance bands are shown in Table II, where spin degeneracy is +not included. +Figure 1(b) shows phonon dispersion curves of La3Te4, there +are 42 different types of vibration modes in the primitive unit +cell, including 3 acoustic modes and 39 optical modes. The +longitudinal (𝜐𝐿𝐴) and transverse (𝜐𝑇 𝐴) speed of sound can be +obtained via fitting acoustic modes at Γ point. The low-energy +peak (optical mode A2) is 9.09 meV, in agreement with Ref. 6. +B. +Ce3Te4 +-3 +-2 +-1 +0 +P +G +N +H +0 +4 +8 +12 +16 +E (eV) +(a) +G +E (meV) +(b) +FIG. 2. (a) The band structure and (b) phonon spectrum of Ce3Te4. +Figure 2 show that the band structure and the phonon spec- +trum of Ce3Te4. Comparison with Figure 1 shows that the en- +ergy band structures and phonon spectra of Ce3Te4 and La3Te4 +are similar, both are direct band gap semiconductor structures. +By comparing the data in Table I, we can clearly find that the +parameters of Ce3Te4 and La3Te4 are basically similar except +for the large difference in lattice constants 𝑎 and optical mode +energy A2. At room temperature, electron-phonon scattering +is weak and impurity scattering dominates the TE transport. +As the temperature increases, the lattice vibration gradually +strengthens and electron-phonon scattering gradually domi- +nates the TE transport. And the energy value of A2 has a +large effect on the TE properties at high temperatures and no + +3 +TABLE I. Parameters obtained from band structure and phonon spectrum of La3Te4 and Ce3Te4, such as lattice constant, band gap, energy +minima, A2 mode and speed of sound. +Materials 𝑎(Å) +𝐸𝑔 +𝐸𝑚𝑖𝑛,1(eV) 𝐸𝑚𝑖𝑛,2(eV) 𝐸𝑚𝑖𝑛,3(eV) A2(meV) 𝜐𝐿𝐴(m/s) 𝜐𝑇 𝐴(m/s) +La3Te4 +9.688 0.99 +-0.316 +-0.039 +-0.018 +9.09 +3357 +1989 +Ce3Te4 +9.542 1.07 +-0.388 +-0.170 +-0.005 +9.65 +3463 +2013 +TABLE II. Effective mass and corresponding degeneracy for each +conduction band and valance band in La3Te4 and Ce3Te4. +(i, j) 𝑚∗ +𝑖, 𝑗,∥(𝑚0) 𝑚∗ +𝑖, 𝑗,⊥ (𝑚0) 𝑚∗ +𝑖, 𝑗 (𝑚0) Degeneracy 𝑁𝑖 +La3Te4 +(1,e) +0.404 +0.3616 +0.389 +2 +(2,e) +1.1108 +0.9601 +1.058 +1 +(3,e) +1.1987 +1.2481 +1.215 +2 +(1,h) +0.3412 +0.4174 +0.341 +1 +(2,h) +1.965 +0.9189 +1.184 +1 +Ce3Te4 +(1,e) +0.7202 +0.6037 +0.6403 +2 +(2,e) +1.8985 +2.3394 +2.1821 +3 +(3,e) +34.0832 +30.8261 +31.8757 +1 +(1,h) +0.5956 +0.9828 +0.8317 +1 +(2,h) +0.3014 +0.3669 +0.3436 +1 +𝑚0 is free electron mass. +effect at room temperature. This is the reason why Ce3−𝑥Te4 +and La3−𝑥Te4 have similar TE properties at low temperatures. +[11] +From Table II, it can be found that the effective mass of the +nearest energy band (3, e) of Ce3Te4 near the Fermi energy +level is much larger than the effective mass of the other en- +ergy bands. This is mainly because compared to La(5𝑑6𝑠2), +Ce(4 𝑓 5𝑑6𝑠2) has one more 4 𝑓 electron which is localized, +and this leads to a large effective mass of its bonding orbitals. +However, 4 𝑓 electron does not contribute to the TE transport +properties.[20, 21] Therefore, we can ignore the contribution +of energy band (3, e) in the simulations. +III. +THERMOELECTRIC TRANSPORT PROPERTIES +Three conduction bands should be considered in calculating +the electron transport due to they are close enough. Besides, +the bipolar transport should also be incorporated since holes +will be excited and the electron-hole pair is formed in con- +duction band at high temperatures. The transport properties +in Th3Te4 are calculated by the multiband BTE under the +RTA.[12] Considering the TE properties of charge carriers in +the lowest conduction band and the highest valence band, each +of these bands is 𝑁-folded degeneracy, the dispersion rela- +tion of each carriers pocket can be expressed considering the +nonparabolicity: +ℏ2𝑘2 +𝑖, 𝑗,∥ +2𝑚∗ +𝑖, 𝑗, ∥ ++ +ℏ2𝑘2 +𝑖, 𝑗,⊥ +2𝑚∗ +𝑖, 𝑗,⊥ += 𝛾(𝐸𝑖, 𝑗) = 𝐸𝑖, 𝑗 + +𝐸2 +𝑖, 𝑗 +𝐸𝑔 +(2) +where ℏ is the reduced Plank constant, 𝐸𝑖, 𝑗 is the energy, and +𝛾(𝐸𝑖, 𝑗) = 𝐸𝑖, 𝑗 (1 + 𝐸𝑖, 𝑗/𝐸𝑔). The density-of-states effective +mass of each band 𝑚∗ +𝑖, 𝑗,𝑑 can be calculate by 𝑚∗ +𝑖, 𝑗,𝑑 = 𝑁 +2 +3 𝑚∗ +𝑖, 𝑗. +For a fixed doping concentration 𝑛𝑑, the chemical potential +𝜇 in the La3Te4 can be determined numerically.[14, 15] As- +suming that all the scattering events are independent, the total +relaxation time of each band (𝜏𝑡𝑜𝑡 +𝑖, 𝑗 ) can be expressed by the +Mathiessen’s rule: +1 +𝜏𝑡𝑜𝑡 +𝑖, 𝑗 += +1 +𝜏𝑖𝑚𝑝 +𝑖, 𝑗 ++ +1 +𝜏 𝑝𝑜 +𝑖, 𝑗, ++ +1 +𝜏𝑑𝑎,𝑙 +𝑖, 𝑗 ++ +1 +𝜏𝑑𝑜,𝑙′ +𝑖, 𝑗 +(3) +where 𝜏𝑖𝑚𝑝 +𝑖, 𝑗 +is the relaxation time of carries-impurity scat- +tering, 𝜏 𝑝𝑜 +𝑖, 𝑗 is that of carries-longitudinal polar optical phonon +scattering, 𝜏𝑑𝑎,𝑙 +𝑖, 𝑗 +is that of carries-deformation acoustic phonon +scattering corresponding to 𝑙th branch of acoustic phonon +mode, and 𝜏𝑑𝑜,𝑙′ +𝑖, 𝑗 +is that of carries-deformation optical phonon +scattering corresponding to 𝑙′th branch of optical phonon +mode, respectively. In principle, the relaxation time for differ- +ent scattering mechanisms can be obtained by Fermi’s golden +rule. The detailed temperature- and energy-dependent expres- +sions for each scattering relaxation time mentioned above can +be found in Refs. 14 and 22. +For bipolar transport, the TE transport coefficients such as +electrical conductivity (𝜎), Seebeck coefficient (S) and elec- +tronic thermal conductivity (𝜅𝑐) can be calculated by solving +the BTE under the RTA. For anisotropic materials, the TE +properties along different directions, which is denoted by 𝜉 = ∥ +or ⊥, can be written as, [14, 15] +𝜎𝜉 = +∑︁ +𝑗 +𝜎𝜉, 𝑗, 𝜎𝜉, 𝑗 = +∑︁ +𝑗 +𝑞2 +𝑗 +3𝜋2 ( 2𝑘𝐵𝑇 +ℏ2 +)3/2𝐹0, 𝜉, 𝑗 +(4) +𝑆 𝜉 = +∑︁ +𝑗 +𝑆 𝜉, 𝑗𝜎𝜉, 𝑗 +𝜎𝜉 +, 𝑆 𝜉, 𝑗 = 𝑘𝐵 +𝑞 𝑗 +( 𝐹1, 𝜉, 𝑗 +𝐹0, 𝜉, 𝑗 +− 𝜂 𝑗,𝜇) +(5) +𝜅𝑐, 𝜉 = +∑︁ +𝑗 +𝜅𝑐, 𝜉, 𝑗 + 𝜎𝜉,𝑒𝜎𝜉,ℎ +𝜎𝜉 +(𝑆 𝜉,𝑒 − 𝑆 𝜉,ℎ)2𝑇, +𝜅𝑐, 𝜉, 𝑗 = 𝑘2 +𝐵𝑇 +3𝜋2 ( 2𝑘𝐵𝑇 +ℏ2 +)3/2(𝐹2, 𝜉, 𝑗 − +𝐹2 +1, 𝜉, 𝑗 +𝐹0, 𝜉, 𝑗 +) +(6) +𝐹𝑛, 𝜉, 𝑗 = +∑︁ +𝑖 +𝑚∗3/2 +𝑖, 𝑗,𝑑 +𝑚∗ +𝑖, 𝑗, 𝜉 +∫ ∞ +0 +𝜂𝑛 +𝑖, 𝑗𝛾 +3 +2 (𝜂𝑖, 𝑗)𝜏𝑡𝑜𝑡 +𝑖, 𝑗 (− 𝜕 𝑓0 +𝜕𝜂𝑖, 𝑗 +)𝑑𝜂𝑖, 𝑗(7) +where 𝑞 𝑗 denotes the charge of carriers, 𝑘𝐵 is the Boltzmann +constant, 𝜂𝑖, 𝑗 = 𝐸𝑖, 𝑗 +𝑘𝐵𝑇 , 𝜂𝑒,𝜇 = 𝜇−𝐸𝑔 +𝑘𝐵𝑇 , 𝜂ℎ,𝜇 = − +𝜇 +𝑘𝐵𝑇 , 𝜂𝑔 = +𝐸𝑔 +𝑘𝐵𝑇 , +and 𝛾(𝜂𝑖, 𝑗) = 𝜂𝑖, 𝑗 (1 + 𝜂𝑖, 𝑗 +𝜂𝑔 ), respectively. 𝑓0 in Eq. (7) is the +equilibrium Fermi-Dirac distribution. + +4 +TABLE III. Fitting parameters used to calculate the transport coeffi- +cients in La3Te4 at 400K. +Parameters +Fitted value +mass density (g cm−3) +6.92[23] +optical phonon energy (meV) +6.2[6] +deformation potential constant (eV) +6.1[24] +high-frequency permittivity (𝜖0) +2.7 +static permittivity (𝜖0) +27[24] +impurity density (1019cm−3) +8[11] +A. +TE properties of La3Te4 +We now turn to calculate the electrical conductivity and the +Seebeck coefficient of La3Te4 based on the band structures of +La3Te4 obtained from first-principles calculations in Sec.II A. +In order to justify the input parameters in our calculation, we +first fit the experimental data of La3Te4 reported by Ref. 19. +The isotropic electrical conductivity along different directions +is averaged to compare to the measured electrical conductiv- +ity. Figure 3 shows that the calculated electrical conductivity +and the Seebeck coefficient as a function of carrier concen- +tration are in good agreement with the experimental results. +Table III presents the reasonable fitted parameters adopted in +our calculations. An increase of electrical conductivity and +a decrease of the Seebeck coefficient with increasing carrier +concentration comes from the 𝜎 ∝ 𝑛𝑑 and 𝑆 ∝ 𝑛−2/3 +𝑑 +. +0 +1 +2 +3 +4 +0 +50 +100 +150 +200 +|S| (mV/K) +nd(1021cm-3) + theory + exp. +0.0 +0.5 +1.0 +1.5 +2.0 +s (105 S/m) +FIG. 3. Calculated Seebeck coefficient (a) and electrical conductivity +(b) of La3Te4 as a function of carrier concentration. The experimental +data are extracted from Ref. 19. +On this basis, we can further investigate the TE proper- +ties of La3Te4 as a function of temperature, as shown in Fig- +ure 4. At low temperatures, impurity scattering is dominant +for the TE properties and other scattering is less influential. +As the temperature increases, the lattice vibration becomes +strong, intensifying electron-phonon scattering. In this case, +the electron-deformation acoustic (optical) phonon scattering +0 +1 +2 +3 +4 +400 +600 +800 +1000 +20 +40 +60 +80 +100 +� +exp. +� +� w=6.2 meV +� +linear dependence +� +� w=9.1 mev +s (105 S/m) +(a) +|S| (mV/K) +T(K) +(b) +FIG. 4. Calculated Seebeck coefficient (a) and electrical conductivity +(b) of La3Te4 as a function of temperature. The constant optical +branch phonon energies (red dashed and green dotted lines) do not +describe well their electrical conductivity experimental data. For the +linear dependence, the optical branched phonon energy fits well with +the experimental values of electrical conductivity. For the Seebeck +coefficients, the fit results remain largely consistent for the three +different modes. The experimental data are extracted from Ref.7. +will dominate the TE properties. Following the Einstein model +to approximate all optical branches as constant frequencies, the +numerical simulation results do not fit the experiment well, as +shown by the red dashed line and the green dotted line in Figure +4 (a). When the optical phonon frequency is small, the numer- +ical simulation value is smaller than the experimental value +in the high temperature region. Conversely, when the optical +phonon frequency is larger, the theoretical value will gradually +approach the experimental value as the temperature increases. +This is due to the fact that, like phonon-phonon scattering, +electron-phonon scattering also has normal scattering (N pro- +cess) and inversion scattering (U process). According to the +Debye model, the phonon energy (meV) is about a few thou- +sandths of the electron energy on the Fermi surface. Therefore, +the change of the electron energy is almost negligible due to +electron-phonon scattering. Although electron-phonon scat- +tering can be considered as completely elastic scattering, it +changes the direction of electron motion, which has a signifi- +cant effect on electrical conductivity. At low temperatures, the +electron scattering angle is small because only low-frequency +phonon modes are excited, which has limited effect on the con- + +5 +ductivity. As the temperature rises, more vibrational modes +of phonons will be excited. The angle of phonon and elec- +tron scattering differs for different modes, which will also +have different effects on the conductivity. Here, for simplicity, +we assume a linear dependence of the energy of the different +modes of phonons scattered with electrons on temperature as, +ℏ𝜔 = ℏ𝜔0 + 3.412 × 10−3(𝑇 − 𝑇0), where ℏ𝜔0 = 6.2meV, +𝑇0 = 400K. The calculated results using linear dependence are +in good agreement with the experimental values, as shown by +the blue solid line in Figure 4(a). The Seebeck coefficient is +mainly measured by the average energy magnitude of carriers, +which is related to the density of states near the Fermi surface. +And the effect of electron-phonon scattering for the average +carrier energy can be neglected. Therefore, the Seebeck co- +efficients fitted by the three different methods are essentially +comparable, as shown in the Figure 4(b). +B. +TE properties of Ce3Te4 +Considering that the local state electrons do not contribute to +the electron transport, we only consider the contribution of the +five lowest conduction bands and the two highest valence bands +to the TE transport properties of Ce3Te4. Figure 5 depicts the +calculated electrical conductivity and Seebeck coefficient ver- +sus temperature based on the BTE under the RTA. The fitting +parameters used in the calculations are shown in Table IV. +We can find that the calculated values are consistent with the +experimental data, which indicates that the fitting parameters +are chosen reasonably. In addition, it shows a decrease in elec- +trical conductivity and an increase of the Seebeck coefficient +with increasing temperature. This is mainly due to the increase +of carrier scattering intensity at higher temperatures and the +increase of the average energy carried by carriers. +200 +220 +240 +260 +14 +16 +18 +20 +22 +24 +|S| (mV/K) +T (K) + theory + exp. +0 +1 +2 +3 +4 +5 +s (105 S/m) +FIG. 5. Calculated Seebeck coefficient (a) and electrical conductivity +(b) of Ce3Te4 as a function of temperature. The experimental data +are extracted from Ref. 11. +TABLE IV. Fitting parameters used to calculate the transport coeffi- +cients in Ce3Te4 at 400K. +Parameters +Fitted value +carriers concentration (1021cm−3) +4.6[11, 13] +mass density (g cm−3) +7.12[23] +optical phonon energy (meV) +6.2 +deformation potential constant (eV) +6.1 +high-frequency permittivity (𝜖0) +2.7 +static permittivity (𝜖0) +27 +impurity density (1019cm−3) +5 [11] +C. +Comparison of TE properties of La3Te4 and Ce3Te4 +At room temperature, experimental and theoretical reports +have shown that the Ce3Te4 has similar TE transport properties +as La3Te4, such as electrical conductivity, Seebeck coefficient, +and power factor. [7, 11] We can obtain the same conclusion +by numerical simulation using BTE under the RTA as shown in +Figure 6. Since the effective mass of Ce3Te4 is larger than that +of La3Te4, the electrical conductivity of Ce3Te4 is reasonably +smaller than that of La3Te4. However, at low temperatures, +the electrical conductivity of La3Te4 and Ce3Te4 are approxi- +mately equal, which indicates that the average relaxation time +of Ce3Te4 is larger than that of La3Te4. The effect of scatter- +ing will be greater with increasing of temperature. In the high +temperature region, more phonon modes are excited, which +leads to a rise in the number of phonons and allows phonons +to participate in TE transport. At this point, electron-phonon +scattering, which consists of the polar optical phonons, the de- +formed acoustic phonons, and the deformed optical phonons, +dominates TE transport. As shown in Figure 6(a) and (b), the +Seebeck coefficient of Ce3Te4 is comparable to that of La3Te4 +at high temperatures, but its electrical conductivity is higher +than that of La3Te4. For example, the electrical conductiv- +ity of Ce3Te4 and La3Te4 at 𝑇 = 1000K are 1.07×105 and +0.942×105 S/m, respectively. This is mainly due to the dif- +ference in the optical branch phonon energy of A2 mode of +La3Te4 and Ce3Te4, as shown in Table I. From the phonon +spectrum, we find that the optical branch phonon energy of +A2 mode of Ce3Te4 is slightly larger than that of La3Te4. The +magnitude of the optical branch phonon energy will determine +the magnitude of relaxation time of the electron-deformation +optical phonon scattering. A larger energy will correspond +to a larger relaxation time. In the case of high temperatures, +electron-phonon scattering plays a dominant role, which has +increased the total relaxation time. Therefore, the electrical +conductivity of Ce3Te4 is larger than that of La3Te4 under +other equal conditions. Moreover, the power factor of Ce3Te4 +is also better than that of La3Te4 at high temperature. +D. +Optimal carrier concentration of Ce3Te4 at various +temperatures +To further investigate the TE properties of Ce3Te4 at high +temperatures, we will consider the TE transport properties of + +6 +0 +1 +2 +3 +4 +-100 +-80 +-60 +-40 +-20 +0 +200 +400 +600 +800 +1000 +0 +2 +4 +6 +8 +10 +(c) +(b) +s (105 S/m) +� +Ce3Te4 +� +La3Te4 +(a) +S (mV/K) +PF (mW/cm×K2) +T(K) +FIG. 6. Comparison of TE properties of La3Te4 and Ce3Te4: Elec- +trical conductivity (a), Seebeck coefficient (b), and Power factor (c) +as a function of temperature. +Ce3Te4 as a function of carrier concentration with various +high temperatures, as shown in Figure 7. Due to 𝜎 ∝ 𝑛𝑑 and +𝜎 ∝ 𝜏𝑡𝑜𝑡 ∝ 1/𝑇, the electrical conductivity will be enhanced +when the carrier concentration increases or the temperature +decreases as shown in Figure 7(a). Due to the coupling rela- +tionship between 𝜎 and 𝑆, the Seebeck coefficient changes in +the opposite direction. However, it can be seen from Figure +7(b) that the trend of Seebeck variation becomes gradually +smoother as the carrier concentration increases. This is be- +cause the bipolar effect will be more pronounced at high tem- +peratures by exciting the intrinsic carrier. In particular, fora +lower carrier concentration, the bipolar effect has a greater im- +pact on the Seebeck coefficient, which can be found to decrease +at a greater rate than that at higher carrier concentrations. Fig- +0.0 +0.4 +0.8 +1.2 +1.6 +-400 +-300 +-200 +-100 +0 +0 +1 +2 +3 +4 +5 +0 +5 +10 +15 +(c) +(b) +s (105 S/m) +� +600K +� +800K +� +1000K +� +1200K +(a) +S (mV/K) +PF (mW/cm×K2) +nd(1021cm-3) +FIG. 7. Calculated TE properties of Ce3Te4: Electrical conductivity +(a), Seebeck coefficient (b), and Power factor (c) as a function of +carrier concentration at various temperatures. +ure 7(c) shows the dependence of the power factor on the +carrier concentration at different temperatures. We found that +the power factor corresponding to the higher temperature is +smaller as the carrier concentration is low. Because of the +inhibitory effect of minority carriers on the TE properties, al- +though the intrinsic excitation increases the minority carrier +concentration and the electrical conductivity, the presence of +the minority carriers can cause a decrease in the Seebeck co- +efficient, which is more than compensate for the increase in +the Seebeck coefficient caused by the increase in conductiv- +ity. As the carrier concentration increases, the effect of the +bipolar diffusion effect on the TE performance generated by +the minority carriers diminishes. In other words, the bipolar + +7 +effect can be suppressed via the heavily doped method. In ad- +dition, the optimal carrier concentration of Ce3Te4 varies for +different temperatures, and the corresponding power factors +are also different. For example, the optimal carrier concentra- +tion is around 0.5 × 1021cm−3 with the peak power factor 9.02 +𝜇Wcm−1K−2 at 𝑇 = 600K; the optimal carrier concentration +is around 0.8 × 1021cm−3 with the peak power factor 10.42 +𝜇Wcm−1K−2 at 𝑇 = 800K; the optimal carrier concentration +is around 1.2 × 1021cm−3 with the peak power factor 11.78 +𝜇Wcm−1K−2 at 𝑇 = 1000K; And the optimal carrier concen- +tration is around 1.6 × 1021cm−3 with the peak power factor +13.07 𝜇Wcm−1K−2 at 𝑇 = 1200K. This is because the carrier +concentration of the intrinsic excitation is strongly correlated +with the temperature, and as the temperature increases, the +optimal carrier concentration shifts upward. +IV. +CONCLUSION +We have incorporated the multiband Boltzmann transport +equations with first-principles calculations on electronic band +structures in order to theoretically investigate TE properties of +Th3Te4 materials such as La3Te4 and Ce3Te4. Our theoreti- +cal calculations are in good agreement with the experimental +data with calculated parameters and several other fitting pa- +rameters. Theoretical results show that for the TE transport +properties at high temperatures, a linear dependence is more +consistent with the experimental results than the constant opti- +cal branch phonon energy describing the electron-deformation +optical branch phonon scattering. In addition, we predict the +TE transport properties of Ce3Te4 at high temperatures and the +optimal carrier concentration at different temperatures, which +is a guideline for experimental aspects. +ACKNOWLEDGMENTS +This work was supported by National Key R&D Program +of China (No. +2017YFB0406004), National Natural Sci- +ence Foundation of China (No. +11890703), Key-Area Re- +search and Development Program of Guangdong Province +(No. 2020B010190004). +[1] L. E. Bell, Science 321, 1457 (2008). +[2] F. J. DiSalvo, Science 285, 703 (1999). +[3] B. Qin, D. Wang, X. Liu, Y. Qin, J.-F. Dong, J. Luo, J.-W. Li, +W. Liu, G. Tan, X. Tang, et al., Science 373, 556 (2021). +[4] M. Jonson and G. Mahan, Phys. Rev. B 21, 4223 (1980). +[5] D. M. Rowe, CRC handbook of thermoelectrics (CRC press, +2018). +[6] R. Viennois, K. Niedziolka, and P. Jund, Phys. Rev. B 88, 174302 +(2013). +[7] A. F. May, J.-P. Fleurial, and G. J. Snyder, Phys. Rev. B 78, +125205 (2008). +[8] C. Wood, A. Lockwood, J. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' ∗ 1Center for Phononics and Thermal Energy Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' School of Physics Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Tongji University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Shanghai 200092,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' China 2Max Plank Institute for Polymer Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Ackermannweg 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 55128 Mainz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Germany 3NNU-SULI Thermal Energy Research Center and Center for Quantum Transport and Thermal Energy Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' School of Physics and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Nanjing Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Nanjing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Jiangsu 210023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' China (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 2023) Th3Te4 materials are potential candidates for commercial thermoelectric (TE) materials at high-temperature due to their superior physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' We incorporate the multiband Boltzmann transport equations with firstprinciples calculations to theoretically investigate the TE properties of Th3Te4 materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' As a demonstration of our method, the TE properties of La3Te4 are similar with that of Ce3Te4 at low-temperature, which is consistent with the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Then we systematically calculate the electrical conductivity, the Seebeck coefficient, and the power factor of the two materials above based on parameters obtained from first-principles calculations as well as several other fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Our results reveal that for the electron–optical-phonon scattering at high temperatures, a linear dependence of optical phonon energy on temperature explains better the experimental results than a constant optical phonon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Based on this, we predict that the TE properties of Ce3Te4 is better than La3Te4 at high temperatures and the optimal carrier concentration corresponding to Ce3Te4 shifts upward with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The optimal carrier concentration of Ce3Te4 is around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='6 × 1021cm−3 with the peak power factor 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='07 𝜇Wcm−1K−2 at 𝑇 = 1200K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' INTRODUCTION Thermoelectric (TE) materials have drawn great interest as solid-state energy converters which can directly convert heat to electricity and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [1–3] The energy conversion efficiency of TE materials is characterized by a dimension- less figure of merit 𝑍𝑇 = 𝜎𝑆2𝑇/𝜅, where 𝜎 is the electrical conductivity, 𝑆 is the Seebeck coefficient, 𝑇 is the absolute temperature, and 𝜅 is the thermal conductivity consisting of electronic 𝜅𝑐 and lattice thermal 𝜅𝑙 conductivity, 𝜅 = 𝜅𝑐 + 𝜅𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' A higher cooling or power generation efficiency of TE devices requires larger 𝑍𝑇 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' In the past decades, the 𝑍𝑇 of TE materials has remained near 1 because 𝜎, 𝑆 and 𝜅𝑐 are cou- pled to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [4] It is difficult to improve the TE properties of materials by optimizing one of the parameters alone while keeping the others constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [5] A larger power factor, defined as 𝜎𝑆2, is also required to gain larger output power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Th3P4 (Th refers to rare earth metals, P refers to sulfur group) has long been of interest due to its superior physical properties, such as superconductivity, mixed valence, strong electronic correlation, magnetic properties, optical proper- ties, and TE properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [6] Th3Te4 is a cubic crystal structure with the space group 𝐼43𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The Te atoms are hexa-aligned with the rare earth metal lanthanide system through a twisted octahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [7] It can be found by stoichiometry that the com- pounds with Th3Te4 structure have good electrical properties due to the presence of one extra electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' At the same time, the presence of vacancies leads to disorder and distortion in the lattice, which enhances phonon scattering and leads to a lower lattice thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [8] The properties of Th3Te4 have previously been investigated by using solid-state diffusion and melt synthesis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [9] However, the melt synthesis method leads to inhomogeneous ∗ zhoujunzhou@njnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='cn sample chemistry and carrier concentration caused by working temperatures up to 2080 ∼ 2280K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' May et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' in 2008 pro- posed a mechanical alloying method to prepare La𝑥−3Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [7] This method can effectively avoid the generation of inhomo- geneous grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The authors estimated the lattice thermal conductivity of La𝑥−3Te4 at 573K as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='4Wm−1K−1 through the free electron Lorentz number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' They also mea- sured a power factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='6Wm−1K−2 and a 𝑍𝑇 value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='1 for La𝑥−3Te4 at 1273K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Recently, Pr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='74Te4 with a 𝑍𝑇 value as high as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='7 was prepared by Cheikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' using a mechanical alloying method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [10] Ce𝑥−3Te4 and La𝑥−3Te4 have similar TE properties in the low temperature region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [11] Using the first- principle, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' found that the Ce3Te4 structure has a 𝛿-peak with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='21eV in the density of states near the Fermi sur- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [12] Therefore, they predicted that Ce3Te4 has excellent TE properties at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Although the localized 𝑓 electrons in Ce3Te4 make the density of states near the Fermi surface sharp, the Seebeck coefficient is not increased by the presence of 𝑓 electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [13] This is also confirmed by exper- imental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' In this paper, the multiband Boltzmann transport equations (BTE) are used to explore and predict the TE transport prop- erties under the relaxation time approximation (RTA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The pa- rameters such as band gap and effective mass of each band are calculated from first-principles calculations to solve the BTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The RTA based on the multiband carrier transport model is also used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [14, 15] In order to demonstrate our method, we study the TE properties of La3Te4 and Ce3Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Based on these results, the optimal carrier concentrations for peak of power factor are predicted for the Ce3Te4 materials at high tempera- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The TE properties of other Th3−𝑥Te4 materials can be studied similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='02763v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='mtrl-sci] 7 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' BAND STRUCTURE AND PHONON SPECTRUM We employ the Vienna ab initio Simulation Package (VASP), [16] which is based on the density function theory (DFT) and generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE), [17, 18] to calculate the band structure and phonon spectrum of Th3Te4 materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The structure of Th3Te4 were relaxed in cell shape, atom posi- tions and volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' A plane-wave energy cutoff of 650 eV and Monkhorst-Pack Γ-centered k-point meshes of 9×9×9 were employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' For La3Te4, we consider the effect of spin-orbit coupling (SOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [19] Besides, due to the localization of f- electrons in Ce3Te4, the on-site Coulomb interaction must be considered to correct the self-interaction for f-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [13] The total energy is converged to less than 10−9 eV/unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' To de- termine the phonon spectrum, a conventional cell is expanded to a 2×2×2 supercell which contains 224 atoms, which further undergoes a structure relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Hellmann-Feynman forces is calculated in relaxed supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Finally, The phonon spec- trum of Th3Te4 are obtained via utilizing the phonopy package combined with VASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' La3Te4 The electronic band structure and phonon spectrum of La3Te4 are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The main parameters for cal- culating the TE properties are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 3 2 1 0 1 P G N H 0 4 8 12 16 E (eV) (a) G E (meV) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' (a) The band structure and (b) phonon spectrum of La3Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The calculated band structure for La3Te4 is consistent with that in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 17, where a direct gap at Γ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='99 eV) was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The energy minima of these bands which relative to E𝐹 are E𝑚𝑖𝑛,1=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='316 eV, E𝑚𝑖𝑛,2=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='039 eV, and E𝑚𝑖𝑛,3=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='018 eV, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The effective mass of each conduction and valance bands near the Γ is obtained by fitting the band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The density-of-state effective mass of each carriers pocket can be expressed as 𝑚∗ 𝑖, 𝑗 = (𝑚∗2 𝑖, 𝑗,∥𝑚∗ 𝑖, 𝑗,⊥) 1 3 , (1) where 𝑚∗ 𝑖, 𝑗, ∥ (𝑚∗ 𝑖, 𝑗,⊥) is the parallel (perpendicular) effective mass near the band edge and 𝑘𝑖, 𝑗, ∥ (𝑘𝑖, 𝑗,⊥) is the corresponding wave vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' i represents the index of electron band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 is the type of carrier, 𝑗 = 𝑒 for electron and 𝑗 = ℎ for hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Effective mass and corresponding degeneracy for each conduction and valance bands are shown in Table II, where spin degeneracy is not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Figure 1(b) shows phonon dispersion curves of La3Te4, there are 42 different types of vibration modes in the primitive unit cell, including 3 acoustic modes and 39 optical modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The longitudinal (𝜐𝐿𝐴) and transverse (𝜐𝑇 𝐴) speed of sound can be obtained via fitting acoustic modes at Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The low-energy peak (optical mode A2) is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='09 meV, in agreement with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Ce3Te4 3 2 1 0 P G N H 0 4 8 12 16 E (eV) (a) G E (meV) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' (a) The band structure and (b) phonon spectrum of Ce3Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Figure 2 show that the band structure and the phonon spec- trum of Ce3Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Comparison with Figure 1 shows that the en- ergy band structures and phonon spectra of Ce3Te4 and La3Te4 are similar, both are direct band gap semiconductor structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' By comparing the data in Table I, we can clearly find that the parameters of Ce3Te4 and La3Te4 are basically similar except for the large difference in lattice constants 𝑎 and optical mode energy A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' At room temperature, electron-phonon scattering is weak and impurity scattering dominates the TE transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' As the temperature increases, the lattice vibration gradually strengthens and electron-phonon scattering gradually domi- nates the TE transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' And the energy value of A2 has a large effect on the TE properties at high temperatures and no 3 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Parameters obtained from band structure and phonon spectrum of La3Te4 and Ce3Te4, such as lattice constant, band gap, energy minima, A2 mode and speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Materials 𝑎(Å) 𝐸𝑔 𝐸𝑚𝑖𝑛,1(eV) 𝐸𝑚𝑖𝑛,2(eV) 𝐸𝑚𝑖𝑛,3(eV) A2(meV) 𝜐𝐿𝐴(m/s) 𝜐𝑇 𝐴(m/s) La3Te4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='018 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='09 3357 1989 Ce3Te4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='542 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='005 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='65 3463 2013 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Effective mass and corresponding degeneracy for each conduction band and valance band in La3Te4 and Ce3Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' (i, j) 𝑚∗ 𝑖, 𝑗,∥(𝑚0) 𝑚∗ 𝑖, 𝑗,⊥ (𝑚0) 𝑚∗ 𝑖, 𝑗 (𝑚0) Degeneracy 𝑁𝑖 La3Te4 (1,e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='404 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='3616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='389 2 (2,e) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='1108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='9601 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='058 1 (3,e) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='1987 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='2481 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='215 2 (1,h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='3412 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='4174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='341 1 (2,h) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='9189 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='184 1 Ce3Te4 (1,e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='7202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='6037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='6403 2 (2,e) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='8985 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='3394 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='1821 3 (3,e) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='0832 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='8261 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='8757 1 (1,h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='5956 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='9828 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='8317 1 (2,h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='3014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='3669 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='3436 1 𝑚0 is free electron mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' effect at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' This is the reason why Ce3−𝑥Te4 and La3−𝑥Te4 have similar TE properties at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [11] From Table II, it can be found that the effective mass of the nearest energy band (3, e) of Ce3Te4 near the Fermi energy level is much larger than the effective mass of the other en- ergy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' This is mainly because compared to La(5𝑑6𝑠2), Ce(4 𝑓 5𝑑6𝑠2) has one more 4 𝑓 electron which is localized, and this leads to a large effective mass of its bonding orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' However, 4 𝑓 electron does not contribute to the TE transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [20, 21] Therefore, we can ignore the contribution of energy band (3, e) in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' THERMOELECTRIC TRANSPORT PROPERTIES Three conduction bands should be considered in calculating the electron transport due to they are close enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Besides, the bipolar transport should also be incorporated since holes will be excited and the electron-hole pair is formed in con- duction band at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The transport properties in Th3Te4 are calculated by the multiband BTE under the RTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [12] Considering the TE properties of charge carriers in the lowest conduction band and the highest valence band, each of these bands is 𝑁-folded degeneracy, the dispersion rela- tion of each carriers pocket can be expressed considering the nonparabolicity: ℏ2𝑘2 𝑖, 𝑗,∥ 2𝑚∗ 𝑖, 𝑗, ∥ + ℏ2𝑘2 𝑖, 𝑗,⊥ 2𝑚∗ 𝑖, 𝑗,⊥ = 𝛾(𝐸𝑖, 𝑗) = 𝐸𝑖, 𝑗 + 𝐸2 𝑖, 𝑗 𝐸𝑔 (2) where ℏ is the reduced Plank constant, 𝐸𝑖, 𝑗 is the energy, and 𝛾(𝐸𝑖, 𝑗) = 𝐸𝑖, 𝑗 (1 + 𝐸𝑖, 𝑗/𝐸𝑔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The density-of-states effective mass of each band 𝑚∗ 𝑖, 𝑗,𝑑 can be calculate by 𝑚∗ 𝑖, 𝑗,𝑑 = 𝑁 2 3 𝑚∗ 𝑖, 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' For a fixed doping concentration 𝑛𝑑, the chemical potential 𝜇 in the La3Te4 can be determined numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 15] As- suming that all the scattering events are independent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' the total relaxation time of each band (𝜏𝑡𝑜𝑡 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 ) can be expressed by the Mathiessen’s rule: 1 𝜏𝑡𝑜𝑡 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 = 1 𝜏𝑖𝑚𝑝 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 + 1 𝜏 𝑝𝑜 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' + 1 𝜏𝑑𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='𝑙 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 + 1 𝜏𝑑𝑜,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='𝑙′ 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 (3) where 𝜏𝑖𝑚𝑝 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 is the relaxation time of carries-impurity scat- tering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜏 𝑝𝑜 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 is that of carries-longitudinal polar optical phonon scattering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜏𝑑𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='𝑙 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 is that of carries-deformation acoustic phonon scattering corresponding to 𝑙th branch of acoustic phonon mode,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' and 𝜏𝑑𝑜,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='𝑙′ 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 is that of carries-deformation optical phonon scattering corresponding to 𝑙′th branch of optical phonon mode,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' In principle, the relaxation time for differ- ent scattering mechanisms can be obtained by Fermi’s golden rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The detailed temperature- and energy-dependent expres- sions for each scattering relaxation time mentioned above can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 14 and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' For bipolar transport, the TE transport coefficients such as electrical conductivity (𝜎), Seebeck coefficient (S) and elec- tronic thermal conductivity (𝜅𝑐) can be calculated by solving the BTE under the RTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' For anisotropic materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' the TE properties along different directions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' which is denoted by 𝜉 = ∥ or ⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' can be written as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 15] 𝜎𝜉 = ∑︁ 𝑗 𝜎𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜎𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 = ∑︁ 𝑗 𝑞2 𝑗 3𝜋2 ( 2𝑘𝐵𝑇 ℏ2 )3/2𝐹0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 (4) 𝑆 𝜉 = ∑︁ 𝑗 𝑆 𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗𝜎𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 𝜎𝜉 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑆 𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 = 𝑘𝐵 𝑞 𝑗 ( 𝐹1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 𝐹0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 − 𝜂 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='𝜇) (5) 𝜅𝑐,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜉 = ∑︁ 𝑗 𝜅𝑐,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 + 𝜎𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='𝑒𝜎𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='ℎ 𝜎𝜉 (𝑆 𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='𝑒 − 𝑆 𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='ℎ)2𝑇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜅𝑐,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 = 𝑘2 𝐵𝑇 3𝜋2 ( 2𝑘𝐵𝑇 ℏ2 )3/2(𝐹2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 − 𝐹2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 𝐹0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 ) (6) 𝐹𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 = ∑︁ 𝑖 𝑚∗3/2 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='𝑑 𝑚∗ 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜉 ∫ ∞ 0 𝜂𝑛 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗𝛾 3 2 (𝜂𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗)𝜏𝑡𝑜𝑡 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 (− 𝜕 𝑓0 𝜕𝜂𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 )𝑑𝜂𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗(7) where 𝑞 𝑗 denotes the charge of carriers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑘𝐵 is the Boltzmann constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜂𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 = 𝐸𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 𝑘𝐵𝑇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜂𝑒,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='𝜇 = 𝜇−𝐸𝑔 𝑘𝐵𝑇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜂ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='𝜇 = − 𝜇 𝑘𝐵𝑇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝜂𝑔 = 𝐸𝑔 𝑘𝐵𝑇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' and 𝛾(𝜂𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗) = 𝜂𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 (1 + 𝜂𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑗 𝜂𝑔 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 𝑓0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' (7) is the equilibrium Fermi-Dirac distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 4 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Fitting parameters used to calculate the transport coeffi- cients in La3Te4 at 400K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Parameters Fitted value mass density (g cm−3) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='92[23] optical phonon energy (meV) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='2[6] deformation potential constant (eV) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='1[24] high-frequency permittivity (𝜖0) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='7 static permittivity (𝜖0) 27[24] impurity density (1019cm−3) 8[11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' TE properties of La3Te4 We now turn to calculate the electrical conductivity and the Seebeck coefficient of La3Te4 based on the band structures of La3Te4 obtained from first-principles calculations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' In order to justify the input parameters in our calculation, we first fit the experimental data of La3Te4 reported by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The isotropic electrical conductivity along different directions is averaged to compare to the measured electrical conductiv- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Figure 3 shows that the calculated electrical conductivity and the Seebeck coefficient as a function of carrier concen- tration are in good agreement with the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Table III presents the reasonable fitted parameters adopted in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' An increase of electrical conductivity and a decrease of the Seebeck coefficient with increasing carrier concentration comes from the 𝜎 ∝ 𝑛𝑑 and 𝑆 ∝ 𝑛−2/3 𝑑 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 0 1 2 3 4 0 50 100 150 200 |S| (mV/K) nd(1021cm-3) theory exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='0 s (105 S/m) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Calculated Seebeck coefficient (a) and electrical conductivity (b) of La3Te4 as a function of carrier concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The experimental data are extracted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' On this basis, we can further investigate the TE proper- ties of La3Te4 as a function of temperature, as shown in Fig- ure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' At low temperatures, impurity scattering is dominant for the TE properties and other scattering is less influential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' As the temperature increases, the lattice vibration becomes strong, intensifying electron-phonon scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' In this case, the electron-deformation acoustic (optical) phonon scattering 0 1 2 3 4 400 600 800 1000 20 40 60 80 100 � exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' � � w=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='2 meV � linear dependence � � w=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='1 mev s (105 S/m) (a) |S| (mV/K) T(K) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Calculated Seebeck coefficient (a) and electrical conductivity (b) of La3Te4 as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The constant optical branch phonon energies (red dashed and green dotted lines) do not describe well their electrical conductivity experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' For the linear dependence, the optical branched phonon energy fits well with the experimental values of electrical conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' For the Seebeck coefficients, the fit results remain largely consistent for the three different modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The experimental data are extracted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' will dominate the TE properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Following the Einstein model to approximate all optical branches as constant frequencies, the numerical simulation results do not fit the experiment well, as shown by the red dashed line and the green dotted line in Figure 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' When the optical phonon frequency is small, the numer- ical simulation value is smaller than the experimental value in the high temperature region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Conversely, when the optical phonon frequency is larger, the theoretical value will gradually approach the experimental value as the temperature increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' This is due to the fact that, like phonon-phonon scattering, electron-phonon scattering also has normal scattering (N pro- cess) and inversion scattering (U process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' According to the Debye model, the phonon energy (meV) is about a few thou- sandths of the electron energy on the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Therefore, the change of the electron energy is almost negligible due to electron-phonon scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Although electron-phonon scat- tering can be considered as completely elastic scattering, it changes the direction of electron motion, which has a signifi- cant effect on electrical conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' At low temperatures, the electron scattering angle is small because only low-frequency phonon modes are excited, which has limited effect on the con- 5 ductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' As the temperature rises, more vibrational modes of phonons will be excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The angle of phonon and elec- tron scattering differs for different modes, which will also have different effects on the conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Here, for simplicity, we assume a linear dependence of the energy of the different modes of phonons scattered with electrons on temperature as, ℏ𝜔 = ℏ𝜔0 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='412 × 10−3(𝑇 − 𝑇0), where ℏ𝜔0 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='2meV, 𝑇0 = 400K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The calculated results using linear dependence are in good agreement with the experimental values, as shown by the blue solid line in Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The Seebeck coefficient is mainly measured by the average energy magnitude of carriers, which is related to the density of states near the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' And the effect of electron-phonon scattering for the average carrier energy can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Therefore, the Seebeck co- efficients fitted by the three different methods are essentially comparable, as shown in the Figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' TE properties of Ce3Te4 Considering that the local state electrons do not contribute to the electron transport, we only consider the contribution of the five lowest conduction bands and the two highest valence bands to the TE transport properties of Ce3Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Figure 5 depicts the calculated electrical conductivity and Seebeck coefficient ver- sus temperature based on the BTE under the RTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The fitting parameters used in the calculations are shown in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' We can find that the calculated values are consistent with the experimental data, which indicates that the fitting parameters are chosen reasonably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' In addition, it shows a decrease in elec- trical conductivity and an increase of the Seebeck coefficient with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' This is mainly due to the increase of carrier scattering intensity at higher temperatures and the increase of the average energy carried by carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 200 220 240 260 14 16 18 20 22 24 |S| (mV/K) T (K) theory exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 0 1 2 3 4 5 s (105 S/m) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Calculated Seebeck coefficient (a) and electrical conductivity (b) of Ce3Te4 as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The experimental data are extracted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Fitting parameters used to calculate the transport coeffi- cients in Ce3Te4 at 400K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Parameters Fitted value carriers concentration (1021cm−3) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='6[11, 13] mass density (g cm−3) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='12[23] optical phonon energy (meV) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='2 deformation potential constant (eV) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='1 high-frequency permittivity (𝜖0) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='7 static permittivity (𝜖0) 27 impurity density (1019cm−3) 5 [11] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Comparison of TE properties of La3Te4 and Ce3Te4 At room temperature, experimental and theoretical reports have shown that the Ce3Te4 has similar TE transport properties as La3Te4, such as electrical conductivity, Seebeck coefficient, and power factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [7, 11] We can obtain the same conclusion by numerical simulation using BTE under the RTA as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Since the effective mass of Ce3Te4 is larger than that of La3Te4, the electrical conductivity of Ce3Te4 is reasonably smaller than that of La3Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' However, at low temperatures, the electrical conductivity of La3Te4 and Ce3Te4 are approxi- mately equal, which indicates that the average relaxation time of Ce3Te4 is larger than that of La3Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The effect of scatter- ing will be greater with increasing of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' In the high temperature region, more phonon modes are excited, which leads to a rise in the number of phonons and allows phonons to participate in TE transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' At this point, electron-phonon scattering, which consists of the polar optical phonons, the de- formed acoustic phonons, and the deformed optical phonons, dominates TE transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' As shown in Figure 6(a) and (b), the Seebeck coefficient of Ce3Te4 is comparable to that of La3Te4 at high temperatures, but its electrical conductivity is higher than that of La3Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' For example, the electrical conductiv- ity of Ce3Te4 and La3Te4 at 𝑇 = 1000K are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='07×105 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='942×105 S/m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' This is mainly due to the dif- ference in the optical branch phonon energy of A2 mode of La3Te4 and Ce3Te4, as shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' From the phonon spectrum, we find that the optical branch phonon energy of A2 mode of Ce3Te4 is slightly larger than that of La3Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' The magnitude of the optical branch phonon energy will determine the magnitude of relaxation time of the electron-deformation optical phonon scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' A larger energy will correspond to a larger relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' In the case of high temperatures, electron-phonon scattering plays a dominant role, which has increased the total relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Therefore, the electrical conductivity of Ce3Te4 is larger than that of La3Te4 under other equal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Moreover, the power factor of Ce3Te4 is also better than that of La3Te4 at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Optimal carrier concentration of Ce3Te4 at various temperatures To further investigate the TE properties of Ce3Te4 at high temperatures, we will consider the TE transport properties of 6 0 1 2 3 4 100 80 60 40 20 0 200 400 600 800 1000 0 2 4 6 8 10 (c) (b) s (105 S/m) � Ce3Te4 � La3Te4 (a) S (mV/K) PF (mW/cm×K2) T(K) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Comparison of TE properties of La3Te4 and Ce3Te4: Elec- trical conductivity (a), Seebeck coefficient (b), and Power factor (c) as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Ce3Te4 as a function of carrier concentration with various high temperatures, as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Due to 𝜎 ∝ 𝑛𝑑 and 𝜎 ∝ 𝜏𝑡𝑜𝑡 ∝ 1/𝑇, the electrical conductivity will be enhanced when the carrier concentration increases or the temperature decreases as shown in Figure 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Due to the coupling rela- tionship between 𝜎 and 𝑆, the Seebeck coefficient changes in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' However, it can be seen from Figure 7(b) that the trend of Seebeck variation becomes gradually smoother as the carrier concentration increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' This is be- cause the bipolar effect will be more pronounced at high tem- peratures by exciting the intrinsic carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' In particular, fora lower carrier concentration, the bipolar effect has a greater im- pact on the Seebeck coefficient, which can be found to decrease at a greater rate than that at higher carrier concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Fig- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='6 400 300 200 100 0 0 1 2 3 4 5 0 5 10 15 (c) (b) s (105 S/m) � 600K � 800K � 1000K � 1200K (a) S (mV/K) PF (mW/cm×K2) nd(1021cm-3) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Calculated TE properties of Ce3Te4: Electrical conductivity (a), Seebeck coefficient (b), and Power factor (c) as a function of carrier concentration at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' ure 7(c) shows the dependence of the power factor on the carrier concentration at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' We found that the power factor corresponding to the higher temperature is smaller as the carrier concentration is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Because of the inhibitory effect of minority carriers on the TE properties, al- though the intrinsic excitation increases the minority carrier concentration and the electrical conductivity, the presence of the minority carriers can cause a decrease in the Seebeck co- efficient, which is more than compensate for the increase in the Seebeck coefficient caused by the increase in conductiv- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' As the carrier concentration increases, the effect of the bipolar diffusion effect on the TE performance generated by the minority carriers diminishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' In other words, the bipolar 7 effect can be suppressed via the heavily doped method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' In ad- dition, the optimal carrier concentration of Ce3Te4 varies for different temperatures, and the corresponding power factors are also different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' For example, the optimal carrier concentra- tion is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='5 × 1021cm−3 with the peak power factor 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='02 𝜇Wcm−1K−2 at 𝑇 = 600K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' the optimal carrier concentration is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='8 × 1021cm−3 with the peak power factor 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='42 𝜇Wcm−1K−2 at 𝑇 = 800K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' the optimal carrier concentration is around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='2 × 1021cm−3 with the peak power factor 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='78 𝜇Wcm−1K−2 at 𝑇 = 1000K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' And the optimal carrier concen- tration is around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='6 × 1021cm−3 with the peak power factor 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content='07 𝜇Wcm−1K−2 at 𝑇 = 1200K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' This is because the carrier concentration of the intrinsic excitation is strongly correlated with the temperature, and as the temperature increases, the optimal carrier concentration shifts upward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' CONCLUSION We have incorporated the multiband Boltzmann transport equations with first-principles calculations on electronic band structures in order to theoretically investigate TE properties of Th3Te4 materials such as La3Te4 and Ce3Te4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Our theoreti- cal calculations are in good agreement with the experimental data with calculated parameters and several other fitting pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' Theoretical results show that for the TE transport properties at high temperatures, a linear dependence is more consistent with the experimental results than the constant opti- cal branch phonon energy describing the electron-deformation optical branch phonon scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' In addition, we predict the TE transport properties of Ce3Te4 at high temperatures and the optimal carrier concentration at different temperatures, which is a guideline for experimental aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by National Key R&D Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 2017YFB0406004), National Natural Sci- ence Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 11890703), Key-Area Re- search and Development Program of Guangdong Province (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' 2020B010190004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} +page_content=' [1] L.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E0T4oBgHgl3EQf6gID/content/2301.02763v1.pdf'} diff --git a/hdE4T4oBgHgl3EQfrg0j/content/tmp_files/2301.05208v1.pdf.txt b/hdE4T4oBgHgl3EQfrg0j/content/tmp_files/2301.05208v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1f4f847c60a5b75bdc1d6a3b6073838126f9faf2 --- /dev/null +++ b/hdE4T4oBgHgl3EQfrg0j/content/tmp_files/2301.05208v1.pdf.txt @@ -0,0 +1,1696 @@ +arXiv:2301.05208v1 [math.PR] 12 Jan 2023 +Biased random walk on dynamical percolation +Sebastian Andres1 +Nina Gantert2 +Dominik Schmid3 +Perla Sousi4 +January 13, 2023 +Abstract +We study biased random walks on dynamical percolation on Zd. We establish a law of large +numbers and an invariance principle for the random walk using regeneration times. Moreover, +we verify that the Einstein relation holds, and we investigate the speed of the walk as a func- +tion of the bias. While for d = 1 the speed is increasing, we show that in general this fails in +dimension d ≥ 2. As our main result, we establish two regimes of parameters, separated by an +explicit critical curve, such that the speed is either eventually strictly increasing or eventually +strictly decreasing. This is in sharp contrast to the biased random walk on a static supercritical +percolation cluster, where the speed is known to be eventually zero. +Keywords and phrases. Dynamical percolation, biased random walk, regeneration times. +MSC 2020 subject classifications. Primary 60K35, 60K37. +1 +Introduction +In this paper, we introduce and study biased random walks in dynamically evolving environments. +The model of random walks on dynamical percolation was introduced in [23] by Peres, Stauffer and +Steif, and has the following description. +Fix a locally finite graph G = (V, E) and an initial state η ∈ {0, 1}E(G) of the edges. We say +that an edge e is open at time t if ηt(e) = 1, and closed otherwise. +For parameters µ ≥ 0 +and p ∈ [0, 1], we consider the dynamics (ηt)t≥0 with η0 = η, where each edge e in the graph is +assigned an independent Poisson process of rate µ. If there is a point of the Poisson process at +time t, we refresh the state of e in ηt, i.e. we declare e open with probability p and closed with +probability 1 − p, independently of all other edges and previous states of e. +From now on, we focus on the case where the underlying graph is Zd with d ≥ 1. We define a +continuous-time random walk (Xt)t≥0 in the environment (ηt)t≥0 with bias parameter λ > 0 as +follows: set X0 = 0 and assign a rate 1 Poisson clock to the particle. We also set for λ > 0 +Zλ := eλ + e−λ + 2d − 2. +(1.1) +Whenever the clock rings at time t and the random walker is currently at a site x, we choose one +of the neighbours y of x with probability +p(x, x ± ei) = 1 +Zλ +for i ∈ {2, . . . , d}, +1University of Manchester, United Kingdom, sebastian.andres@manchester.ac.uk +2Technical University of Munich, Germany, nina.gantert@tum.de +3University of Bonn, Germany, d.schmid@uni-bonn.de +4University of Cambridge, United Kingdom, p.sousi@statslab.cam.ac.uk +1 + +p(x, x + e1) = eλ +Zλ +, +p(x, x − e1) = e−λ +Zλ +. +If ηt({x, y}) = 1, the random walker moves from x to y, and it stays at x, otherwise. We will call +the process (Xt, ηt)t≥0 a λ-biased random walk on dynamical percolation with parameters µ and p. +Note that (ηt)t≥0 and (Xt, ηt)t≥0 are Markov processes, while (Xt)t≥0 is not. Moreover, (ηt)t≥0 +has the Bernoulli-p-product measure πp on {0, 1}E(Zd) as its unique invariant distribution, and we +assume in the following that η = η0 ∼ πp. +In this paper our focus is on the speed of the first coordinate of the walk as a function of the bias. +The motivation to study this question comes from the two different regimes one observes in the +case of a biased random walk on a static percolation cluster. It was first shown in [8] and [28] that +when p > pc and X is a λ-biased random walk on the infinite percolation cluster, then there exist +λ1 < λ2 so that when λ > λ2, the speed is 0, while for λ < λ1, the speed is strictly positive. A +few years later it was proved by [16] that there is a sharp transition, i.e. there exists λ∗ so that +for all λ > λ∗ the speed is equal to 0, while for λ < λ∗ the speed is strictly positive. Motivated +by these results, in this paper we study the speed in the dynamical setting and we establish that +for all choices of the parameters, the speed is always strictly positive and it satisfies an Einstein +relation as we show in Theorem 1.2 below. Our second main result concerns the monotonicity of +the speed as a function of the bias in dimensions d ≥ 2, where we observe two different regimes. +Before stating our results we recall an invariance principle established in [23, Theorem 3.1] in the +unbiased case. Unless otherwise stated, our probability measure is taking averages not only over +the walk but over the environment as well. +Theorem 1.1 ([23, Theorem 3.1]). For d ≥ 1, µ > 0, p ∈ (0, 1) and λ = 0, there exists σ ∈ (0, ∞) +so that +�Xkt +√ +k +� +t∈[0,1] +(d) +→ (σBt)t∈[0,1] +in D[0, 1] as k → ∞, where (Bt)t≥0 is a standard Brownian motion. +We now present our first result on the speed of the biased random walk (Xt)t≥0 for fixed environment +parameters µ > 0 and p ∈ (0, 1). +Theorem 1.2. Let d ≥ 1 and let (Xt, ηt)t≥0 be a λ-biased random walk on dynamical percolation +on Zd with parameters µ > 0 and p ∈ (0, 1). Then for all λ > 0, there exists v(λ) = vµ,p(λ) such +that almost surely +lim +t→∞ +Xt +t = (v(λ), 0, . . . , 0) . +(1.2) +Moreover, the function λ �→ v(λ) is strictly positive for all λ > 0, continuously differentiable and +satisfies +lim +λ→0 v′(λ) = σ2, +where σ is as in Theorem 1.1. +The last statement in the theorem above is known as Einstein relation. Moreover, as we will see +in Proposition 3.2, an invariance principle also holds in the biased case and the proof follows along +the same lines as the proof of Theorem 3.1 in [23]. +2 + +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +Eventually +Eventually +monotone decreasing +monotone increasing +p +µ +Figure 1: Plot of the different regimes in Theorem 1.3 for large λ. +When d = 1, using the obvious coupling between two walks with different bias parameters, it is +immediate to see that the speed is always monotone increasing in the bias and in fact in Section 4.1 +we also establish that in d = 1 the speed is strictly increasing as a function of the bias. It is thus +natural to ask what happens for d ≥ 2. While the speed turns out to be monotone increasing +in λ > 0 for certain regimes of µ > 0 and p ∈ (0, 1) in dimensions d ≥ 2 as we show in Section 5, +our main result is an explicit criterion deciding whether the speed is eventually strictly increasing +or decreasing. +Theorem 1.3 (Monotonicity of the speed for d ≥ 2). Consider the biased random walk on dynam- +ical percolation on Zd for d ≥ 2. For all p ∈ (0, 1) and µ > 0, there exists some λ0 = λ0(µ, d) such +that the following hold. +(1) The speed v(λ) is strictly increasing for all λ ≥ λ0 provided that µ2 > p(1 − p). +(2) The speed v(λ) is strictly decreasing for all λ ≥ λ0 provided that µ2 < p(1 − p). +Remark 1.4. Note that this is in contrast to the biased random walk on a static super-critical +percolation cluster, where the speed is known to be zero for large values of λ; see [8, 16, 28]. The +criterion for the eventual monotonicity of the speed, identified in Theorem 1.3, is visualised in +Figure 1. Moreover, this suggests different shapes of the speed functions; see Figure 2. +1.1 +Related work +Biased random walks in random media were investigated intensively over the last years, we refer +to [1, 2, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 20, 28] for a non-exhaustive list and to [5] for a +survey. The most prominent examples are biased random walks on Galton-Watson trees and biased +random walks on supercritical percolation clusters, see [20] and [8, 28] respectively. Our work is +in particular motivated by the study of biased random walks on (static) supercritical percolation +clusters. This model was introduced in [4]. Due to traps in the cluster, the speed of the walk is zero +for large values of the bias. Simulations show that the speed is a unimodal function of the bias, +first increasing until the maximum is achieved and then decreasing and eventually becoming zero. +It was proved in [28] and [8] that for small values of the bias, the speed is strictly positive. Finally, +in the breakthrough paper [16], it was shown that there is a critical value separating the positive +speed regime from the zero speed regime. In the case of Galton-Watson trees with leaves, this phase +transition was known earlier and there is an explicit formula for the critical value, see [20]. When +3 + +there are no “hard traps”, the speed of the biased walk should be strictly positive, and one may +ask if it is increasing as a function of the bias. In the case of a Galton-Watson tree without leaves, +it is conjectured that the speed is indeed an increasing function of the bias. While this conjecture +still remains open, it was proved that the speed is eventually increasing, see [6] and [1]. The same +argument as in [6] gives that for biased random walk among uniformly elliptic i.i.d. conductances, +the speed is eventually increasing as a function of the bias. However, it was shown in [7] that +for some laws of the conductances, the speed is not increasing for all values of the bias, i.e. there +exist λ1 < λ2 such that v(λ1) > v(λ2). +In the presence of hard traps, a central limit theorem is expected to hold for small values of the +bias, in a strict subset of the positive speed regime. This was proved for biased random walk on +supercritical percolation clusters in [28] and [8] and for random walks on Galton-Watson trees with +leaves in [20]. For other models such results have been established for example in [18] and [11]. +In an environment without hard traps, there are examples where a central limit theorem holds for +all values of the bias, see for instance [24, 25]. In our case, this also turns out to be true, see +Proposition 3.2. +The literature on (unbiased) random walks in time-dependent random environments is too vast +to give a review, we just point to two papers which are relevant in our setup, namely [3, 10], see +also the references therein. Unbiased random walks on dynamical percolation have been studied in +particular in terms of their mixing times, see [19, 21, 22, 23, 26]. +1.2 +Overview of proof ideas +The proof of the first part of Theorem 1.2 follows by using a suitably defined sequence of regen- +eration times (τi), i.e. a sequence of random times such that the evolution of the walk and the +environment between [τi, τi+1] is independent for different choices of i. A key property of the re- +generation times that we define is that their distribution only depends on the parameter µ, but +not on the bias parameter λ and the percolation parameter p. Using these regeneration times and +a law of large numbers we get the existence of the speed. Moreover, we find an expression for the +speed in terms of an infinite series which allows to give a simple expression for its derivative as we +show in Lemma 3.4 and the Einstein relation. In particular, the speed is strictly increasing for all λ +sufficiently small. +We now give an overview of the main ideas behind the proof of our main result, Theorem 1.3, giving a +necessary and sufficient condition for the speed to be eventually in λ strictly increasing or decreasing. +There are two main ingredients. First, in Proposition 4.1, we obtain an asymptotic expression for +the speed which is valid for all bias parameters λ sufficiently large. Next, in Lemma 4.3, we give an +asymptotic bound on the derivative of the speed for large λ. The proof of Theorem 1.3 is a direct +consequence of these two results. +In order to prove Proposition 4.1, we start with a detailed analysis of the speed in the one dimen- +sional case. In order to analyse the case d ≥ 2, we rely on the regeneration times and compare the +first coordinate of the walk with a time-changed one dimensional walk in a suitably defined evolving +environment. To be more precise, we construct a coupling which keeps the first coordinates of the +two walks together until the second time that the d dimensional walk jumps in a direction other +than e1. +In order to prove Lemma 4.3 we rely on a comparison between walkers with different bias parameters +using marked Poisson point processes. A key task is to develop an asymptotic expression for the +derivative of the speed on the scale e−λ, with the constants only depending on µ and p. +4 + +λ +v(λ) +λ +v(λ) +λ +v(λ) +Figure 2: The different pictures show three possible shapes for v(λ) which are in accordance with +Theorem 1.3. +1.3 +Organisation +In Section 2 we define our sequence of regeneration times that will be used in the rest of the paper. +In Section 3 we prove Theorem 1.2. In Section 4 we prove Theorem 1.3 and we also study the +one dimensional case in Section 4.1 where we establish the strict monotonicity of the speed for all +values of λ. Finally, in Section 5 we prove that the speed in d ≥ 2 is strictly increasing when µ or p +are sufficiently large. +2 +Regeneration times for the biased random walk +Throughout the paper we write Pp,µ for the probability measure corresponding to dynamical per- +colation with parameters µ and p. For λ ≥ 0, we write Pλ for the semi-direct product of Pp,µ with +the law of the λ-biased random walk starting at the origin. We write Eλ for the expectation with +respect to Pλ. +In order to prove Theorem 1.2 we need to define a sequence of regeneration times for the random +walk on dynamical percolation. +A similar definition of regeneration times was used by Peres, +Stauffer and Steif in [23]. Here we use the definition given in [19] which works for general underlying +graphs and was used in [19] to compare mixing and hitting times for random walks on dynamical +percolation in terms of the respective quantities for the static graph. For the biased random walk +on Zd, we have the following construction following [19, Section 3]. +We fix an enumeration (ei)i∈N of the edges in E according to an arbitrary rule. Then for each +edge ei, we create an infinite number of copies denoted ei,1, ei,2, . . . . +We now define a process +(It)t≥0, where for every t ≥ 0, It is a subset of edges and copies of edges that we refer to as the +infected set. Let I0 = ∅. Suppose that for some t ≥ 0, the Poisson clock associated to the random +walk (Xt)t≥0 rings and that the walker examines the edge ei for some i ∈ N. If no copy of ei is +contained in It−, we set +It := It− ∪ {ei,1} . +(2.1) +Otherwise, we add to It the copy ei,j of ei with the smallest index j such that ei,j /∈ It−. +Next, for all t ≥ 0, we assign the lexicographic ordering ⪯ to the edges in It using the ordering of +the edges of E, i.e. for ei,j, ek,l ∈ It we have +ei,j ⪯ ek,l +⇔ +(i ≤ k) ∨ ((i = k) ∧ (j ≤ l)) +(2.2) +Further, let (Nt)t≥0 be a Poisson process with time dependent intensity µ|It|. Whenever a clock of +this process rings at time t, we choose an index uniformly at random from {1, . . . , |It|} and remove +5 + +Xt +Xt +Figure 3: Visualisation of the dynamical percolation cluster and the infected set of edges It. The +edges in red on the left correspond to open edges. In the right picture, blue edges correspond to +closed edges, which are infected at time t, red edges correspond to open edges that are in It and +dotted red edges correspond to open edges that are not in It. +the copy of the edge with this index in It according to the ordering ⪯. Moreover, if the picked edge +is of the form ei,1 for some i ∈ N, we refresh the state of the edge ei in the environment ηt, i.e. we +set ηt(ei) = 1 with probability p, and ηt(ei) = 0, otherwise. +For all edges ej with ej,1 /∈ It, we use independent rate µ Poisson clocks to determine when the +state of the edge in (ηt)t≥0 is refreshed. Note that with this construction (Xt, ηt)t≥0 has indeed the +correct transition rates, see Figure 3 for an illustration. +Recall that we start from η0 ∼ πp, X0 = 0, and that we set I0 = ∅. Let τ0 := 0. For every i ∈ N, +we set +τi+1 := inf{t > τi : It = ∅ and It′ ̸= ∅ for some t′ ∈ (τi, t)} . +(2.3) +Let N0 := N ∪ {0} and note that the times (τi)i∈N0 are indeed regeneration times for the process +(Xt)t≥0, i.e. (τi −τi−1)i∈N are i.i.d. and the random walk increments (Xτi −Xτi−1)i∈N are i.i.d. Note +also that the law of (τi)i∈N0 only depends on the parameter µ and not on λ or p. +Observe that the process (|It|)t≥0 is a continuous-time birth-and-death chain on N0 with transition +rates q(i − 1, i) = 1 and q(i, i − 1) = µi for all i ∈ N. The following lemma is the content of [19, +Lemma 3.5]. +Lemma 2.1. For all p ∈ (0, 1) and µ, λ > 0, the increments (τi−τi−1)i∈N are i.i.d., have exponential +tails and satisfy Eλ[τ1] = e1/µ. +In particular, note that the law of the regeneration times does not depend on λ > 0. Therefore, +with a slight abuse of notation, we will write P instead of Pλ when considering events involving +only the regeneration times. For every t ≥ 0 we let U(t) be the number of attempted jumps of +the walker X up to time t, which follows the Poisson distribution with parameter t. We have the +following result on U(τ1). +Lemma 2.2. For every µ > 0 and p ∈ (0, 1), there exists a positive constant cµ satisfying cµ → ∞ +as µ → ∞ so that for all m ≥ 2 +P(U(τ1) ≥ m) ≤ e−cµm. +Proof. The fact that the random variable U(τ1) has exponential tails is an immediate consequence +of Lemma 2.1 and the exponential concentration of a Poisson random variable around its mean. It +6 + +remains to show that we can choose cµ such that cµ → ∞ as µ → ∞. Let µ > 1. Recall the birth +and death chain (|It|)t≥0, and let (Sk)k≥0 with S0 = 0 denote its jump chain. Further, let +˜τ0 := inf{n ≥ 1: Sn = 0} +(2.4) +be first return time of (Sk)k≥0 to the origin and observe that 2U(τ1) = ˜τ0. Note that the process +(|It|)t≥0 is dominated from above by a biased random walk on {0, 1, . . . } with transition rates +q(i, i − 1) = µ and q(i − 1, i) = 1 for all i ∈ N. Hence, we get for all θ > 0 that the process (Mk)k∈N +defined by +Mk := eθSk · f(θ)k−1 +with f(θ) := (µ + 1)/(eθ + e−θµ) is a super-martingale. Since almost surely M1 = eθ, and ˜τ0 +has exponential tails, we can apply the optional stopping theorem together with Fatou’s lemma to +obtain +eθ = E[M1] ≥ E[M˜τ0] = E +� +exp(θS˜τ0)f(θ)˜τ0−1� += E +� +(f(θ))˜τ0−1� +for all θ > 0. Take θ = log log µ for µ > 0 sufficiently large such that f(θ) ≥ 1. Then, we get that +for all m ≥ 2 by Markov’s inequality +P(U(τ1) ≥ m) = P(˜τ0 − 1 ≥ 2m − 1) ≤ E +� +(f(θ))˜τ0−1� +f(θ)2m−1 +≤ eθf(θ)−(2m−1). +(2.5) +Since f(θ) ≥ 1 +2 log µ for all µ > 0 sufficiently large, we conclude. +3 +Speed and Einstein relation +We will now show Theorem 1.2 in two steps. First, we prove in Proposition 3.1 a law of large +numbers for the biased random walk on dynamical percolation and in Proposition 3.2 we prove an +invariance principle. Both proofs use similar arguments to [23, Theorem 3.1]. In Proposition 3.5 +we show that the speed in the e1-direction is strictly positive. +Proposition 3.1. Recall the sequence of regeneration times (τi)i∈N from (2.3). Then, Pλ-almost +surely, +lim +t→∞ +Xt +t = (v(λ), 0, . . . , 0) = Eλ[Xτ1] +E[τ1] +. +(3.1) +Proof. We first show that there exists a positive constant C (depending on µ) so that for all λ > 0, +Eλ[∥Xτ1∥1] ≤ C. +(3.2) +Recall that for every t ≥ 0 we write U(t) for the total number of times that an edge was added to +the infected set during the time interval [0, t]. Then +∥Xτ1∥1 ≤ U(τ1). +By Lemma 2.2 we get that U(τ1) has exponential tails, and hence this proves (3.2). +Since the increments (Xτi − Xτi−1) are i.i.d. and (3.2) holds, we can apply the strong law of large +numbers to obtain that almost surely +lim +k→∞ +Xτk +k += lim +k→∞ +1 +k +k +� +i=1 +(Xτi − Xτi−1) = Eλ[Xτ1] . +(3.3) +7 + +To prove a law of large numbers for (Xt)t≥0, for every t ≥ 0 writing k = k(t) = ⌊t/E[τ1]⌋ we get +Xt +t = Xt − XkE[τ1] +t ++ XkE[τ1] − Xτk +t ++ Xτk +t . +It now follows that the first fraction on the right hand side above converges to 0 as t → ∞ almost +surely. Using that τk/k → E[τ1] as k → ∞ almost surely, it follows easily that the second fraction +on the right hand side above converges to 0 almost surely. Finally using (3.3) we get that the third +fraction converges to Eλ[Xτ1] /E[τ1] almost surely as t → ∞ and this concludes the proof. +At this point, let us state some consequences of Proposition 3.1. +The next proposition follows in exactly the same way as the proof of Theorem 3.1 in [23] when +λ = 0. The only difference from [23] is the definition of the regeneration times, but the way they +are used for the invariance principle is the same as in [23]. We give a sketch of the proof, as we +need the expression for the diffusivity of the Brownian motion in order to establish the Einstein +relation in Proposition 3.5. +Proposition 3.2. The biased random walk (Xt)t≥0 = ((X1 +t , . . . , Xd +t ))t≥0 on dynamical percolation +satisfies an invariance principle +�X1 +kt − v(λ)kt +√ +k +� +t∈[0,1] +(d) +→ (σBt)t∈[0,1] +(3.4) +in D[0, 1] as k → ∞, where (Bt)t≥0 denotes a standard Brownian motion and σ2 = σ2(d, µ, p, λ) = +Varλ(X1 +τ1)(E[τ1])−1. +Sketch of the proof. Since the arguments are similar to the ones in Theorem 3.1 in [23], we will +only outline the key steps of the proof. As Eλ[τ 2 +1 ] < ∞, by a similar tightness argument as in +Theorem 4.1 of [27], it suffices to consider the convergence in (3.4) only for t = 1. Since (τn)n∈N is +a sequence of regeneration times, we have that as n → ∞, +X1 +τn − v(λ)nE[τ1] +� +n Var(X1τ1) +(d) +→ N +(3.5) +where N is standard normal random variable. Next, we define +ℓ(k) := max{ℓ ∈ N: τℓ ≤ k} +(3.6) +to be the index of the last regeneration time before k. We write now +X1 +k − v(λ)k +√ +k += +X1 +k − X1 +τℓ(k) +√ +k ++ +X1 +τℓ(k) − v(λ)τℓ(k) +√ +k ++ v(λ)(τℓ(k) − k) +√ +k +. +(3.7) +For all 0 < s < t we write U[s, t] for the number of edges added to the infected set between times +s and t. Note that U[s, t] is a Poisson random variable of parameter t − s. Then +|X1 +k − X1 +τℓ(k)| ≤ U[τℓ(k), τℓ(k)+1] . +(3.8) +Recall from Lemma 2.1 that the regeneration times have exponential tails. Since +k − τℓ(k) +√ +k +(d) +→ 0 +and +τℓ(k)+1 − τℓ(k) +√ +k +(d) +→ 0 +for k → ∞ +(3.9) +as the numerator of the second fraction converges to the size-biased distribution of τ1, we see +from (3.8) and (3.9) that the first and third term on the right-hand side of (3.7) converge to 0 in +probability, and by using (3.5) for the second term we conclude. +8 + +Remark 3.3. Since the above proof works for all λ ≥ 0, it follows that the diffusivity σ2 for +λ = 0 is the same as in Theorem 1.1. Moreover, note that we only prove an annealed invariance +principle in Proposition 3.2, but conjecture that also a quenched invariance principle holds; see [3] +for sufficiently large values of µ > 0. +Recall from Proposition 3.1 that the speed v(λ) is equal to Eλ[Xτ1]/E[τ1] and E[τ1] does not depend +on λ. Moreover, by considering the Radon-Nikodym derivative of the law of a λ-biased random +walk path with respect to the unbiased one, we obtain +Eλ[X1 +τ1] = E0 +� +X1 +τ1eλX1 +τ1 +� 2d +Zλ +�U(τ1)� +=: f(λ). +For every t ≥ 0 we let Rt, respectively Lt, be the number of steps to the right, respectively to the +left, that X1 performs by time t. Setting R = Rτ1 and L = Lτ1 we can write +f(λ) = +� +m∈N +� +k+ℓ≤m +(k − ℓ)eλ(k−ℓ) +� 2d +Zλ +�m +P0((R, L) = (k, ℓ), U(τ1) = m), +(3.10) +where P0 denotes the law of the symmetric simple random walk on dynamical percolation. +Lemma 3.4. Let µ > 0 and p ∈ (0, 1). Then the speed v(λ) is continuously differentiable in λ > 0 +and the derivative satisfies +v′(λ) = +1 +E[τ1] · +� +Eλ +� +(X1 +τ1)2� +− eλ − e−λ +Zλ +· Eλ +� +X1 +τ1 · U(τ1) +�� +. +In particular, we have +lim +λ→0 v′(λ) = σ2, +where σ2 is the variance from Theorem 1.1. +Proof. We first prove that for all λ > 0, +lim +δ→0 +f(λ + δ) − f(λ) +δ += +� +m∈N +� +k+ℓ≤m +(k − ℓ)eλ(k−ℓ) +� 2d +Zλ +�m � +k − ℓ − m · Z′ +λ +Zλ +� +P0((R, L) = (k, ℓ), U(τ1) = m), +(3.11) +where Z′ +λ := eλ−e−λ. Note that the sum appearing above divided by E[τ1] is equal to the expression +for the derivative given in the statement of the lemma. A direct calculation shows that +f(λ + δ) − f(λ) +δ += +� +m∈N +� +k+ℓ≤m +(k − ℓ)eλ(k−ℓ) +� 2d +Zλ +�m +g(δ) · P0((R, L) = (k, ℓ), U(τ1) = m), +where the function g is defined via +g(δ) = eδ(k−ℓ)Zm +λ − Zm +λ+δ +δZm +λ+δ +. +There exists a positive constant c = cd so that for all λ and δ we have +1 ≥ +Zλ +Zλ+δ +≥ 1 − cδ. +9 + +By taking δ < 1/c, and considering whether δ < 1/m or δ ≥ 1/m, we see that there is a positive +constant C = Cd so that +|g(δ)| ≤ Cm. +Therefore, we obtain that uniformly for δ < 1/c, +���� +f(λ + δ) − f(λ) +δ +���� ≤ +� +m∈N +� +k+ℓ≤m +Cm2eλ(k−ℓ) +� 2d +Zλ +�m +P0((R, L) = (k, ℓ), U(τ1) = m) += C · Eλ +� +(U(τ1))2� +< ∞, +where for the last bound we used Lemma 2.2, since the distribution of U(τ1) is independent of λ. We +can thus apply the dominated convergence theorem which allows us to differentiate the summands +in (3.10) with respect to λ to get (3.11). Applying the dominated convergence theorem again we +see that all the terms appearing in the expression for v′(λ) are continuous functions in λ, and hence +this finishes the proof. +Proposition 3.5. Fix p ∈ (0, 1) and µ > 0. Then the speed function λ �→ v(λ) is strictly positive +for all λ > 0. +Proof. To see that the speed v(λ) = vµ,p(λ) is strictly positive for all µ, λ > 0 and p ∈ (0, 1], +note that P0((R, L) = (k, ℓ), U(τ1) = m) = P0((R, L) = (ℓ, k), U(τ1) = m) for all (k, ℓ) ∈ Z2 by +symmetry of the random walk and the environment law. Thus, we can write +f(λ) = +� +m∈N +� +k+ℓ≤m +(k − ℓ)eλ(k−ℓ) +� 2d +Zλ +�m +P0((R, L) = (k, ℓ), U(τ1) = m) += +� +m∈N +� +k>ℓ +k+ℓ≤m +(k − ℓ) +� 2d +Zλ +�m � +eλ(k−ℓ) − e−λ(k−ℓ)� +P0((R, L) = (k, ℓ), U(τ1) = m). +(3.12) +Since all the terms of the sum in (3.12) are positive, we can infer a strictly positive speed using +that P0((R, L) = (2, 1)) > 0. +Proof of Theorem 1.2. Theorem 1.2 is now an immediate consequence of Propositions 3.1 and 3.5 +and Lemma 3.4. +4 +Monotonicity of the speed +In this section we prove Theorem 1.3. In order to do so, we first establish an asymptotic expression +for the speed that is valid for large values of the bias λ. We recall the definition from (1.1) of +Zλ = eλ + e−λ + 2d − 2. +Proposition 4.1. For d ≥ 1, let (X, η) be a λ-biased random walk on dynamical percolation on Zd +with parameters µ > 0 and p ∈ (0, 1). There exists some λ0 = λ0(µ, d) such that for all λ > λ0, +v(λ) = +µp +1 − p + µ − +(2d − 2)p +(1 − p + µ)2 (µ2 − p(1 − p))Z−1 +λ ++ O(e−2λ), +where the implicit constant in O depends on µ and d. +10 + +Remark 4.2. Note that the speed v(λ) converges to µp(1 − p + µ)−1 as λ → ∞ in agreement with +the 1-dimensional case as we will see in Proposition 4.5. +The above proposition proves the monotonicity of v(λ) along arithmetic progressions for large λ. +In order to prove Theorem 1.3 we also need to obtain a control on the derivative of v(λ) that is +valid for large values of λ. +Lemma 4.3. Let d ≥ 1. Then for all µ > 0 and p ∈ (0, 1) there exists λ0 = λ0(µ) and positive +constants cµ, Cµ,p so that for all λ ≥ λ0 we have +��v′(λ) − Cµ,p exp(−λ) +�� ≤ cµ exp(−2λ) . +(4.1) +We now have all the tools needed in order to conclude the proof of Theorem 1.3. We defer the +proofs of Proposition 4.1 and Lemma 4.3 to Sections 4.2 and 4.3 respectively. +Proof of Theorem 1.3. Using Lemma 4.3, it suffices to study the constant Cµ,p in (4.1) and show +that Cµ,p < 0 when µ2 > p(1 − p) as well as Cµ,p > 0 when µ2 < p(1 − p). Since we know that the +speed is continuously differentiable by Theorem 1.2, we get that for all s > 0 large enough +v(2s) − v(s) = +� 2s +s +v′(t)dt = Cµ,p exp(−s) + O(exp(−2s)) . +(4.2) +Taking now s = λ sufficiently large, we get from Proposition 4.1 that +Cµ,p = +(2d − 2)p +(1 − p + µ)2 (µ2 − p(1 − p)) , +(4.3) +allowing us to conclude, since we get +v′(λ) = Cµ,pe−λ + O(e−2λ), +and hence the sign of v′(λ) agrees with the sign of Cµ,p for all λ sufficiently large. +4.1 +Speed for d = 1 +In this section we focus on dimension 1, where one can use the obvious coupling between two +random walks with different biases to obtain that the speed is increasing as a function of the bias. +We investigate the limiting speed and the rate of convergence to the limit for large λ. +In the following proposition we establish strict monotonicity as well as an explicit form for the +speed in the totally asymmetric biased random walk case, where the random walk only attempts +jumps to the right. +Lemma 4.4. Let (X, η) be a totally asymmetric biased random walk in dynamical percolation on +Z with parameters µ and p that jumps to the right at rate 1. Then the speed v satisfies +v = +µp +1 − p + µ. +Proof. Suppose X0 = 0 and η0 ∼ π. Let S be the first time that X jumps along the edge e = {0, 1}. +Then v = E[S]−1. To compute E[S] we describe a way to realise the evolution of the state of e +over time. Let (ξi)i∈N be i.i.d. Bernoulli-p-distributed random variables. Let (Ti)i∈N and (Sj)j∈N +11 + +denote the rate µ and rate 1 Poisson clocks that determine when the edge e is updated and when +the random walk attempts a move to the right, respectively. Taking T0 := 0, we assign to each +interval [Ti−1, Ti] the label ξi. If ξi = 1, the edge e is open and otherwise it is closed. We let +Z := inf {j ∈ N: Sj ∈ [Ti−1, Ti] for some i ∈ N with ξi = 1} . +(4.4) +In particular, we have S = SZ. By the memoryless property of the exponential distribution we get +P(Z = j|Z > j − 1) = +µp +µ + 1 +and +P(Z = 1) = p. +(4.5) +Therefore, for all j > 1, we have that +P(Z = j) = P(Z = j|Z > j − 1)(1 − p) +j−1 +� +i=2 +P(Z > i|Z > i − 1) += (1 − p) +� +1 − +µp +µ + 1 +�j−2 µp +µ + 1. +It then follows that +E[Z] = 1 − p + µ +µp +. +(4.6) +Let (χi)i∈N with χi = Si+1 − Si denote the inter-arrival times of the Poisson process (Sj)j∈N. +In particular, (χi)i∈N are i.i.d. Exponential-1-distributed random variables. Observe that Z is a +stopping time with respect to (χi)i∈N, taking the enlarged filtration which contains also the process +(Ti)i∈N which is independent of (χi)i∈N. Therefore, applying Wald’s identity and using (4.6), we +conclude +E[S] = E[χ1]E[Z] = 1 − p + µ +µp +and this finishes the proof. +Proposition 4.5 (Monotonicity and asymptotic speed for d = 1). Let (Xt, ηt)t≥0 be a biased +random walk in dynamical percolation on Z with parameters µ and p. Then the speed function v(λ) +from (1.2) is strictly increasing for all λ > 0 and satisfies +lim +λ→∞ vµ,p(λ) = +µp +1 − p + µ +(4.7) +for all choices of p ∈ (0, 1) and µ > 0. +Proof. We start by arguing that the speed is strictly increasing in λ > 0. We construct a coupling P +between a λ1-biased random walk (Xt, ηt)t≥0 and a λ2-biased random walk ( � +Xt, �ηt)t≥0 on dynamical +percolation on Z with 0 < λ1 < λ2. We take the same environment for both walks and we let them +attempt jumps in the following way: whenever the two random walks are at the same location, we +couple them by using the same exponential 1 clocks to determine the jump times and then moving +them both to the right with probability eλ1/(eλ1 + e−λ1), moving � +X to the right and X to the left +with probability eλ2/(eλ2 + e−λ2) − eλ1/(eλ1 + e−λ1) and moving them both to the left otherwise. +If the two walks are in different locations, we let them attempt jumps in the common environment +using independent exponential 1 clocks. +Recall the construction of the infected set from Section 2 and the definition of copies of edges. We +define the following modified infected set (It)t≥0, where for every t ≥ 0, It is a subset of edges +12 + +and copies of edges. Suppose that for some t ≥ 0, both random walks are at the same position. If +the two random walks examine the same edge ei for some i ∈ N and no copy of ei is contained in +It−, we set +It := It− ∪ {ei,1}. +Otherwise, we add to It the copy ei,j of ei with the smallest index j such that ei,j /∈ It−. If the +random walks are at the same position, but examine different edges, we add both edges or copies +of them with the smallest index as above to the modified infected set. When the two random walks +are at different positions, recall that according to the coupling, the two random walks perform +jumps according to independent exponential 1 clocks. Whenever an edge is examined by one of the +two random walks, we add this edge or a copy of it to the modified infected set. +Let (N t)t≥0 be a Poisson process with time dependent intensity µ|It|. Whenever a clock of this +process rings at time t, we choose an index uniformly at random from {1, . . . , |It|} and remove +the copy of the edge with this index in It according to the ordering ⪯ of edges from Section 2. If +the picked edge is of the form ei,1 for some i ∈ N, we also refresh the state of the edge ei in the +common environment ηt for the two walkers, i.e. we set ηt(ei) = 1 with probability p, and ηt(ei) = 0, +otherwise. For all edges ej with ej,1 /∈ It, we use independent rate µ Poisson clocks to determine +when the respective edge is updated in the environment for the two random walks. +Note that under this coupling P the biased random walks on dynamical percolation (Xt, ηt)t≥0 +and ( � +Xt, �ηt)t≥0 have marginally the correct law. We define +τ := inf{t > 0: It = ∅ and It′ ̸= ∅ for some t′ ∈ (0, t)}. +(4.8) +Since the process (|It|)t≥0 is dominated from above by a biased random walk on {0, 1, . . . } with +transition rates q(i, i − 1) = µi and q(i − 1, i + 1) = 2 for all i ∈ N, a similar argument as in +Lemma 2.1 (see also the proof of Lemma 2.2) shows that that the random variable τ has all finite +moments. Applying now the same arguments as in the proof of Proposition 3.1, we get that +v(λ1) = E[Xτ] +E[τ] +and +v(λ2) = E[ � +Xτ] +E[τ] , +(4.9) +where we write E for the expectation with respect to P. Note that P(Xτ ≤ � +Xτ) = 1. Considering +the event that both random walks jump into different directions, and the respective edges get +removed from the modified infected set before another jump occurs, we also get +P(Xτ < � +Xτ) ≥ +� +eλ2 +eλ2 + e−λ2 − +eλ1 +eλ1 + e−λ1 +� +· +� +µ +µ + 1 +�2 +> 0 +as λ1 < λ2. This immediately implies that v(λ1) < v(λ2), hence establishing strict monotonicity. +Next, we investigate the speed when λ → ∞. Since the P-probability for (Xt) to attempt a jump +in the −e1 direction until time τ goes to 0 as λ1 → ∞, and the speed is increasing in λ, +lim +λ→∞ v(λ) = ¯v, +(4.10) +where v denotes the speed of the totally biased random walk, i.e. whenever the clock associated to +the random walker at x rings, it attempts a jump to x + 1. Lemma 4.4 concludes the proof. +13 + +4.2 +Asymptotic expression for the speed +In this section we prove Proposition 4.1. In order to do so, we first construct a coupling between +(X, η) and a one-dimensional biased random walk Y in a suitably defined evolving environment ξ +on Z for which we can calculate an asymptotic expression for the speed using Lemma 4.4. +For x ∈ Z abusing notation we write x+e1 for the edge (x, x+1). Recall that Zλ = eλ+e−λ+2d−2. +Definition 4.6 (Coupling between (X, η) and (Y, ξ)). Let �µ = µ + p(2d − 2)Z−1 +λ . Both X and Y +start from 0. We let the environment η evolve according to dynamical percolation on Zd with +parameters µ, p and η0 ∼ πp. The edges to the left of 0 in the environment ξ update according to +dynamical percolation with parameters �µ, p. +Let P1, P2 and P3 be three independent Poisson processes of parameters eλZ−1 +λ , e−λZ−1 +λ +and (2d− +2)Z−1 +λ , respectively. At the points of P1 or P2 both X and Y attempt a jump to the right or +the left respectively and we add the corresponding edges (the lowest numbered copies not in the +infected sets as in Definition 2.3) to their respective infected sets and we say that the two edges +are a match. At the points of P3 the walk X attempts a jump in one of the 2d − 2 directions other +than e1 and −e1 chosen uniformly at random and we add the corresponding edge to the infected +set of X only. We now explain how to remove edges from the infected sets: we pick an edge from +the infected set of X to be removed in the same way as in Definition 2.3 (each edge is being picked +at rate µ) and we also remove its match if it exists from the infected set of Y . We then update +the corresponding edges in η and ξ in the same way as in Definition 2.3 (i.e. if the edges are of the +form ei,1 for some i). +Below whenever we say that we stop the coupling, afterwards we continue (X, η) and (Y, ξ) by +letting them attempt jumps at the points of P1, P2 and P3 (the latter only for X) and each edge +of Y in its infected set refreshes also at the points of an additional Poisson process �P of parameter +p(2d − 2)Z−1 +λ . If an edge of the infected set of Y refreshes according to this Poisson process, then +we do not remove it from the infected set. However, if that edge is of the form ei,1 for some i, then +we update its state in ξ. +Let (Ti) be the jump times of P3 and let S be the first point of P2. We stop the coupling at +time S ∧ T2. +For every edge e, we let E(e) be the first time that the state of e is examined +by Y and C(e) be the first time the edge e is crossed by Y . +When E(e) < T1 ∧ S, then for +times s ∈ [E(e), C(e) ∧ T1 ∧ S] we set ξs(e) = ηs(e). At time T1, the walk X attempts a jump in +one of the 2d − 2 directions other than e1 and −e1 chosen uniformly at random. For each edge e +such that E(e) ∈ (T1, T2 ∧ S) and for times t ∈ [E(e), C(e) ∧ T2 ∧ S] we set ξt(e) = ηt(Xt + e1). +During the time interval (C(e) ∧ T2 ∧ S, T2 ∧ S), we refresh the edge e in the environment ξ also +at the points of an additional independent Poisson process �P of parameter p(2d − 2)Z−1 +λ . Note as +mentioned above that these updates do not affect the infected set. We let (τi) be the successive +times at which the infected set of X becomes equal to the empty set. Then by the definition of the +process Y we see that also the infected set of Y becomes empty at times τi for all i. +Remark 4.7. We note that in the above coupling once an edge e has been examined by Y , it +then refreshes at rate �µ. Indeed, up until the first point of P3 it updates at rate µ. If the edge +that X examined at time T1 is open, which happens with probability p (and hence the rate of this +happening is p(2d − 2)Z−1 +λ ), since this is the first time that edge is being examined, then the state +of the edge e in ξ refreshes to XT1 +e1 which is distributed according to Ber(p) (again we are using +that the edge XT1 + e1 has not been examined before). Using the sequence (τi) we see that the +speed vY of Y is given by +vY (λ) = E[Yτ1] +E[τ1] . +14 + +Lemma 4.8. For all p ∈ (0, 1) and µ > 0, there exist constants λ0, c > 0 such that for all λ ≥ λ0, +��vY (λ) − v(λ) +�� ≤ c exp(−2λ). +(4.11) +Proof. Recall that S is the first point of P2 and (Ti) are the points of P3. Let A be the event that +the coupling stops before time τ1, i.e. +A = {S < τ1} ∪ {T2 < τ1}. +Then we have +|vY (λ) − v(λ)| ≤ +1 +E[τ1] · E +� +|X1 +τ1 − Yτ1|1(A) +� +We write U(t) for the total number of points of P1 ∪ P2 ∪ P3 that have arrived before time t. Then +we obtain +E +� +|X1 +τ1 − Yτ1|1(A) +� +≤ 2 E[1(A) · U(τ1)] . +A key observation is that τ1 and U(τ1) only depend on P1 ∪ P2 ∪ P3 and the evolution of the size +of the infected set, which increases at the points of the Poisson process P1 ∪ P2 ∪ P3 and decreases +at an independent rate µ. This together with the thinning property of Poisson processes yields +that, conditional on U(τ1), the numbers of points in P2[0, τ1] and P3[0, τ1] follow the binomial +distribution with parameters (U(τ1), e−λZ−1 +λ ) and (U(τ1), (2d − 2)Z−1 +λ ) respectively. Using this we +then get +E[1(A) · U(τ1)] ≤ E +� +U(τ1) +� +1 − +� +1 − e−λZ−1 +λ +�U(τ1)�� ++ E +� +U(τ1) +� +1 − +� +1 − (2d − 2)Z−1 +λ +�U(τ1) − U(τ1)(2d − 2)Z−1 +λ (1 − (2d − 2)Z−1 +λ )U(τ1)−1�� +. +Using that for all x ∈ (0, 1) and a ∈ N we have (1 − x)a ≥ 1 − ax, we get +E[1(A) · U(τ1)] ≤ E +� +(U(τ1))2e−λZ−1 +λ ++ U(τ1)(U(τ1) − 1)(2d − 2)2Z−2 +λ +� +. +Since Z−1 +λ += O(e−λ) and by Lemma 2.2 we have for a positive constant Cµ that E +� +(U(τ1))2� +≤ +Cµ < ∞, it follows that +E[1(A) · U(τ1)] ≤ O(e−2λ) +with the implicit constants depending only on µ, and hence this concludes the proof. +Proof of Proposition 4.1. Let (�Y , �ξ) be a biased random walk on dynamical percolation on Z +with parameters �µ, p that jumps to the right at rate eλZ−1 +λ +and to the left at rate e−λZ−1 +λ . Then +the speed of Y is the same as the speed of �Y , since to determine it we only need to know the state +of every edge after the first time the walk examines it. +Let δ = δ(λ) = (2d − 2)Z−1 +λ +and consider the process +(Y t, ξt) := (�Yt(1−δ)−1, ξt(1−δ)−1), ∀ t ≥ 0. +Then (Y , ξ) is a one-dimensional biased random walk in dynamical percolation with parameters +(p, µ, λ), where +µ := µ + pδ +1 − δ . +15 + +We write vY for the speed of Y . Let Z be a random walk on dynamical percolation on Z with +parameters µ, p that only attempts jumps to the right at rate 1. Using Lemma 4.4 we get that the +speed of Z is given by +vZ = +µp +1 − p + µ. +We now want to compare the speed of Z to the speed of Y . We couple Z and Y by letting them +evolve together until the first time that Y attempts a jump to the left. Afterwards, they both +attempt jumps at the same times and they use the same Poisson process to remove edges from +their infected sets. We let τ1 be their first regeneration time. Let A be the event that the walker Y +attempts at least one jump to the left before time τ1 and let U(τ1) be the total number of jump +attempts before time τ1. Therefore we deduce +���� +µp +1 − p + µ − vY (λ) +���� ≤ +1 +E[τ1]E +� +|Zτ1 − Y τ1|1(A) +� +≤ +2 +E[τ1]E[U(τ1)1(A)] +≤ +2 +E[τ1]E +� +(U(τ1))2 · +e−λ +eλ + e−λ +� +≤ C′ exp(−2λ), +where C′ is a constant only depending on µ and where for the third inequality we used a union +bound and for the last one we used that U(τ1) has exponential tails by Lemma 2.2. +Using now that vY (λ) = (1 − δ)vY (λ) and a straightforward calculation we finally conclude that +for λ sufficiently large +vY (λ) = +µp +1 − p + µ − +(2d − 2)p +(1 − p + µ)2 (µ2 − p(1 − p))Z−1 +λ ++ O(e−2λ), +(4.12) +where the implicit constant in O depends only on µ and d. This together with Lemma 4.8 finishes +the proof. +4.3 +Asymptotic derivative of the speed +In this section we prove Lemma 4.3 by constructing a coupling between two walks with different +bias parameters. +Let ε > 0 and let (Xλ +t , ηt)t≥0 and (Xλ+ε +t +, ηt)t≥0 be λ-biased (respectively (λ + ε)-biased) random +walks on dynamical percolation in Zd with parameters µ and p. +Definition 4.9 (Coupling between Xλ and Xλ+ε). We start both walks from 0 and we let them +both attempt jumps at the points of a Poisson process P = (Pt)t≥0 of rate 1. We also let both envi- +ronments evolve together until the first very bad point defined below and afterwards we couple the +environments by using the same rate µ Poisson process for the removal of edges of their respective +infected sets. +Whenever a clock of P rings indicating the jump attempt at time t of both walkers, we sample a +random variable U uniformly on [0, 1] and proceed as follows: +(1) If U < (2d−2)/Zλ+ε, then we let both walkers attempt a jump into one of the 2d−2 directions +different from e1 and −e1 chosen uniformly at random. +(2) If U ∈ [(2d − 2)/Zλ+ε, (2d − 2)/Zλ], then we let the Xλ walk attempt a jump into one of +the 2d − 2 directions different from e1 and −e1 chosen uniformly at random, while we let the +Xλ+ε walk attempt a jump in the e1 direction. +16 + +(3) If U ∈ [(2d − 2)/Zλ, (2d − 2)/Zλ + e−λ−ε/Zλ+ε], then we let both walkers attempt a jump in +the −e1 direction. +(4) If U ∈ [(2d − 2)/Zλ + e−λ−ε/Zλ+ε, 1 − eλ/Zλ], then we let the Xλ walk attempt a jump in +the −e1 direction, while we let the Xλ+ε walk attempt a jump in the e1 direction. +(5) If U > 1 − eλ/Zλ, then we let both walkers attempt a jump in the e1 direction. +In the following, we let (Ti)i∈N be the points of the Poisson process (Pt)t≥0 and we colour each +point independently according to the outcome of the corresponding random variable U in the above +coupling. We say that a point is good if the corresponding random variable U satisfies (5), we say +that a point is bad if U satisfies (1) or (3), and we say that a point is very bad if U satisfies (2) +or (4). Notice that good, bad, and very bad points are again independent Poisson point processes +of intensities qg := eλZ−1 +λ +for good points, qb := (2d − 2 + e−λ−ε)/Zλ+ε for bad points, and +qvb := eλ+ε +Zλ+ε +− eλ +Zλ +> 0 +(4.13) +for very bad points. Note that there exist constants c1, c2, c3 > 0 and λ0 > 0 so that for all ε ∈ (0, 1) +and λ ≥ λ0 we get +|qb − (2d − 2) exp(−λ − ε)| ≤ c1 exp(−2λ) +(4.14) +and +|qvb − ε(2d − 2) exp(−λ)| ≤ c2ε exp(−2λ) + c3ε2 exp(−λ). +(4.15) +Moreover, note that the above coupling between the two random walkers ensures that they stay +together until the first very bad point and both infected sets have the same size at all times. +Therefore, both processes have the same sequence of regeneration times. We let τ1 be their first +regeneration time. +Let G be the event that there is no bad point up to time τ1 and for every ℓ ∈ N let Vℓ be the event +that Tℓ is the unique very bad point of U(τ1). Let R be the event that at the first very bad point +the walk Xλ attempts a move in one of 2d − 2 directions. We write U(t) for the number of points +of the Poisson process P that have arrived by time t. +We write for all x ∈ Zd +|x|1 := x · e1 . +(4.16) +Lemma 4.10. There exists a positive constant c = cd so that the following holds. Let p ∈ (0, 1) +and µ > 0. For all k ∈ N and ℓ ≤ k we have +P(Gc | U(τ1) = k, Vℓ) ≤ (k − 1) · qb +and +P(Rc | U(τ1) = k, Vℓ, G) ≤ ce−λ. +(4.17) +Moreover, there exist functions f = fµ,p, g = gµ,p : N × N → R+, which in particular do not depend +on λ and ε, such that +E +� +|Xλ+ε +τ1 +|1 +��� U(τ1) = k, Vℓ, G +� += f(k, ℓ) +and +E +� +|Xλ +τ1|1 +��� U(τ1) = k, Vℓ, G, R +� += g(k, ℓ). +Proof. Since the distribution of U(τ1) is independent of the colouring of the Poisson process P, it +follows that conditionally on U(τ1) = k and Vℓ, every point Ti for i ≤ k with i ̸= ℓ has probability +qb of being a bad point. Using this together with a union bound we deduce +P(Gc | U(τ1) = k, Vℓ) ≤ (k − 1) · qb. +17 + +Using again the independence between U(τ1) and the colouring, we obtain +P(Rc | U(τ1) = k, Vℓ, G) = 1 − eλZ−1 +λ +− (2d − 2)Z−1 +λ +− e−λ−εZ−1 +λ+ε +qvb +≤ ce−λ +for a suitable choice of c, completing the proof of (4.17). Recall that (Ti) are the points of P. We +notice that on the event {U(τ1) = k} ∩ Vℓ ∩ G ∩ R we can write +|Xλ+ε +τ1 +|1 = +k +� +i=1 +1(ηTi(Xλ+ε +Ti− , Xλ+ε +Ti− + e1) = 1) +and +|Xλ +τ1|1 = +k +� +i=1 +i̸=ℓ +1(ηTi(Xλ +Ti−, Xλ +Ti− + e1) = 1). +Using the independence between the Poisson process P = (Ti) and the colouring of each point as +good, bad or very bad together with the definition of the regeneration time τ1 which is independent +of the colouring of the process P (because even if we examine the same edge multiple times we still +add a copy of it to the infected set), we see that +L((T1, . . . , Tk), η | U(τ1) = k, Vℓ, G, R) = L((T1, . . . , Tk), η | U(τ1) = k). +In particular, this shows that the conditional law of ((T1, . . . , Tk), η) given U(τ1) = k, Vℓ, G, R is +independent of λ. We note that under this conditioning, Xλ+ε becomes a walk that only attempts +jumps to the right at the times T1, . . . , Tk and Xλ attempts jumps to the right at the times Ti for +i ≤ k and i ̸= ℓ and attempts a jump to one of 2d − 2 directions at time Tℓ. Therefore, we deduce +that there exist functions f = fµ,p and g = gµ,p independent of λ and ε so that +E +� +|Xλ+ε +τ1 +|1 +��� G, U(τ1) = k, Vℓ +� += f(k, ℓ) +and +E +� +|Xλ +τ1|1 +��� G, U(τ1) = k, Vℓ, R +� += g(k, ℓ) +and this concludes the proof. +We are now ready to prove Lemma 4.3. +Proof of Lemma 4.3. We start the proof by recalling from Proposition 3.1 that +v(λ + ε) − v(λ) = E[τ1]−1E +� +|Xλ+ε +τ1 +|1 − |Xλ +τ1|1 +� +. +Let Aε be the event that there exists a very bad point before τ1. Then we have +v(λ + ε) − v(λ) = E[τ1]−1E +� +|Xλ+ε +τ1 +|1 − |Xλ +τ1|1 +��� Aε +� +P(Aε) . +(4.18) +Recall that U(t) stands for the number of points of the Poisson process P of rate 1 up to time t. Since +the assignment of good/bad/very bad points to the points of the Poisson process is independent of +the value of τ1, we get +P(Aε) = E +� +1 − (1 − qvb)U(τ1)� +. +Since 1 − (1 − qvb)U(τ1) ≤ qvb · U(τ1) by the dominated convergence theorem and L’Hˆopital’s rule, +recalling the approximation of qvb from (4.15), we obtain +lim +ε→0 +P(Aε) +ε += E +� +lim +ε→0 +1 +ε +� +1 − +� +1 − ε(2d − 2) exp(−λ) + O(εe−2λ + ε2) +�U(τ1)�� += (2d − 2)e−λ · E[U(τ1)] + O(e−2λ), +(4.19) +18 + +where the implicit constant only depends on µ and d. We next prove that there exists a positive +constant �Cµ,p,d depending only on µ, p and d such that +lim +ε→0 E +� +|Xλ+ε +τ1 +|1 − |Xλ +τ1|1 +��� Aε +� += �Cµ,p,d + O(e−λ), +where the implicit constant in O depends only on µ and d. We define �Aε to be the event that there +is a unique very bad point up to time τ1. First, we note that +P +� +�Aε +��� Aε +� += E +� +U(τ1) · qvb · (1 − qvb)U(τ1)−1� +E +� +1 − (1 − qvb)U(τ1)� +, +(4.20) +and using similar arguments as above we get that +lim +ε→0 P +� +�Aε +��� Aε +� += 1. +(4.21) +We now have +E +� +|Xλ+ε +τ1 +|1 +��� Aε +� += E +� +|Xλ+ε +τ1 +|1 +��� �Aε +� +P +� +�Aε +��� Aε +� ++ E +� +|Xλ+ε +τ1 +|1 +��� �Ac +ε ∩ Aε +� +P +� +�Ac +ε +��� Aε +� +(4.22) +and similarly for Xλ +τ1. As |Xλ+ε +τ1 +| ≤ U(τ1), we get similarly as above +lim sup +ε→0 +E +� +|Xλ+ε +τ1 +|1 +��� �Ac +ε ∩ Aε +� +≤ lim +ε→0 E +� +U(τ1) +��� �Ac +ε ∩ Aε +� += E +� +(U(τ1))2(U(τ1) − 1) +� +E[U(τ1)(U(τ1) − 1)] +. +Since E[U(τ1)] > 1 and using that U(τ1) has exponential tails by Lemma 2.2 together with (4.21) +gives that the second term appearing in the sum in (4.22) converges to 0 as ε → 0. For the first +expectation appearing on the right hand side of (4.22) we have +E +� +|Xλ+ε +τ1 +|1 +��� �Aε +� += +� +k +� +ℓ≤k +E +� +|Xλ+ε +τ1 +|1 +��� U(τ1) = k, Vℓ +� +P +� +U(τ1) = k, Vℓ +��� �Aε +� +(4.23) +and similarly for Xλ +τ1. For each k and ℓ ≤ k we have +E +� +|Xλ+ε +τ1 +|1 +��� U(τ1) = k, Vℓ +� += E +� +|Xλ+ε +τ1 +|1 +��� U(τ1) = k, Vℓ, G +� ++ +� +E +� +|Xλ+ε +τ1 +|1 +��� U(τ1) = k, Vℓ, Gc� +− E +� +|Xλ+ε +τ1 +|1 +��� U(τ1) = k, Vℓ, G +�� +P(Gc | U(τ1) = k, Vℓ) . +Let us remark that in the case of Xλ we also add the event R to the intersection above. Using +again the obvious bound |Xλ+ε +τ1 +|1 ≤ U(τ1), all four statements of Lemma 4.10 and equations (4.14) +and (4.15) we get +E +� +|Xλ+ε +τ1 +|1 +��� U(τ1) = k, Vℓ +� += f(k, ℓ) + O(k2 · e−λ) +and +E +� +|Xλ +τ1|1 +��� U(τ1) = k, Vℓ +� += g(k, ℓ) + O(k2 · e−λ), +where the implicit constants in the terms O above only depend on µ and d. Plugging these back +into (4.23) we deduce +E +� +|Xλ+ε +τ1 +|1 +��� �Aε +� += +� +k +� +ℓ≤k +(f(k, ℓ) + O(k2 · e−λ))P +� +U(τ1) = k, Vℓ +��� �Aε +� +(4.24) +19 + +and similarly for Xλ. Using again the independence between U(τ1) and the colouring, we have +P +� +U(τ1) = k, Vℓ +��� �Aε +� += P(U(τ1) = k) · qvb · (1 − qvb)k−1 +E +� +U(τ1) · qvb · (1 − qvb)U(τ1)−1� , +and hence since qvb → 0 as ε → 0 by (4.15), we deduce +lim +ε→0 P +� +U(τ1) = k, Vℓ +��� �Aε +� += P(U(τ1) = k) +E[U(τ1)] +. +(4.25) +Using again the obvious bound |Xλ+ε +τ1 +|1 ≤ U(τ1), and hence also that f(k, ℓ) ≤ k, plugging (4.24) +into (4.22) and using the dominated convergence theorem we can take the limit as ε → 0 and +use (4.21) and (4.25) to obtain +lim +ε→0 E +� +|Xλ+ε +τ1 +|1 +��� Aε +� += +� +k +� +ℓ≤k +f(k, ℓ) · P(U(τ1) = k) +E[U(τ1)] ++ O +� +e−λ · +� +k +k3 · P(U(τ1) = k) +E[U(τ1)] +� += +� +k +� +ℓ≤k +f(k, ℓ) · P(U(τ1) = k) +E[U(τ1)] ++ O(e−λ), +where the implicit constant in O only depends on µ and d and where for the last equality we +used that U(τ1) has exponential tails by Lemma 2.2 again, and hence a finite third moment. The +analogous equality holds for Xλ with f replaced by g. +Therefore, these together with (4.19) +and (4.18) imply that +lim +ε→0 +v(λ + ε) − v(λ) +ε += (2d − 2) · E[U(τ1)] +E[τ1] +· +� +k +� +ℓ≤k +(f(k, ℓ) − g(k, ℓ)) · P(U(τ1) = k) +E[U(τ1)] +· e−λ + O(e−2λ). +This now finishes the proof as f and g are functions that only depend on µ and p and not on λ, +while the implicit constant in O depends only on µ and d. +5 +Strict monotonicity of the speed for large µ or p close to 1 +As already mentioned in the introduction and as we saw in Section 4.1, for d = 1, the function +λ �→ v(λ) is strictly increasing for any fixed choice of the percolation parameters p ∈ (0, 1] and µ > 0 +due to a coupling argument. Let us emphasise that this argument cannot be extended for d ≥ 2, +as Theorem 1.3 demonstrates. However, we identify in the following two regimes of parameters µ +and p in d ≥ 2 dimensions, where the speed is strictly increasing for all λ > 0. +Recall the function f(λ) from (3.10) as well as Zλ from (1.1) and Z′ +λ = eλ − e−λ. Moreover, for +k, ℓ, m ∈ N, we write +fk,ℓ,m(λ) := (k − ℓ)eλ(k−ℓ) +� 2d +Zλ +�m � +k − ℓ − m · Z′ +λ +Zλ +� +P0 +� +(R, L) = (k, ℓ), U(τ1) = m +� +and recall from (3.11) that +f ′(λ) = +� +m∈N +� +k+ℓ≤m +fk,ℓ,m(λ). +Proposition 5.1. Fix p ∈ (0, 1). There exists some constant ˜µ = ˜µ(p) > 0 such that for all µ > ˜µ, +we have that λ �→ vµ,p(λ) is strictly increasing in λ > 0. +20 + +Proof. It suffices to prove that f ′(λ) is strictly positive for all λ > 0. For all i, j ∈ N and m ≥ 2 +using Lemma 2.2 we have +P0 +� +(R, L) = (i, j), U(τ1) = m) ≤ exp(−cµm +� +(5.1) +for some constant cµ with cµ → ∞ as µ → ∞. +By Lemma 2.1 and the construction of the +regeneration time τ1 we get +P0 +� +(R, L) = (1, 0), U(τ1) = 1 +� += P0 +� +(R, L) = (0, 1), U(τ1) = 1 +� +≥ p · 1 +2d · +µ +µ + 1 ≥ p +4d , +(5.2) +for all µ ≥ 1. For every m we let +Am := +� +(k, ℓ) ∈ N2 : k + ℓ ≤ m and k − ℓ ≤ m − 1 +� +. +Using (5.2) and the definition of Zλ, we see that the decay of f1,0,1(λ) in λ is of the same order +as sup(k,l)∈A2 fk,l,2(λ). Hence, for all µ sufficiently large, together with the exponential decay in m +in (5.1), a computation shows that for all λ > 0, +� +m≥2 +� +(k,ℓ)∈Am +|fk,ℓ,m(λ)| ≤ 1 +2(f1,0,1(λ) + f0,1,1(λ)). +Since fm,0,m(λ) ≥ f0,m,m(λ) for all λ > 0 and m ∈ N, we obtain that +f ′(λ) ≥ 1 +2(f1,0,1(λ) + f0,1,1(λ)). +Using again (5.2) we get that +f1,0,1(λ) + f0,1,1(λ) ≥ 2d +Zλ +· (2d − 2)(eλ + e−λ) + 4 +Zλ +· p +4d > 0 +for all λ > 0 and this concludes the proof. +Proposition 5.2. Fix µ > 0. There exists some constant ˜p = ˜p(µ) > 0 such that for all p ∈ (˜p, 1), +we have that λ �→ v(λ) is strictly increasing in λ > 0. +Proof. Let p be sufficiently close to 1 so that µ2 > p(1−p) and let λ0 = λ0(µ) be as in Theorem 1.3. +For all λ > λ0 the speed is strictly increasing by Theorem 1.3. Thus it remains to show that the +speed is strictly increasing for all λ ∈ (0, λ0] for all p sufficiently large. To do this, we will prove +that v′(λ) > 0 for all such λ. +In this proof we want to emphasise the dependence of the speed on the percolation parameter p, +so we write v(λ, p) = v(λ). Observe that when p = 1, the speed v′(λ, 1) ≥ cd,µ · e−λ for all λ > 0, +where cd,µ is a constant depending on d and µ. It thus suffices to prove that for p sufficiently close +to 1, +|v′(λ, p) − v′(λ, 1)| ≤ cd,µ +2 +· e−λ +(5.3) +uniformly for all λ ∈ (0, λ0]. Recall the expression for v′(λ, p) from Lemma 3.4. We want to compare +Eλ,p +� +(X1 +τ1)2� +to Eλ,1 +� +(X1 +τ1)2� +and also Eλ,p +� +X1 +τ1 · U(τ1) +� +to Eλ,1 +� +X1 +τ1 · U(τ1) +� +. To do this, we couple +the walks in the two environments by letting them attempt the same jumps at the same times and +using the same Poisson process of rate µ to remove edges from their infected sets. We let τ1 be their +first regeneration time. We now let κ be the first time when the walk in the p-dynamical percolation +21 + +process attempts a jump along a closed edge. Then up until time κ the two walks are in the same +location. Note that κ stochastically dominates a geometric random variable of parameter 1 − p, +because for all s < t and all edges e we have +Pp(e is open at time t | e is open at time s) ≥ p. +By a union bound we get +P(κ < U(τ1)) ≤ P +� +U(τ1) > +1 +√1 − p +� ++ P +� +κ < +1 +√1 − p +� +≤ Cµ · +� +1 − p, +where Cµ is a constant depending on µ. Using this and the bound |X1 +τ1| ≤ U(τ1) we get +|Eλ,p +� +(X1 +τ1)2� +− Eλ,1 +� +(X1 +τ1)2� +| ≤ 2Eλ,p +� +(U(τ1))21(κ < U(τ1)) +� +. +By the Cauchy-Schwarz inequality and the exponential tails of U(τ1) uniformly in p by Lemma 2.2 +we deduce +Eλ,p +� +(U(τ1))21(κ < U(τ1)) +� +≤ C′ +µ(1 − p)1/4. +Similarly we can bound the remaining terms appearing in v′(λ), and hence we see that taking +p = p(λ0, µ) sufficiently close to 1, we get (5.3) and this concludes the proof. +Acknowledgments. +We thank Frank den Hollander and Remco van der Hofstad for valuable discussions. DS acknowl- +edges the DAAD PRIME program for financial support. +References +[1] E. A¨ıd´ekon. Speed of the biased random walk on a Galton–Watson tree. Probability Theory +and Related Fields, 159(3):597–617, 2014. +[2] M. Axelson-Fisk and O. H¨aggstr¨om. Biased random walk in a one-dimensional percolation +model. Stochastic Process. Appl., 119(10):3395–3415, 2009. +[3] A. Bandyopadhyay and O. Zeitouni. 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Communi- +cations in Mathematical Physics, 240(1-2):123–148, 2003. +24 + diff --git a/hdE4T4oBgHgl3EQfrg0j/content/tmp_files/load_file.txt b/hdE4T4oBgHgl3EQfrg0j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..92553f2f0efb275a1bbe31d414e850428faeae80 --- /dev/null +++ b/hdE4T4oBgHgl3EQfrg0j/content/tmp_files/load_file.txt @@ -0,0 +1,950 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf,len=949 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='05208v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='PR] 12 Jan 2023 Biased random walk on dynamical percolation Sebastian Andres1 Nina Gantert2 Dominik Schmid3 Perla Sousi4 January 13, 2023 Abstract We study biased random walks on dynamical percolation on Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We establish a law of large numbers and an invariance principle for the random walk using regeneration times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Moreover, we verify that the Einstein relation holds, and we investigate the speed of the walk as a func- tion of the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' While for d = 1 the speed is increasing, we show that in general this fails in dimension d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' As our main result, we establish two regimes of parameters, separated by an explicit critical curve, such that the speed is either eventually strictly increasing or eventually strictly decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' This is in sharp contrast to the biased random walk on a static supercritical percolation cluster, where the speed is known to be eventually zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Keywords and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Dynamical percolation, biased random walk, regeneration times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' MSC 2020 subject classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Primary 60K35, 60K37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 1 Introduction In this paper, we introduce and study biased random walks in dynamically evolving environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The model of random walks on dynamical percolation was introduced in [23] by Peres, Stauffer and Steif, and has the following description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Fix a locally finite graph G = (V, E) and an initial state η ∈ {0, 1}E(G) of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We say that an edge e is open at time t if ηt(e) = 1, and closed otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For parameters µ ≥ 0 and p ∈ [0, 1], we consider the dynamics (ηt)t≥0 with η0 = η, where each edge e in the graph is assigned an independent Poisson process of rate µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' If there is a point of the Poisson process at time t, we refresh the state of e in ηt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' we declare e open with probability p and closed with probability 1 − p, independently of all other edges and previous states of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' From now on, we focus on the case where the underlying graph is Zd with d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We define a continuous-time random walk (Xt)t≥0 in the environment (ηt)t≥0 with bias parameter λ > 0 as follows: set X0 = 0 and assign a rate 1 Poisson clock to the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We also set for λ > 0 Zλ := eλ + e−λ + 2d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1) Whenever the clock rings at time t and the random walker is currently at a site x, we choose one of the neighbours y of x with probability p(x, x ± ei) = 1 Zλ for i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' , d}, 1University of Manchester, United Kingdom, sebastian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='andres@manchester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='uk 2Technical University of Munich, Germany, nina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='gantert@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='de 3University of Bonn, Germany, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='schmid@uni-bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='de 4University of Cambridge, United Kingdom, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='sousi@statslab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='uk 1 p(x, x + e1) = eλ Zλ , p(x, x − e1) = e−λ Zλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' If ηt({x, y}) = 1, the random walker moves from x to y, and it stays at x, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We will call the process (Xt, ηt)t≥0 a λ-biased random walk on dynamical percolation with parameters µ and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Note that (ηt)t≥0 and (Xt, ηt)t≥0 are Markov processes, while (Xt)t≥0 is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Moreover, (ηt)t≥0 has the Bernoulli-p-product measure πp on {0, 1}E(Zd) as its unique invariant distribution, and we assume in the following that η = η0 ∼ πp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In this paper our focus is on the speed of the first coordinate of the walk as a function of the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The motivation to study this question comes from the two different regimes one observes in the case of a biased random walk on a static percolation cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' It was first shown in [8] and [28] that when p > pc and X is a λ-biased random walk on the infinite percolation cluster, then there exist λ1 < λ2 so that when λ > λ2, the speed is 0, while for λ < λ1, the speed is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' A few years later it was proved by [16] that there is a sharp transition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' there exists λ∗ so that for all λ > λ∗ the speed is equal to 0, while for λ < λ∗ the speed is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Motivated by these results, in this paper we study the speed in the dynamical setting and we establish that for all choices of the parameters, the speed is always strictly positive and it satisfies an Einstein relation as we show in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Our second main result concerns the monotonicity of the speed as a function of the bias in dimensions d ≥ 2, where we observe two different regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Before stating our results we recall an invariance principle established in [23, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1] in the unbiased case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Unless otherwise stated, our probability measure is taking averages not only over the walk but over the environment as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 ([23, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For d ≥ 1, µ > 0, p ∈ (0, 1) and λ = 0, there exists σ ∈ (0, ∞) so that �Xkt √ k � t∈[0,1] (d) → (σBt)t∈[0,1] in D[0, 1] as k → ∞, where (Bt)t≥0 is a standard Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We now present our first result on the speed of the biased random walk (Xt)t≥0 for fixed environment parameters µ > 0 and p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let d ≥ 1 and let (Xt, ηt)t≥0 be a λ-biased random walk on dynamical percolation on Zd with parameters µ > 0 and p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then for all λ > 0, there exists v(λ) = vµ,p(λ) such that almost surely lim t→∞ Xt t = (v(λ), 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' , 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2) Moreover, the function λ �→ v(λ) is strictly positive for all λ > 0, continuously differentiable and satisfies lim λ→0 v′(λ) = σ2, where σ is as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The last statement in the theorem above is known as Einstein relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Moreover, as we will see in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2, an invariance principle also holds in the biased case and the proof follows along the same lines as the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='8 1 Eventually Eventually monotone decreasing monotone increasing p µ Figure 1: Plot of the different regimes in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3 for large λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' When d = 1, using the obvious coupling between two walks with different bias parameters, it is immediate to see that the speed is always monotone increasing in the bias and in fact in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 we also establish that in d = 1 the speed is strictly increasing as a function of the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' It is thus natural to ask what happens for d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' While the speed turns out to be monotone increasing in λ > 0 for certain regimes of µ > 0 and p ∈ (0, 1) in dimensions d ≥ 2 as we show in Section 5, our main result is an explicit criterion deciding whether the speed is eventually strictly increasing or decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3 (Monotonicity of the speed for d ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Consider the biased random walk on dynam- ical percolation on Zd for d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For all p ∈ (0, 1) and µ > 0, there exists some λ0 = λ0(µ, d) such that the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (1) The speed v(λ) is strictly increasing for all λ ≥ λ0 provided that µ2 > p(1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (2) The speed v(λ) is strictly decreasing for all λ ≥ λ0 provided that µ2 < p(1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Note that this is in contrast to the biased random walk on a static super-critical percolation cluster, where the speed is known to be zero for large values of λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' see [8, 16, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The criterion for the eventual monotonicity of the speed, identified in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3, is visualised in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Moreover, this suggests different shapes of the speed functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 Related work Biased random walks in random media were investigated intensively over the last years, we refer to [1, 2, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 20, 28] for a non-exhaustive list and to [5] for a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The most prominent examples are biased random walks on Galton-Watson trees and biased random walks on supercritical percolation clusters, see [20] and [8, 28] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Our work is in particular motivated by the study of biased random walks on (static) supercritical percolation clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' This model was introduced in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Due to traps in the cluster, the speed of the walk is zero for large values of the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Simulations show that the speed is a unimodal function of the bias, first increasing until the maximum is achieved and then decreasing and eventually becoming zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' It was proved in [28] and [8] that for small values of the bias, the speed is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Finally, in the breakthrough paper [16], it was shown that there is a critical value separating the positive speed regime from the zero speed regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In the case of Galton-Watson trees with leaves, this phase transition was known earlier and there is an explicit formula for the critical value, see [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' When 3 there are no “hard traps”, the speed of the biased walk should be strictly positive, and one may ask if it is increasing as a function of the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In the case of a Galton-Watson tree without leaves, it is conjectured that the speed is indeed an increasing function of the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' While this conjecture still remains open, it was proved that the speed is eventually increasing, see [6] and [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The same argument as in [6] gives that for biased random walk among uniformly elliptic i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' conductances, the speed is eventually increasing as a function of the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' However, it was shown in [7] that for some laws of the conductances, the speed is not increasing for all values of the bias, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' there exist λ1 < λ2 such that v(λ1) > v(λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In the presence of hard traps, a central limit theorem is expected to hold for small values of the bias, in a strict subset of the positive speed regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' This was proved for biased random walk on supercritical percolation clusters in [28] and [8] and for random walks on Galton-Watson trees with leaves in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For other models such results have been established for example in [18] and [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In an environment without hard traps, there are examples where a central limit theorem holds for all values of the bias, see for instance [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In our case, this also turns out to be true, see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The literature on (unbiased) random walks in time-dependent random environments is too vast to give a review, we just point to two papers which are relevant in our setup, namely [3, 10], see also the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Unbiased random walks on dynamical percolation have been studied in particular in terms of their mixing times, see [19, 21, 22, 23, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 Overview of proof ideas The proof of the first part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 follows by using a suitably defined sequence of regen- eration times (τi), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' a sequence of random times such that the evolution of the walk and the environment between [τi, τi+1] is independent for different choices of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' A key property of the re- generation times that we define is that their distribution only depends on the parameter µ, but not on the bias parameter λ and the percolation parameter p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using these regeneration times and a law of large numbers we get the existence of the speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Moreover, we find an expression for the speed in terms of an infinite series which allows to give a simple expression for its derivative as we show in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4 and the Einstein relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In particular, the speed is strictly increasing for all λ sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We now give an overview of the main ideas behind the proof of our main result, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3, giving a necessary and sufficient condition for the speed to be eventually in λ strictly increasing or decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' There are two main ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' First, in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1, we obtain an asymptotic expression for the speed which is valid for all bias parameters λ sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Next, in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3, we give an asymptotic bound on the derivative of the speed for large λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3 is a direct consequence of these two results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In order to prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1, we start with a detailed analysis of the speed in the one dimen- sional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In order to analyse the case d ≥ 2, we rely on the regeneration times and compare the first coordinate of the walk with a time-changed one dimensional walk in a suitably defined evolving environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' To be more precise, we construct a coupling which keeps the first coordinates of the two walks together until the second time that the d dimensional walk jumps in a direction other than e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In order to prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3 we rely on a comparison between walkers with different bias parameters using marked Poisson point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' A key task is to develop an asymptotic expression for the derivative of the speed on the scale e−λ, with the constants only depending on µ and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 4 λ v(λ) λ v(λ) λ v(λ) Figure 2: The different pictures show three possible shapes for v(λ) which are in accordance with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3 Organisation In Section 2 we define our sequence of regeneration times that will be used in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In Section 3 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In Section 4 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3 and we also study the one dimensional case in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 where we establish the strict monotonicity of the speed for all values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Finally, in Section 5 we prove that the speed in d ≥ 2 is strictly increasing when µ or p are sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 2 Regeneration times for the biased random walk Throughout the paper we write Pp,µ for the probability measure corresponding to dynamical per- colation with parameters µ and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For λ ≥ 0, we write Pλ for the semi-direct product of Pp,µ with the law of the λ-biased random walk starting at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We write Eλ for the expectation with respect to Pλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 we need to define a sequence of regeneration times for the random walk on dynamical percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' A similar definition of regeneration times was used by Peres, Stauffer and Steif in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Here we use the definition given in [19] which works for general underlying graphs and was used in [19] to compare mixing and hitting times for random walks on dynamical percolation in terms of the respective quantities for the static graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For the biased random walk on Zd, we have the following construction following [19, Section 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We fix an enumeration (ei)i∈N of the edges in E according to an arbitrary rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then for each edge ei, we create an infinite number of copies denoted ei,1, ei,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We now define a process (It)t≥0, where for every t ≥ 0, It is a subset of edges and copies of edges that we refer to as the infected set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let I0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Suppose that for some t ≥ 0, the Poisson clock associated to the random walk (Xt)t≥0 rings and that the walker examines the edge ei for some i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' If no copy of ei is contained in It−, we set It := It− ∪ {ei,1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1) Otherwise, we add to It the copy ei,j of ei with the smallest index j such that ei,j /∈ It−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Next, for all t ≥ 0, we assign the lexicographic ordering ⪯ to the edges in It using the ordering of the edges of E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' for ei,j, ek,l ∈ It we have ei,j ⪯ ek,l ⇔ (i ≤ k) ∨ ((i = k) ∧ (j ≤ l)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2) Further, let (Nt)t≥0 be a Poisson process with time dependent intensity µ|It|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Whenever a clock of this process rings at time t, we choose an index uniformly at random from {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' , |It|} and remove 5 Xt Xt Figure 3: Visualisation of the dynamical percolation cluster and the infected set of edges It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The edges in red on the left correspond to open edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In the right picture, blue edges correspond to closed edges, which are infected at time t, red edges correspond to open edges that are in It and dotted red edges correspond to open edges that are not in It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' the copy of the edge with this index in It according to the ordering ⪯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Moreover, if the picked edge is of the form ei,1 for some i ∈ N, we refresh the state of the edge ei in the environment ηt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' we set ηt(ei) = 1 with probability p, and ηt(ei) = 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For all edges ej with ej,1 /∈ It, we use independent rate µ Poisson clocks to determine when the state of the edge in (ηt)t≥0 is refreshed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Note that with this construction (Xt, ηt)t≥0 has indeed the correct transition rates, see Figure 3 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Recall that we start from η0 ∼ πp, X0 = 0, and that we set I0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let τ0 := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For every i ∈ N, we set τi+1 := inf{t > τi : It = ∅ and It′ ̸= ∅ for some t′ ∈ (τi, t)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3) Let N0 := N ∪ {0} and note that the times (τi)i∈N0 are indeed regeneration times for the process (Xt)t≥0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (τi −τi−1)i∈N are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' and the random walk increments (Xτi −Xτi−1)i∈N are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Note also that the law of (τi)i∈N0 only depends on the parameter µ and not on λ or p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Observe that the process (|It|)t≥0 is a continuous-time birth-and-death chain on N0 with transition rates q(i − 1, i) = 1 and q(i, i − 1) = µi for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The following lemma is the content of [19, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For all p ∈ (0, 1) and µ, λ > 0, the increments (τi−τi−1)i∈N are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=', have exponential tails and satisfy Eλ[τ1] = e1/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In particular, note that the law of the regeneration times does not depend on λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Therefore, with a slight abuse of notation, we will write P instead of Pλ when considering events involving only the regeneration times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For every t ≥ 0 we let U(t) be the number of attempted jumps of the walker X up to time t, which follows the Poisson distribution with parameter t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We have the following result on U(τ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For every µ > 0 and p ∈ (0, 1), there exists a positive constant cµ satisfying cµ → ∞ as µ → ∞ so that for all m ≥ 2 P(U(τ1) ≥ m) ≤ e−cµm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The fact that the random variable U(τ1) has exponential tails is an immediate consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 and the exponential concentration of a Poisson random variable around its mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' It 6 remains to show that we can choose cµ such that cµ → ∞ as µ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let µ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Recall the birth and death chain (|It|)t≥0, and let (Sk)k≥0 with S0 = 0 denote its jump chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Further, let ˜τ0 := inf{n ≥ 1: Sn = 0} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4) be first return time of (Sk)k≥0 to the origin and observe that 2U(τ1) = ˜τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Note that the process (|It|)t≥0 is dominated from above by a biased random walk on {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' } with transition rates q(i, i − 1) = µ and q(i − 1, i) = 1 for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Hence, we get for all θ > 0 that the process (Mk)k∈N defined by Mk := eθSk · f(θ)k−1 with f(θ) := (µ + 1)/(eθ + e−θµ) is a super-martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Since almost surely M1 = eθ, and ˜τ0 has exponential tails, we can apply the optional stopping theorem together with Fatou’s lemma to obtain eθ = E[M1] ≥ E[M˜τ0] = E � exp(θS˜τ0)f(θ)˜τ0−1� = E � (f(θ))˜τ0−1� for all θ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Take θ = log log µ for µ > 0 sufficiently large such that f(θ) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then, we get that for all m ≥ 2 by Markov’s inequality P(U(τ1) ≥ m) = P(˜τ0 − 1 ≥ 2m − 1) ≤ E � (f(θ))˜τ0−1� f(θ)2m−1 ≤ eθf(θ)−(2m−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='5) Since f(θ) ≥ 1 2 log µ for all µ > 0 sufficiently large, we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 3 Speed and Einstein relation We will now show Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' First, we prove in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 a law of large numbers for the biased random walk on dynamical percolation and in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 we prove an invariance principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Both proofs use similar arguments to [23, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='5 we show that the speed in the e1-direction is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Recall the sequence of regeneration times (τi)i∈N from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then, Pλ-almost surely, lim t→∞ Xt t = (v(λ), 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' , 0) = Eλ[Xτ1] E[τ1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We first show that there exists a positive constant C (depending on µ) so that for all λ > 0, Eλ[∥Xτ1∥1] ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2) Recall that for every t ≥ 0 we write U(t) for the total number of times that an edge was added to the infected set during the time interval [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then ∥Xτ1∥1 ≤ U(τ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 we get that U(τ1) has exponential tails, and hence this proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Since the increments (Xτi − Xτi−1) are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2) holds, we can apply the strong law of large numbers to obtain that almost surely lim k→∞ Xτk k = lim k→∞ 1 k k � i=1 (Xτi − Xτi−1) = Eλ[Xτ1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3) 7 To prove a law of large numbers for (Xt)t≥0, for every t ≥ 0 writing k = k(t) = ⌊t/E[τ1]⌋ we get Xt t = Xt − XkE[τ1] t + XkE[τ1] − Xτk t + Xτk t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' It now follows that the first fraction on the right hand side above converges to 0 as t → ∞ almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using that τk/k → E[τ1] as k → ∞ almost surely, it follows easily that the second fraction on the right hand side above converges to 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Finally using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3) we get that the third fraction converges to Eλ[Xτ1] /E[τ1] almost surely as t → ∞ and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' At this point, let us state some consequences of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The next proposition follows in exactly the same way as the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 in [23] when λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The only difference from [23] is the definition of the regeneration times, but the way they are used for the invariance principle is the same as in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We give a sketch of the proof, as we need the expression for the diffusivity of the Brownian motion in order to establish the Einstein relation in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The biased random walk (Xt)t≥0 = ((X1 t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' , Xd t ))t≥0 on dynamical percolation satisfies an invariance principle �X1 kt − v(λ)kt √ k � t∈[0,1] (d) → (σBt)t∈[0,1] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4) in D[0, 1] as k → ∞, where (Bt)t≥0 denotes a standard Brownian motion and σ2 = σ2(d, µ, p, λ) = Varλ(X1 τ1)(E[τ1])−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Sketch of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Since the arguments are similar to the ones in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 in [23], we will only outline the key steps of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' As Eλ[τ 2 1 ] < ∞, by a similar tightness argument as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 of [27], it suffices to consider the convergence in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4) only for t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Since (τn)n∈N is a sequence of regeneration times, we have that as n → ∞, X1 τn − v(λ)nE[τ1] � n Var(X1τ1) (d) → N (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='5) where N is standard normal random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Next, we define ℓ(k) := max{ℓ ∈ N: τℓ ≤ k} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='6) to be the index of the last regeneration time before k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We write now X1 k − v(λ)k √ k = X1 k − X1 τℓ(k) √ k + X1 τℓ(k) − v(λ)τℓ(k) √ k + v(λ)(τℓ(k) − k) √ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='7) For all 0 < s < t we write U[s, t] for the number of edges added to the infected set between times s and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Note that U[s, t] is a Poisson random variable of parameter t − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then |X1 k − X1 τℓ(k)| ≤ U[τℓ(k), τℓ(k)+1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='8) Recall from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 that the regeneration times have exponential tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Since k − τℓ(k) √ k (d) → 0 and τℓ(k)+1 − τℓ(k) √ k (d) → 0 for k → ∞ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='9) as the numerator of the second fraction converges to the size-biased distribution of τ1, we see from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='9) that the first and third term on the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='7) converge to 0 in probability, and by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='5) for the second term we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 8 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Since the above proof works for all λ ≥ 0, it follows that the diffusivity σ2 for λ = 0 is the same as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Moreover, note that we only prove an annealed invariance principle in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2, but conjecture that also a quenched invariance principle holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' see [3] for sufficiently large values of µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Recall from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 that the speed v(λ) is equal to Eλ[Xτ1]/E[τ1] and E[τ1] does not depend on λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Moreover, by considering the Radon-Nikodym derivative of the law of a λ-biased random walk path with respect to the unbiased one, we obtain Eλ[X1 τ1] = E0 � X1 τ1eλX1 τ1 � 2d Zλ �U(τ1)� =: f(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For every t ≥ 0 we let Rt, respectively Lt, be the number of steps to the right, respectively to the left, that X1 performs by time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Setting R = Rτ1 and L = Lτ1 we can write f(λ) = � m∈N � k+ℓ≤m (k − ℓ)eλ(k−ℓ) � 2d Zλ �m P0((R, L) = (k, ℓ), U(τ1) = m), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='10) where P0 denotes the law of the symmetric simple random walk on dynamical percolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let µ > 0 and p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then the speed v(λ) is continuously differentiable in λ > 0 and the derivative satisfies v′(λ) = 1 E[τ1] · � Eλ � (X1 τ1)2� − eλ − e−λ Zλ Eλ � X1 τ1 · U(τ1) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In particular, we have lim λ→0 v′(λ) = σ2, where σ2 is the variance from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We first prove that for all λ > 0, lim δ→0 f(λ + δ) − f(λ) δ = � m∈N � k+ℓ≤m (k − ℓ)eλ(k−ℓ) � 2d Zλ �m � k − ℓ − m · Z′ λ Zλ � P0((R, L) = (k, ℓ), U(τ1) = m), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='11) where Z′ λ := eλ−e−λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Note that the sum appearing above divided by E[τ1] is equal to the expression for the derivative given in the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' A direct calculation shows that f(λ + δ) − f(λ) δ = � m∈N � k+ℓ≤m (k − ℓ)eλ(k−ℓ) � 2d Zλ �m g(δ) · P0((R, L) = (k, ℓ), U(τ1) = m), where the function g is defined via g(δ) = eδ(k−ℓ)Zm λ − Zm λ+δ δZm λ+δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' There exists a positive constant c = cd so that for all λ and δ we have 1 ≥ Zλ Zλ+δ ≥ 1 − cδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 9 By taking δ < 1/c, and considering whether δ < 1/m or δ ≥ 1/m, we see that there is a positive constant C = Cd so that |g(δ)| ≤ Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Therefore, we obtain that uniformly for δ < 1/c, ���� f(λ + δ) − f(λ) δ ���� ≤ � m∈N � k+ℓ≤m Cm2eλ(k−ℓ) � 2d Zλ �m P0((R, L) = (k, ℓ), U(τ1) = m) = C · Eλ � (U(τ1))2� < ∞, where for the last bound we used Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2, since the distribution of U(τ1) is independent of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We can thus apply the dominated convergence theorem which allows us to differentiate the summands in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='10) with respect to λ to get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Applying the dominated convergence theorem again we see that all the terms appearing in the expression for v′(λ) are continuous functions in λ, and hence this finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Fix p ∈ (0, 1) and µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then the speed function λ �→ v(λ) is strictly positive for all λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' To see that the speed v(λ) = vµ,p(λ) is strictly positive for all µ, λ > 0 and p ∈ (0, 1], note that P0((R, L) = (k, ℓ), U(τ1) = m) = P0((R, L) = (ℓ, k), U(τ1) = m) for all (k, ℓ) ∈ Z2 by symmetry of the random walk and the environment law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Thus, we can write f(λ) = � m∈N � k+ℓ≤m (k − ℓ)eλ(k−ℓ) � 2d Zλ �m P0((R, L) = (k, ℓ), U(τ1) = m) = � m∈N � k>ℓ k+ℓ≤m (k − ℓ) � 2d Zλ �m � eλ(k−ℓ) − e−λ(k−ℓ)� P0((R, L) = (k, ℓ), U(τ1) = m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='12) Since all the terms of the sum in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='12) are positive, we can infer a strictly positive speed using that P0((R, L) = (2, 1)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 is now an immediate consequence of Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='5 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 4 Monotonicity of the speed In this section we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In order to do so, we first establish an asymptotic expression for the speed that is valid for large values of the bias λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We recall the definition from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1) of Zλ = eλ + e−λ + 2d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For d ≥ 1, let (X, η) be a λ-biased random walk on dynamical percolation on Zd with parameters µ > 0 and p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' There exists some λ0 = λ0(µ, d) such that for all λ > λ0, v(λ) = µp 1 − p + µ − (2d − 2)p (1 − p + µ)2 (µ2 − p(1 − p))Z−1 λ + O(e−2λ), where the implicit constant in O depends on µ and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 10 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Note that the speed v(λ) converges to µp(1 − p + µ)−1 as λ → ∞ in agreement with the 1-dimensional case as we will see in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The above proposition proves the monotonicity of v(λ) along arithmetic progressions for large λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3 we also need to obtain a control on the derivative of v(λ) that is valid for large values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then for all µ > 0 and p ∈ (0, 1) there exists λ0 = λ0(µ) and positive constants cµ, Cµ,p so that for all λ ≥ λ0 we have ��v′(λ) − Cµ,p exp(−λ) �� ≤ cµ exp(−2λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1) We now have all the tools needed in order to conclude the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We defer the proofs of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3 to Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3, it suffices to study the constant Cµ,p in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1) and show that Cµ,p < 0 when µ2 > p(1 − p) as well as Cµ,p > 0 when µ2 < p(1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Since we know that the speed is continuously differentiable by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2, we get that for all s > 0 large enough v(2s) − v(s) = � 2s s v′(t)dt = Cµ,p exp(−s) + O(exp(−2s)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2) Taking now s = λ sufficiently large, we get from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 that Cµ,p = (2d − 2)p (1 − p + µ)2 (µ2 − p(1 − p)) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3) allowing us to conclude, since we get v′(λ) = Cµ,pe−λ + O(e−2λ), and hence the sign of v′(λ) agrees with the sign of Cµ,p for all λ sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 Speed for d = 1 In this section we focus on dimension 1, where one can use the obvious coupling between two random walks with different biases to obtain that the speed is increasing as a function of the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We investigate the limiting speed and the rate of convergence to the limit for large λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In the following proposition we establish strict monotonicity as well as an explicit form for the speed in the totally asymmetric biased random walk case, where the random walk only attempts jumps to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let (X, η) be a totally asymmetric biased random walk in dynamical percolation on Z with parameters µ and p that jumps to the right at rate 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then the speed v satisfies v = µp 1 − p + µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Suppose X0 = 0 and η0 ∼ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let S be the first time that X jumps along the edge e = {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then v = E[S]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' To compute E[S] we describe a way to realise the evolution of the state of e over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let (ξi)i∈N be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Bernoulli-p-distributed random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let (Ti)i∈N and (Sj)j∈N 11 denote the rate µ and rate 1 Poisson clocks that determine when the edge e is updated and when the random walk attempts a move to the right, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Taking T0 := 0, we assign to each interval [Ti−1, Ti] the label ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' If ξi = 1, the edge e is open and otherwise it is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We let Z := inf {j ∈ N: Sj ∈ [Ti−1, Ti] for some i ∈ N with ξi = 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4) In particular, we have S = SZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' By the memoryless property of the exponential distribution we get P(Z = j|Z > j − 1) = µp µ + 1 and P(Z = 1) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='5) Therefore, for all j > 1, we have that P(Z = j) = P(Z = j|Z > j − 1)(1 − p) j−1 � i=2 P(Z > i|Z > i − 1) = (1 − p) � 1 − µp µ + 1 �j−2 µp µ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' It then follows that E[Z] = 1 − p + µ µp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='6) Let (χi)i∈N with χi = Si+1 − Si denote the inter-arrival times of the Poisson process (Sj)j∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In particular, (χi)i∈N are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Exponential-1-distributed random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Observe that Z is a stopping time with respect to (χi)i∈N, taking the enlarged filtration which contains also the process (Ti)i∈N which is independent of (χi)i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Therefore, applying Wald’s identity and using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='6), we conclude E[S] = E[χ1]E[Z] = 1 − p + µ µp and this finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='5 (Monotonicity and asymptotic speed for d = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let (Xt, ηt)t≥0 be a biased random walk in dynamical percolation on Z with parameters µ and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then the speed function v(λ) from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2) is strictly increasing for all λ > 0 and satisfies lim λ→∞ vµ,p(λ) = µp 1 − p + µ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='7) for all choices of p ∈ (0, 1) and µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We start by arguing that the speed is strictly increasing in λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We construct a coupling P between a λ1-biased random walk (Xt, ηt)t≥0 and a λ2-biased random walk ( � Xt, �ηt)t≥0 on dynamical percolation on Z with 0 < λ1 < λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We take the same environment for both walks and we let them attempt jumps in the following way: whenever the two random walks are at the same location, we couple them by using the same exponential 1 clocks to determine the jump times and then moving them both to the right with probability eλ1/(eλ1 + e−λ1), moving � X to the right and X to the left with probability eλ2/(eλ2 + e−λ2) − eλ1/(eλ1 + e−λ1) and moving them both to the left otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' If the two walks are in different locations, we let them attempt jumps in the common environment using independent exponential 1 clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Recall the construction of the infected set from Section 2 and the definition of copies of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We define the following modified infected set (It)t≥0, where for every t ≥ 0, It is a subset of edges 12 and copies of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Suppose that for some t ≥ 0, both random walks are at the same position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' If the two random walks examine the same edge ei for some i ∈ N and no copy of ei is contained in It−, we set It := It− ∪ {ei,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Otherwise, we add to It the copy ei,j of ei with the smallest index j such that ei,j /∈ It−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' If the random walks are at the same position, but examine different edges, we add both edges or copies of them with the smallest index as above to the modified infected set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' When the two random walks are at different positions, recall that according to the coupling, the two random walks perform jumps according to independent exponential 1 clocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Whenever an edge is examined by one of the two random walks, we add this edge or a copy of it to the modified infected set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let (N t)t≥0 be a Poisson process with time dependent intensity µ|It|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Whenever a clock of this process rings at time t, we choose an index uniformly at random from {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' , |It|} and remove the copy of the edge with this index in It according to the ordering ⪯ of edges from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' If the picked edge is of the form ei,1 for some i ∈ N, we also refresh the state of the edge ei in the common environment ηt for the two walkers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' we set ηt(ei) = 1 with probability p, and ηt(ei) = 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For all edges ej with ej,1 /∈ It, we use independent rate µ Poisson clocks to determine when the respective edge is updated in the environment for the two random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Note that under this coupling P the biased random walks on dynamical percolation (Xt, ηt)t≥0 and ( � Xt, �ηt)t≥0 have marginally the correct law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We define τ := inf{t > 0: It = ∅ and It′ ̸= ∅ for some t′ ∈ (0, t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='8) Since the process (|It|)t≥0 is dominated from above by a biased random walk on {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' } with transition rates q(i, i − 1) = µi and q(i − 1, i + 1) = 2 for all i ∈ N, a similar argument as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 (see also the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2) shows that that the random variable τ has all finite moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Applying now the same arguments as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1, we get that v(λ1) = E[Xτ] E[τ] and v(λ2) = E[ � Xτ] E[τ] , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='9) where we write E for the expectation with respect to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Note that P(Xτ ≤ � Xτ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Considering the event that both random walks jump into different directions, and the respective edges get removed from the modified infected set before another jump occurs, we also get P(Xτ < � Xτ) ≥ � eλ2 eλ2 + e−λ2 − eλ1 eλ1 + e−λ1 � � µ µ + 1 �2 > 0 as λ1 < λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' This immediately implies that v(λ1) < v(λ2), hence establishing strict monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Next, we investigate the speed when λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Since the P-probability for (Xt) to attempt a jump in the −e1 direction until time τ goes to 0 as λ1 → ∞, and the speed is increasing in λ, lim λ→∞ v(λ) = ¯v, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='10) where v denotes the speed of the totally biased random walk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' whenever the clock associated to the random walker at x rings, it attempts a jump to x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4 concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 Asymptotic expression for the speed In this section we prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In order to do so, we first construct a coupling between (X, η) and a one-dimensional biased random walk Y in a suitably defined evolving environment ξ on Z for which we can calculate an asymptotic expression for the speed using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For x ∈ Z abusing notation we write x+e1 for the edge (x, x+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Recall that Zλ = eλ+e−λ+2d−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='6 (Coupling between (X, η) and (Y, ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let �µ = µ + p(2d − 2)Z−1 λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Both X and Y start from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We let the environment η evolve according to dynamical percolation on Zd with parameters µ, p and η0 ∼ πp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The edges to the left of 0 in the environment ξ update according to dynamical percolation with parameters �µ, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let P1, P2 and P3 be three independent Poisson processes of parameters eλZ−1 λ , e−λZ−1 λ and (2d− 2)Z−1 λ , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' At the points of P1 or P2 both X and Y attempt a jump to the right or the left respectively and we add the corresponding edges (the lowest numbered copies not in the infected sets as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3) to their respective infected sets and we say that the two edges are a match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' At the points of P3 the walk X attempts a jump in one of the 2d − 2 directions other than e1 and −e1 chosen uniformly at random and we add the corresponding edge to the infected set of X only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We now explain how to remove edges from the infected sets: we pick an edge from the infected set of X to be removed in the same way as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3 (each edge is being picked at rate µ) and we also remove its match if it exists from the infected set of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We then update the corresponding edges in η and ξ in the same way as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' if the edges are of the form ei,1 for some i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Below whenever we say that we stop the coupling, afterwards we continue (X, η) and (Y, ξ) by letting them attempt jumps at the points of P1, P2 and P3 (the latter only for X) and each edge of Y in its infected set refreshes also at the points of an additional Poisson process �P of parameter p(2d − 2)Z−1 λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' If an edge of the infected set of Y refreshes according to this Poisson process, then we do not remove it from the infected set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' However, if that edge is of the form ei,1 for some i, then we update its state in ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let (Ti) be the jump times of P3 and let S be the first point of P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We stop the coupling at time S ∧ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For every edge e, we let E(e) be the first time that the state of e is examined by Y and C(e) be the first time the edge e is crossed by Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' When E(e) < T1 ∧ S, then for times s ∈ [E(e), C(e) ∧ T1 ∧ S] we set ξs(e) = ηs(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' At time T1, the walk X attempts a jump in one of the 2d − 2 directions other than e1 and −e1 chosen uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For each edge e such that E(e) ∈ (T1, T2 ∧ S) and for times t ∈ [E(e), C(e) ∧ T2 ∧ S] we set ξt(e) = ηt(Xt + e1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' During the time interval (C(e) ∧ T2 ∧ S, T2 ∧ S), we refresh the edge e in the environment ξ also at the points of an additional independent Poisson process �P of parameter p(2d − 2)Z−1 λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Note as mentioned above that these updates do not affect the infected set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We let (τi) be the successive times at which the infected set of X becomes equal to the empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then by the definition of the process Y we see that also the infected set of Y becomes empty at times τi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We note that in the above coupling once an edge e has been examined by Y , it then refreshes at rate �µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Indeed, up until the first point of P3 it updates at rate µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' If the edge that X examined at time T1 is open, which happens with probability p (and hence the rate of this happening is p(2d − 2)Z−1 λ ), since this is the first time that edge is being examined, then the state of the edge e in ξ refreshes to XT1 +e1 which is distributed according to Ber(p) (again we are using that the edge XT1 + e1 has not been examined before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using the sequence (τi) we see that the speed vY of Y is given by vY (λ) = E[Yτ1] E[τ1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 14 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For all p ∈ (0, 1) and µ > 0, there exist constants λ0, c > 0 such that for all λ ≥ λ0, ��vY (λ) − v(λ) �� ≤ c exp(−2λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Recall that S is the first point of P2 and (Ti) are the points of P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let A be the event that the coupling stops before time τ1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' A = {S < τ1} ∪ {T2 < τ1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then we have |vY (λ) − v(λ)| ≤ 1 E[τ1] · E � |X1 τ1 − Yτ1|1(A) � We write U(t) for the total number of points of P1 ∪ P2 ∪ P3 that have arrived before time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then we obtain E � |X1 τ1 − Yτ1|1(A) � ≤ 2 E[1(A) · U(τ1)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' A key observation is that τ1 and U(τ1) only depend on P1 ∪ P2 ∪ P3 and the evolution of the size of the infected set, which increases at the points of the Poisson process P1 ∪ P2 ∪ P3 and decreases at an independent rate µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' This together with the thinning property of Poisson processes yields that, conditional on U(τ1), the numbers of points in P2[0, τ1] and P3[0, τ1] follow the binomial distribution with parameters (U(τ1), e−λZ−1 λ ) and (U(τ1), (2d − 2)Z−1 λ ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using this we then get E[1(A) · U(τ1)] ≤ E � U(τ1) � 1 − � 1 − e−λZ−1 λ �U(τ1)�� + E � U(τ1) � 1 − � 1 − (2d − 2)Z−1 λ �U(τ1) − U(τ1)(2d − 2)Z−1 λ (1 − (2d − 2)Z−1 λ )U(τ1)−1�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using that for all x ∈ (0, 1) and a ∈ N we have (1 − x)a ≥ 1 − ax, we get E[1(A) · U(τ1)] ≤ E � (U(τ1))2e−λZ−1 λ + U(τ1)(U(τ1) − 1)(2d − 2)2Z−2 λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Since Z−1 λ = O(e−λ) and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 we have for a positive constant Cµ that E � (U(τ1))2� ≤ Cµ < ∞, it follows that E[1(A) · U(τ1)] ≤ O(e−2λ) with the implicit constants depending only on µ, and hence this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let (�Y , �ξ) be a biased random walk on dynamical percolation on Z with parameters �µ, p that jumps to the right at rate eλZ−1 λ and to the left at rate e−λZ−1 λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then the speed of Y is the same as the speed of �Y , since to determine it we only need to know the state of every edge after the first time the walk examines it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let δ = δ(λ) = (2d − 2)Z−1 λ and consider the process (Y t, ξt) := (�Yt(1−δ)−1, ξt(1−δ)−1), ∀ t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then (Y , ξ) is a one-dimensional biased random walk in dynamical percolation with parameters (p, µ, λ), where µ := µ + pδ 1 − δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 15 We write vY for the speed of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let Z be a random walk on dynamical percolation on Z with parameters µ, p that only attempts jumps to the right at rate 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4 we get that the speed of Z is given by vZ = µp 1 − p + µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We now want to compare the speed of Z to the speed of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We couple Z and Y by letting them evolve together until the first time that Y attempts a jump to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Afterwards, they both attempt jumps at the same times and they use the same Poisson process to remove edges from their infected sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We let τ1 be their first regeneration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let A be the event that the walker Y attempts at least one jump to the left before time τ1 and let U(τ1) be the total number of jump attempts before time τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Therefore we deduce ���� µp 1 − p + µ − vY (λ) ���� ≤ 1 E[τ1]E � |Zτ1 − Y τ1|1(A) � ≤ 2 E[τ1]E[U(τ1)1(A)] ≤ 2 E[τ1]E � (U(τ1))2 · e−λ eλ + e−λ � ≤ C′ exp(−2λ), where C′ is a constant only depending on µ and where for the third inequality we used a union bound and for the last one we used that U(τ1) has exponential tails by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using now that vY (λ) = (1 − δ)vY (λ) and a straightforward calculation we finally conclude that for λ sufficiently large vY (λ) = µp 1 − p + µ − (2d − 2)p (1 − p + µ)2 (µ2 − p(1 − p))Z−1 λ + O(e−2λ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='12) where the implicit constant in O depends only on µ and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' This together with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='8 finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3 Asymptotic derivative of the speed In this section we prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3 by constructing a coupling between two walks with different bias parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let ε > 0 and let (Xλ t , ηt)t≥0 and (Xλ+ε t , ηt)t≥0 be λ-biased (respectively (λ + ε)-biased) random walks on dynamical percolation in Zd with parameters µ and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='9 (Coupling between Xλ and Xλ+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We start both walks from 0 and we let them both attempt jumps at the points of a Poisson process P = (Pt)t≥0 of rate 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We also let both envi- ronments evolve together until the first very bad point defined below and afterwards we couple the environments by using the same rate µ Poisson process for the removal of edges of their respective infected sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Whenever a clock of P rings indicating the jump attempt at time t of both walkers, we sample a random variable U uniformly on [0, 1] and proceed as follows: (1) If U < (2d−2)/Zλ+ε, then we let both walkers attempt a jump into one of the 2d−2 directions different from e1 and −e1 chosen uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (2) If U ∈ [(2d − 2)/Zλ+ε, (2d − 2)/Zλ], then we let the Xλ walk attempt a jump into one of the 2d − 2 directions different from e1 and −e1 chosen uniformly at random, while we let the Xλ+ε walk attempt a jump in the e1 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 16 (3) If U ∈ [(2d − 2)/Zλ, (2d − 2)/Zλ + e−λ−ε/Zλ+ε], then we let both walkers attempt a jump in the −e1 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (4) If U ∈ [(2d − 2)/Zλ + e−λ−ε/Zλ+ε, 1 − eλ/Zλ], then we let the Xλ walk attempt a jump in the −e1 direction, while we let the Xλ+ε walk attempt a jump in the e1 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (5) If U > 1 − eλ/Zλ, then we let both walkers attempt a jump in the e1 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In the following, we let (Ti)i∈N be the points of the Poisson process (Pt)t≥0 and we colour each point independently according to the outcome of the corresponding random variable U in the above coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We say that a point is good if the corresponding random variable U satisfies (5), we say that a point is bad if U satisfies (1) or (3), and we say that a point is very bad if U satisfies (2) or (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Notice that good, bad, and very bad points are again independent Poisson point processes of intensities qg := eλZ−1 λ for good points, qb := (2d − 2 + e−λ−ε)/Zλ+ε for bad points, and qvb := eλ+ε Zλ+ε − eλ Zλ > 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='13) for very bad points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Note that there exist constants c1, c2, c3 > 0 and λ0 > 0 so that for all ε ∈ (0, 1) and λ ≥ λ0 we get |qb − (2d − 2) exp(−λ − ε)| ≤ c1 exp(−2λ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='14) and |qvb − ε(2d − 2) exp(−λ)| ≤ c2ε exp(−2λ) + c3ε2 exp(−λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='15) Moreover, note that the above coupling between the two random walkers ensures that they stay together until the first very bad point and both infected sets have the same size at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Therefore, both processes have the same sequence of regeneration times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We let τ1 be their first regeneration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let G be the event that there is no bad point up to time τ1 and for every ℓ ∈ N let Vℓ be the event that Tℓ is the unique very bad point of U(τ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let R be the event that at the first very bad point the walk Xλ attempts a move in one of 2d − 2 directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We write U(t) for the number of points of the Poisson process P that have arrived by time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We write for all x ∈ Zd |x|1 := x · e1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='16) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' There exists a positive constant c = cd so that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let p ∈ (0, 1) and µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For all k ∈ N and ℓ ≤ k we have P(Gc | U(τ1) = k, Vℓ) ≤ (k − 1) · qb and P(Rc | U(τ1) = k, Vℓ, G) ≤ ce−λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='17) Moreover, there exist functions f = fµ,p, g = gµ,p : N × N → R+, which in particular do not depend on λ and ε, such that E � |Xλ+ε τ1 |1 ��� U(τ1) = k, Vℓ, G � = f(k, ℓ) and E � |Xλ τ1|1 ��� U(τ1) = k, Vℓ, G, R � = g(k, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Since the distribution of U(τ1) is independent of the colouring of the Poisson process P, it follows that conditionally on U(τ1) = k and Vℓ, every point Ti for i ≤ k with i ̸= ℓ has probability qb of being a bad point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using this together with a union bound we deduce P(Gc | U(τ1) = k, Vℓ) ≤ (k − 1) · qb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 17 Using again the independence between U(τ1) and the colouring, we obtain P(Rc | U(τ1) = k, Vℓ, G) = 1 − eλZ−1 λ − (2d − 2)Z−1 λ − e−λ−εZ−1 λ+ε qvb ≤ ce−λ for a suitable choice of c, completing the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Recall that (Ti) are the points of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We notice that on the event {U(τ1) = k} ∩ Vℓ ∩ G ∩ R we can write |Xλ+ε τ1 |1 = k � i=1 1(ηTi(Xλ+ε Ti− , Xλ+ε Ti− + e1) = 1) and |Xλ τ1|1 = k � i=1 i̸=ℓ 1(ηTi(Xλ Ti−, Xλ Ti− + e1) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using the independence between the Poisson process P = (Ti) and the colouring of each point as good, bad or very bad together with the definition of the regeneration time τ1 which is independent of the colouring of the process P (because even if we examine the same edge multiple times we still add a copy of it to the infected set), we see that L((T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' , Tk), η | U(τ1) = k, Vℓ, G, R) = L((T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' , Tk), η | U(τ1) = k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In particular, this shows that the conditional law of ((T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' , Tk), η) given U(τ1) = k, Vℓ, G, R is independent of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We note that under this conditioning, Xλ+ε becomes a walk that only attempts jumps to the right at the times T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' , Tk and Xλ attempts jumps to the right at the times Ti for i ≤ k and i ̸= ℓ and attempts a jump to one of 2d − 2 directions at time Tℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Therefore, we deduce that there exist functions f = fµ,p and g = gµ,p independent of λ and ε so that E � |Xλ+ε τ1 |1 ��� G, U(τ1) = k, Vℓ � = f(k, ℓ) and E � |Xλ τ1|1 ��� G, U(τ1) = k, Vℓ, R � = g(k, ℓ) and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We are now ready to prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We start the proof by recalling from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 that v(λ + ε) − v(λ) = E[τ1]−1E � |Xλ+ε τ1 |1 − |Xλ τ1|1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let Aε be the event that there exists a very bad point before τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then we have v(λ + ε) − v(λ) = E[τ1]−1E � |Xλ+ε τ1 |1 − |Xλ τ1|1 ��� Aε � P(Aε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='18) Recall that U(t) stands for the number of points of the Poisson process P of rate 1 up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Since the assignment of good/bad/very bad points to the points of the Poisson process is independent of the value of τ1, we get P(Aε) = E � 1 − (1 − qvb)U(τ1)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Since 1 − (1 − qvb)U(τ1) ≤ qvb · U(τ1) by the dominated convergence theorem and L’Hˆopital’s rule, recalling the approximation of qvb from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='15), we obtain lim ε→0 P(Aε) ε = E � lim ε→0 1 ε � 1 − � 1 − ε(2d − 2) exp(−λ) + O(εe−2λ + ε2) �U(τ1)�� = (2d − 2)e−λ · E[U(τ1)] + O(e−2λ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='19) 18 where the implicit constant only depends on µ and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We next prove that there exists a positive constant �Cµ,p,d depending only on µ, p and d such that lim ε→0 E � |Xλ+ε τ1 |1 − |Xλ τ1|1 ��� Aε � = �Cµ,p,d + O(e−λ), where the implicit constant in O depends only on µ and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We define �Aε to be the event that there is a unique very bad point up to time τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' First, we note that P � �Aε ��� Aε � = E � U(τ1) · qvb · (1 − qvb)U(τ1)−1� E � 1 − (1 − qvb)U(τ1)� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='20) and using similar arguments as above we get that lim ε→0 P � �Aε ��� Aε � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='21) We now have E � |Xλ+ε τ1 |1 ��� Aε � = E � |Xλ+ε τ1 |1 ��� �Aε � P � �Aε ��� Aε � + E � |Xλ+ε τ1 |1 ��� �Ac ε ∩ Aε � P � �Ac ε ��� Aε � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='22) and similarly for Xλ τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' As |Xλ+ε τ1 | ≤ U(τ1), we get similarly as above lim sup ε→0 E � |Xλ+ε τ1 |1 ��� �Ac ε ∩ Aε � ≤ lim ε→0 E � U(τ1) ��� �Ac ε ∩ Aε � = E � (U(τ1))2(U(τ1) − 1) � E[U(τ1)(U(τ1) − 1)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Since E[U(τ1)] > 1 and using that U(τ1) has exponential tails by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='21) gives that the second term appearing in the sum in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='22) converges to 0 as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For the first expectation appearing on the right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='22) we have E � |Xλ+ε τ1 |1 ��� �Aε � = � k � ℓ≤k E � |Xλ+ε τ1 |1 ��� U(τ1) = k, Vℓ � P � U(τ1) = k, Vℓ ��� �Aε � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='23) and similarly for Xλ τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For each k and ℓ ≤ k we have E � |Xλ+ε τ1 |1 ��� U(τ1) = k, Vℓ � = E � |Xλ+ε τ1 |1 ��� U(τ1) = k, Vℓ, G � + � E � |Xλ+ε τ1 |1 ��� U(τ1) = k, Vℓ, Gc� − E � |Xλ+ε τ1 |1 ��� U(τ1) = k, Vℓ, G �� P(Gc | U(τ1) = k, Vℓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let us remark that in the case of Xλ we also add the event R to the intersection above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using again the obvious bound |Xλ+ε τ1 |1 ≤ U(τ1), all four statements of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='10 and equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='15) we get E � |Xλ+ε τ1 |1 ��� U(τ1) = k, Vℓ � = f(k, ℓ) + O(k2 · e−λ) and E � |Xλ τ1|1 ��� U(τ1) = k, Vℓ � = g(k, ℓ) + O(k2 · e−λ), where the implicit constants in the terms O above only depend on µ and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Plugging these back into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='23) we deduce E � |Xλ+ε τ1 |1 ��� �Aε � = � k � ℓ≤k (f(k, ℓ) + O(k2 · e−λ))P � U(τ1) = k, Vℓ ��� �Aε � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='24) 19 and similarly for Xλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using again the independence between U(τ1) and the colouring, we have P � U(τ1) = k, Vℓ ��� �Aε � = P(U(τ1) = k) · qvb · (1 − qvb)k−1 E � U(τ1) · qvb · (1 − qvb)U(τ1)−1� , and hence since qvb → 0 as ε → 0 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='15), we deduce lim ε→0 P � U(τ1) = k, Vℓ ��� �Aε � = P(U(τ1) = k) E[U(τ1)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='25) Using again the obvious bound |Xλ+ε τ1 |1 ≤ U(τ1), and hence also that f(k, ℓ) ≤ k, plugging (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='24) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='22) and using the dominated convergence theorem we can take the limit as ε → 0 and use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='25) to obtain lim ε→0 E � |Xλ+ε τ1 |1 ��� Aε � = � k � ℓ≤k f(k, ℓ) · P(U(τ1) = k) E[U(τ1)] + O � e−λ · � k k3 · P(U(τ1) = k) E[U(τ1)] � = � k � ℓ≤k f(k, ℓ) · P(U(τ1) = k) E[U(τ1)] + O(e−λ), where the implicit constant in O only depends on µ and d and where for the last equality we used that U(τ1) has exponential tails by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 again, and hence a finite third moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' The analogous equality holds for Xλ with f replaced by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Therefore, these together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='19) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='18) imply that lim ε→0 v(λ + ε) − v(λ) ε = (2d − 2) · E[U(τ1)] E[τ1] � k � ℓ≤k (f(k, ℓ) − g(k, ℓ)) · P(U(τ1) = k) E[U(τ1)] e−λ + O(e−2λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' This now finishes the proof as f and g are functions that only depend on µ and p and not on λ, while the implicit constant in O depends only on µ and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 5 Strict monotonicity of the speed for large µ or p close to 1 As already mentioned in the introduction and as we saw in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1, for d = 1, the function λ �→ v(λ) is strictly increasing for any fixed choice of the percolation parameters p ∈ (0, 1] and µ > 0 due to a coupling argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let us emphasise that this argument cannot be extended for d ≥ 2, as Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3 demonstrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' However, we identify in the following two regimes of parameters µ and p in d ≥ 2 dimensions, where the speed is strictly increasing for all λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Recall the function f(λ) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='10) as well as Zλ from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1) and Z′ λ = eλ − e−λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Moreover, for k, ℓ, m ∈ N, we write fk,ℓ,m(λ) := (k − ℓ)eλ(k−ℓ) � 2d Zλ �m � k − ℓ − m · Z′ λ Zλ � P0 � (R, L) = (k, ℓ), U(τ1) = m � and recall from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='11) that f ′(λ) = � m∈N � k+ℓ≤m fk,ℓ,m(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Fix p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' There exists some constant ˜µ = ˜µ(p) > 0 such that for all µ > ˜µ, we have that λ �→ vµ,p(λ) is strictly increasing in λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 20 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' It suffices to prove that f ′(λ) is strictly positive for all λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For all i, j ∈ N and m ≥ 2 using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 we have P0 � (R, L) = (i, j), U(τ1) = m) ≤ exp(−cµm � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1) for some constant cµ with cµ → ∞ as µ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1 and the construction of the regeneration time τ1 we get P0 � (R, L) = (1, 0), U(τ1) = 1 � = P0 � (R, L) = (0, 1), U(τ1) = 1 � ≥ p · 1 2d · µ µ + 1 ≥ p 4d , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2) for all µ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For every m we let Am := � (k, ℓ) ∈ N2 : k + ℓ ≤ m and k − ℓ ≤ m − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2) and the definition of Zλ, we see that the decay of f1,0,1(λ) in λ is of the same order as sup(k,l)∈A2 fk,l,2(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Hence, for all µ sufficiently large, together with the exponential decay in m in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='1), a computation shows that for all λ > 0, � m≥2 � (k,ℓ)∈Am |fk,ℓ,m(λ)| ≤ 1 2(f1,0,1(λ) + f0,1,1(λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Since fm,0,m(λ) ≥ f0,m,m(λ) for all λ > 0 and m ∈ N, we obtain that f ′(λ) ≥ 1 2(f1,0,1(λ) + f0,1,1(λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using again (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2) we get that f1,0,1(λ) + f0,1,1(λ) ≥ 2d Zλ (2d − 2)(eλ + e−λ) + 4 Zλ p 4d > 0 for all λ > 0 and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Fix µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' There exists some constant ˜p = ˜p(µ) > 0 such that for all p ∈ (˜p, 1), we have that λ �→ v(λ) is strictly increasing in λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Let p be sufficiently close to 1 so that µ2 > p(1−p) and let λ0 = λ0(µ) be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' For all λ > λ0 the speed is strictly increasing by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Thus it remains to show that the speed is strictly increasing for all λ ∈ (0, λ0] for all p sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' To do this, we will prove that v′(λ) > 0 for all such λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' In this proof we want to emphasise the dependence of the speed on the percolation parameter p, so we write v(λ, p) = v(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Observe that when p = 1, the speed v′(λ, 1) ≥ cd,µ · e−λ for all λ > 0, where cd,µ is a constant depending on d and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' It thus suffices to prove that for p sufficiently close to 1, |v′(λ, p) − v′(λ, 1)| ≤ cd,µ 2 e−λ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3) uniformly for all λ ∈ (0, λ0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Recall the expression for v′(λ, p) from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We want to compare Eλ,p � (X1 τ1)2� to Eλ,1 � (X1 τ1)2� and also Eλ,p � X1 τ1 · U(τ1) � to Eλ,1 � X1 τ1 · U(τ1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' To do this, we couple the walks in the two environments by letting them attempt the same jumps at the same times and using the same Poisson process of rate µ to remove edges from their infected sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We let τ1 be their first regeneration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We now let κ be the first time when the walk in the p-dynamical percolation 21 process attempts a jump along a closed edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Then up until time κ the two walks are in the same location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Note that κ stochastically dominates a geometric random variable of parameter 1 − p, because for all s < t and all edges e we have Pp(e is open at time t | e is open at time s) ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' By a union bound we get P(κ < U(τ1)) ≤ P � U(τ1) > 1 √1 − p � + P � κ < 1 √1 − p � ≤ Cµ · � 1 − p, where Cµ is a constant depending on µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Using this and the bound |X1 τ1| ≤ U(τ1) we get |Eλ,p � (X1 τ1)2� − Eλ,1 � (X1 τ1)2� | ≤ 2Eλ,p � (U(τ1))21(κ < U(τ1)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' By the Cauchy-Schwarz inequality and the exponential tails of U(τ1) uniformly in p by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='2 we deduce Eλ,p � (U(τ1))21(κ < U(τ1)) � ≤ C′ µ(1 − p)1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Similarly we can bound the remaining terms appearing in v′(λ), and hence we see that taking p = p(λ0, µ) sufficiently close to 1, we get (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content='3) and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' We thank Frank den Hollander and Remco van der Hofstad for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' DS acknowl- edges the DAAD PRIME program for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' A¨ıd´ekon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Speed of the biased random walk on a Galton–Watson tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Probability Theory and Related Fields, 159(3):597–617, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Axelson-Fisk and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' H¨aggstr¨om.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' Biased random walk in a one-dimensional percolation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE4T4oBgHgl3EQfrg0j/content/2301.05208v1.pdf'} +page_content=' 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b/i9E0T4oBgHgl3EQf7AKY/content/tmp_files/2301.02771v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9d0d1826c7210ef3559330f98b54ef564781ddb --- /dev/null +++ b/i9E0T4oBgHgl3EQf7AKY/content/tmp_files/2301.02771v1.pdf.txt @@ -0,0 +1,821 @@ +Accepted by 2022 IEEE Globecom conference, ©2022 IEEE +Hierarchical Reinforcement Learning for +RIS-Assisted Energy-Efficient RAN +Hao Zhou1, Long Kong1, Medhat Elsayed2, Majid Bavand2, Raimundas Gaigalas3, +Steve Furr2, and Melike Erol-Kantarci1, Senior Member, IEEE +1School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, Ontario, Canada +2 Ericsson Canada, Ottawa, Ontario, Canada +3 Ericsson Sweden, Stockholm County, Sweden +Emails: {hzhou98, lkong2, melike.erolkantarci}@uottawa.ca +{medhat.elsayed,majid.bavand,raimundas.gaigalas,steve.furr}@ericsson.com +Abstract—Reconfigurable intelligent surface (RIS) is emerging +as a promising technology to boost the energy efficiency (EE) +of 5G beyond and 6G networks. Inspired by this potential, in +this paper, we investigate the RIS-assisted energy-efficient radio +access networks (RAN). In particular, we combine RIS with sleep +control techniques, and develop a hierarchical reinforcement +learning (HRL) algorithm for network management. In HRL, the +meta-controller decides the on/off status of the small base stations +(SBSs) in heterogeneous networks, while the sub-controller can +change the transmission power levels of SBSs to save energy. The +simulations show that the RIS-assisted sleep control can achieve +significantly lower energy consumption, higher throughput, and +more than doubled energy efficiency than no-RIS conditions. +Index Terms—Reconfigurable Intelligent Surfaces (RIS), Hier- +archical Reinforcement Learning (HRL), energy efficiency (EE), +radio access network (RAN). +I. INTRODUCTION +In line with previous generations of mobile wireless tech- +nologies, 5G is currently on the road to mass deployment. +Meanwhile, the energy efficiency of 5G has been a significant +research area in academia and industry [1]. As stated in [2], +one of the widely considered approaches for energy efficiency +has been the sleep control technique. Sleep control refers to +selectively turning radio transceivers or base stations (BSs) +to sleep mode. Different than 4G, recurring transmission of +always-on signals to guarantee network coverage, the 5G new +radio (NR) standard allows to deploy the sleep mode. +More recently, reconfigurable intelligent surfaces (RISs) are +proposed and considered as key enablers for future wireless +communications [3]. RIS is essentially an electronically oper- +ated metasurface controlled by programmable software, which +is physically equivalent to digitally controllable scatterers and +software defined surface [4]. A large number of small, low- +cost, and passive artificial “meta-atoms” integrated into the +RIS can smartly change the reflection direction towards any +desired users by tuning a series of phase shifters. Accordingly, +RISs have been designed for various scenarios and applications +including 6G, internet of things (IoT), smart cities [5], etc. The +main benefit of RIS lies in its capability of shaping the wireless +propagation environments by adjusting the signal reflections +[3]. Through this, the signal quality and connectivity can be +substantially improved. Furthermore, the energy consumption +of RIS is extremely low, which is a favourable property +compared to traditional relaying [6]. RIS’s capability and low- +power consumption features motivate us to investigate the RIS- +aided energy-efficient RAN. +Moreover, machine learning has been generally applied +for wireless network management for its advantage in han- +dling dynamic environment [7]. For example, in reinforcement +learning (RL), the optimization problem can be transformed +to the unified Markov decision process (MDPs), which avoids +the complexity of defining a dedicated optimization model. In +this paper, the main contribution is that we propose a novel hi- +erarchical reinforcement learning (HRL) architecture for RIS- +assisted sleep control in heterogeneous networks. Compared +with conventional RL that includes one standalone agent, HRL +defines a meta-controller and a sub-controller, which enables +higher exploration efficiency by the hierarchical architecture +[8]. In particular, we improve the energy efficiency (EE) in +two ways: we consider the macro base station (MBS) as the +meta-controller to implement the sleep control of small base +stations (SBSs) to save energy, and SBSs as sub-controllers +to decide its own transmission power level to reduce energy +consumption. Besides, RIS is deployed to improve the signal +propagation environment and increase the channel capac- +ity. Finally, the simulations show that combining RIS with +sleep control can achieve lower energy consumption, higher +throughput, and more than doubled EE than the standalone +sleep control strategy. +II. RELATED WORK +A. Machine learning based sleep control +The flourishing machine learning techniques offer promising +opportunities for network control and management [9]. The +deep Q-network is deployed in [10] for the sleep control of +renewable energy-powered BSs, where the SBSs can share +their energy by a micro-grid. The neural network is applied +in [11] to predict both traffic demand and energy production, +and then the prediction results are used for sleep control of +BSs. Similarly, [12] deploys deep neural networks to predict +the traffic patterns, and actor-critic reinforcement learning is +used for dynamic sleep control. Different from aforementioned +works, here we apply the HRL algorithm, including a meta- +controller for SBS sleep control and sub-controllers for the +1 +arXiv:2301.02771v1 [eess.SP] 7 Jan 2023 + +Accepted by 2022 IEEE Globecom conference, ©2022 IEEE +transmission power control. This hierarchical control strategy +allows more efficient exploration of the environment, and +mitigates the long convergence issue of conventional RL [8]. +B. RIS related researches +RIS, being an appealing approach, has become a hot topic +for researchers from both wireless communication and signal +processing communities [13]. The machine learning-enabled +RIS-assisted wireless communication systems have been under +exploration in terms of channel modelling [4], [14], chan- +nel estimation, EE [15], etc. More specifically, the authors +in [4] and [14] applied the unsupervised machine learning +tool, namely, the expectation-maximization (EM) algorithm to +model the RIS-assisted wireless communication links. It is also +demonstrated that the machine learning methods are able to +provide better performance than central limit theorem-based +approaches [4] [14]. Besides, Lee et al. in [15] deployed the +deep reinforcement learning to improve the EE of the RIS- +aided cellular communication systems, but the sleep control is +not involved. To the best knowledge of the authors, no work +before has ever investigated the EE problem with sleep control +and RIS embedded in the cellular communication systems. +III. NETWORK AND SYSTEM MODEL +As illustrated in Fig.1, we consider a heterogeneous network +that includes one MBS and several SBSs. The SBS may switch +to the sleep mode when the traffic demand drops, which will +reduce the energy consumption. It is assumed the MBS can +take over the active user equipment (UEs) that were previously +associated with those small cells. On the other hand, high- +density buildings in the urban area lead to high penetration +loss and lower received signal-to-interference-plus-noise ratio +(SINR) for direct transmissions. To this end, we deploy RIS to +reflect the signal from MBS and mitigate the high penetration +loss of direct transmissions. In this work, it is worth noting that +we save energy in two ways: (i) the meta-controller decides the +on/off status of SBSs when traffic load changes, and (ii) the +sub-controller decides the transmission power of active SBSs. +This hierarchical architecture enables a higher management +efficiency, which will be introduced in detail in Section IV. +A. RIS-Assisted Channel Model +It is assumed that UEs can receive the signal from BSs by +direct and indirect transmissions. The direct link is considered +as non-line-of-sight (NLOS) transmission due to the dense +buildings in the urban area. The baseband equivalent channel +between BS-UE is given by: +HBk = gB,khB,k, +(1) +where gB,k is the path loss from BS to UE, and hB,k is a +complex Gaussian distributed random vector with 0 mean and +unit variance, i.e., CN (0, 1). +The indirect link consists of the BS-RIS and RIS-UE links. +Given the fact that RIS is designed to be deployed on the top +Fig. 1. RIS-aided heterogeneous network. +or surface of tall buildings, the BS-RIS link is assumed to be +line-of-sight (LOS) transmission: +HBR = gBR[h1, h2, ..., hN] ∈ C1×N, +(2) +where N is the number of RIS elements, gBR is the path loss +between BS and RIS, and hN = exp +� +−2jπdN +λ +� +is the phase +difference, herein j = √−1, dN is the distance between BS +and RIS element N, λ is the signal wavelength. +Then the RIS will reflect the signal to UEs via a phase +shift vector θ = [θ1, θ2, ..., θN], we define a diagonal matrix +accordingly: +Θ = diag(β1ejθ1, β2ejθ2, ..., βNejθN ) ∈ CN×N, +(3) +where βN is the amplitude reflection coefficient and β ∈ [0, 1]. +Considering the complex environment in the UE side, the +RIS-UE link is presumed to be NLOS transmission, and we +have +HRk = gR,k[h′ +1,k, h′ +2,k, ..., h′ +N,k] ∈ C1×N, +(4) +where gR,k is the path loss between RIS and UE k, and h′ +N,k +is the small scale fading of the signal reflected by the RIS +element N. Path loss is given by gRk = d−αRk/2 +Rk +, where dRk +is the distance from RIS to UE k, and αRk is the pathloss +exponent. +Finally, the channel gain from BS j to UE k is: +Gj,k = |HB,k + HRkΘHT +BR|2. +(5) +For the phase shift control, inspired by [6], we assume +the channel state informations (CSIs) are perfectly shared +between BS and RIS. The RIS phase shift is calculated by: +θN = arg(HBk)−arg(gBRhNgR,kh′ +N,k) to give every term in +HRkΘHT +BR the same phase as HBk, then the total received +signal will be strengthened. +For a downlink transmission between BS j and UE k, the +transmission rate is: +Cj,k =bj,k log2 (1+ +� +r∈Rj,k pj,raj,k,rGj,k,r +bj,kN0 + +� +j′∈J−j +� +k′∈Kj′ +� +r′∈Rj′ +pj′,r′aj′,k′,r′Gj′,k′,r′ +� +� +� , (6) +2 + +Agent +Extrinsic reward +Meta-controller +Goals +Critic +Intrinsic +High-level policies +reward +Sub-controller +Actions +Heterogeneous network environment +Macro-BS +Small +Throughput +RIS +cell +capacity +RIS +? +Sleeping SBS +SBS +Small +Throughput +cell +capacity +Direct transmission +RIS assisted transmissionAccepted by 2022 IEEE Globecom conference, ©2022 IEEE +where Rj,k is the set of resource blocks (RBs) allocated to +UE k by BS j [16], bj,k is the total bandwidth allocated to +UE k, N0 is the power spectral density of noise, and pj,r is +the transmission power of RB r allocated by BS j. aj,k,r is a +binary indicator. aj,k,r = 1 if the RB r is allocated to UE k; +otherwise aj,k,r = 0. Gj,k,r denotes the channel gain between +BS j and UE k. J−j denotes the set of BSs except BS j, Kj′ +is the UE set of BS j′, and Rj′ is the RB set in BS j′. In +this work, the RBs are allocated by the proportional fairness +method [7]. +B. Energy Consumption Model +The energy consumption model for the BS is: +Pin = +� P0 + δpPout, +0 < Pout ≤ Pmax, +Psleep, +Pout = 0, +(7) +where P0 is the fixed power consumption, δp is the slope of +load-dependent power consumption, Pout is the transmission +power, Pmax is the maximum transmission power, and Psleep +is the constant power consumption in sleep mode [17]. +C. Problem Formulation +The overall objective is to maximize the total EE, achieve +the desired SINR for UEs, and prevent the BSs from overload- +ing. Here the overloading means that current traffic demand +has exceeded the transmission capability of one BS, and then +the attached UEs may experience a long delay. The problem +formulation is given by: +max +Pj +� +j∈J +� +k∈Kj Wj,k +� +j∈JPj +− φnod, +(8) +s.t. +(5) (6) (7), +(8a) +� +k∈Kj +� +r∈Rj +aj,k,r ≤ |Rj|, +(8b) +� +k∈Kj +aj,k,r ≤ 1, +(8c) +SINRthr ≤ SINRj,k, +(8d) +where Wj,k is the throughput of UE k in BS j, Pj is the +power consumption of BS j, and nod is the number of BSs +that are overloaded. We apply φ as a penalty factor to prevent +overloading. Equation (8a) is the system operation constraint, +equation (8b) indicates the number of available RBs can +not exceed |Rj|, equation (8c) means one RB can only be +allocated to at most one UE, and equation (8d) is the SINR +threshold constraint of UEs. +On one hand, turning off SBSs can greatly reduce energy +consumption. But it will also increase the risk of MBS +overloading, since the MBS has to take over the UEs of +the sleeping small cells. Therefore, to maximize the total +objective, we have to intelligently control the on/off status +of SBSs to reduce the energy cost and overload risk, and +following we will introduce an HRL based architecture. +IV. HIERARCHICAL REINFORCEMENT LEARNING FOR +ENERGY-EFFICIENT RAN +A. Hierarchical Reinforcement Learning +In traditional RL, the problem is defined by an MDP < +S, A, T, R >, where S is the set of states, A is the set of +actions, T is the transition probability with T : S × A × +S, and R is the reward function. Then, one standalone agent +will interact with the environment to maximize its long-term +expected reward [18]. +By contrast, in HRL, the agent consists of two controllers, +namely meta-controller and sub-controller [19]. Accordingly, +the MDP is rewritten by < S, A, T, R, G>, where Gindicates +the set of goals. Based on current state s ∈ S, the meta- +controller will generate high-level goals g ∈ +G for sub- +controllers. Then, these goals are transformed to high-level +policies by the critic. Consequently, the sub-controller chooses +low-level actions a ∈ A according to high-level policies, and +receives an intrinsic reward rin. Finally, the meta-controller +will receive an extrinsic reward rex from the environment, and +select new goals g′ for the sub-controller. The idea behind the +HRL is to introduce hierarchy architecture in RL. In particular, +the meta-controller will produce high-level policies to guide +the low-level action selection of the sub-controller. Compared +with traditional RL, HRL is considered as a more efficient +learning method due to the hierarchical architecture, and by +dividing sub-goals it allows better management of multiple +functionalities in RAN. +B. MDP Definition +To transform the problem formulation showed by equation +(8) into the HRL notation, the following MDP for sub- +controllers and meta-controller are defined. +Each SBS is regarded as a sub-controller, the MDP is +defined by: +• State: The state ssub of SBS j is defined by its traffic +load ratio ssub = {dSBS}, which is given by: +dSBS = +� +k∈Kj Dj,k +Dmax +j +, +(9) +where Kj indicates the set of UEs that are served by +SBS j, Dj,k is the traffic demand of UE k. Dmax +j +is +the max traffic load of SBS j, which is considered as +a constant value to normalize the current traffic load +of SBS. Meanwhile, note that the transmission demand +of UEs often shows strong statistical regularity, and we +assume the daily traffic load follows the patterns in [20]. +• Action: Based on ssub, the SBS may change its trans- +mission power PSBS to adapt the traffic demand. Then, +the action is defined by asub = {PSBS}. +• Intrinsic reward: The intrinsic reward of SBS is: +rin = +� +k∈Kj Wj,k +PSBS +− φnod, +(10) +where Wj,k, φ and nod have been defined in equation +(8). rin aims at maximizing its own EE and preventing +overloading. +3 + +Accepted by 2022 IEEE Globecom conference, ©2022 IEEE +The meta-controller is responsible for the high-level policies +for the agent. The MBS is defined as the meta-controller, and +its MDPs are: +• State: The state of meta-controller consists of the traffic +load ratio of SBSs: +smeta = {dSBS,j}, j ∈ JSBS, +(11) +where dSBS,j is the load ratio of SBS j, and JSBS is the +set of SBSs. +• Goals for sub-controller: With the traffic load status of +the SBSs, MBS can generate high-level policies for the +SBSs. The goals gmeta are turning on/off the SBSs: +gmeta = {qSBS,j}, j ∈ JSBS, +(12) +where qSBS,j is a binary variable to indicate the on/off +status of SBS j. qSBS,j = 1 means keeping the SBS j +active, otherwise qSBS,j = 0 denotes turning off SBS j +to save energy. +• Extrinsic reward: The meta-controller focuses more on +the overall performance of the whole cell. Accordingly, +the extrinsic reward is given by the objective of the +problem formulation in equation (8): +rex = +� +j∈J +� +k∈Kj Wj,k +� +j∈JPj +− φnod, +(13) +C. Q-value Update and Goal Selections +In this section, we introduce how to update the Q-values of +controllers, and the action and goal selection strategies. +The Q-values of meta-controller is updated by: +Qnew +meta(smeta, gmeta) = Qold +meta(smeta, gmeta)+ +α(rex + γ max +g +Qmeta(s′ +meta, g) − Qold +meta(smeta, gmeta)), +(14) +where s′ +meta denotes the next state, α is the learning rate, +and γ is the discount function (0 < α, γ < 1). Qold +meta +and Qnew +meta denote old and new Q-values for meta-controller, +which means the accumulated reward brought by state-goal +pair (smeta, gmeta). Then we use the ϵ-greedy policy for goal +selection: +π(smeta) = +� +arg max +g +Q(smeta, g), +rand > ϵ, +random goal selection, +rand ≤ ϵ. +(15) +where rand is a random number between 0 and 1, and ϵ < 1. +ϵ-greedy policy can balance the exploration and exploitation +of goals to maximize the long-term reward. +Similarly, for the sub-controller, the Q-values are updated: +Qnew +sub (ssub, gmeta, asub) = Qold +sub(ssub, gmeta, asub)+ +α(rin + γ max +a +Qsub(s′ +sub, g′ +meta, a) − Qold +sub(ssub, gmeta, asub)), +(16) +where s′ +sub is the next state, g′ +meta is the next goal generated +by meta-controller, Qnew +sub and Qold +sub are new and old Q-values +for sub-controller, respectively. We still use ϵ-greedy policy +for the action selection of sub-controller: +π(ssub) = +� +arg max +a +Q(ssub, gmeta, a), +rand > ϵ, +random action selection, +rand ≤ ϵ. +(17) +The HRL based sleep and transmission power control is +summarized in Algorithm 1. +Algorithm 1 HRL algorithm for SBS sleep and power control +h +1: Initialize: Wireless network and HRL parameters. +2: for episode=1 to Total do +3: +for MBS do +4: +With probability ϵ choose goals randomly, otherwise +select gmeta by arg +max +g +Q(smeta, g) (Shown by +equation (15)). +5: +for Each active SBS do +6: +With probability ϵ choose asub randomly, other- +wise select asub by arg +max +a +Q(ssub, gmeta, a) +(Shown by equation (17)). +7: +Calculating intrinsic reward rin, updating state +ssub and Q-values by equation (16). +8: +end for +9: +MBS calculates extrinsic reward rex, updating state +smeta and Q-values by equation (14). +10: +end for +11: end for +12: Output: Optimal SBS sleep and transmission power con- +trol strategy. +V. PERFORMANCE EVALUATION +A. Simulation Settings +In the simulations, we consider a dense urban environment +in the MATLAB simulation platform, where there are 4 SBSs +and 4 RISs. The coverage radius of MBS and SBS are +400m and 80m, respectively. The cell includes 50 randomly +distributed UEs. The fixed power consumption of MBS and +SBS are 130W and 75W, and the load-dependent power con- +sumption slope is 4.7 and 2.6 for MBS and SBS, respectively +[17]. We assume a deep sleep mode at the SBS with 0 power +consumption. Each RIS has 10 reflecting elements with 3 bits +Fig. 2. Daily traffic load pattern of residential area. +4 + +1 +0.8 +0.4 +0.2 +0 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +21 +23 +Time/HourAccepted by 2022 IEEE Globecom conference, ©2022 IEEE +phase shift resolution. We assume the RIS power consumption +is very low and it is not included in the power consumption. +The path loss exponent for LOS and NLOS are 2.5 and 3.5, +respectively [21]. The available bandwidth for each BS is +bR = 20 MHz. The traffic pattern is presumed to follow Fig. +2, which is a typical residential area traffic pattern [20]. The +initial learning rate is 0.95, and we decay the learning rate +after every several episodes for a stable learning performance, +and the discount factor is 0.3. The simulation is repeated for +10 runs in MATLAB, and we present the average results with +95% confidence interval. +B. Simulation Results +In this section, we include 4 cases: (1) no RIS and no +sleep control (typical-cell), (2) sleep control without RIS +(sleep-only), (3) RIS without sleep control (RIS-only), and +(4) combining RIS with sleep control (RIS-sleep). We apply +conventional Q-learning for case (1) to (3), and HRL for our +proposed case (4). +Fig. 3 to 5 first present the total power consumption of the +BSs, average throughput per UEs, and EE against peak traffic +load for the 4 cases, respectively. One can observe that: (i) +typical-cell, as a benchmark here, presents the highest power +consumption and lowest EE; (ii) comparison between typical- +cell and sleep-only in Fig. 3 demonstrates that sleep control +can significantly reduce the power consumption; (iii) the EE +results of typical-cell and RIS-only in Fig. 4 shows that RIS +is highly beneficial to the average throughput. +More specifically, as shown in Fig. 4, when the traffic load +is lower than 4 Mbps, the existing channel capacity is already +huge enough to serve the UEs. However, when the peak traffic +load becomes higher than 5 Mbps, RIS-only and RIS-sleep +show a higher throughput than other two cases, which can be +explained by RIS’s capability to improve the SINR of UEs. +When it comes to the EE metric, as shown by Fig. 5, +the case 4, namely RIS-sleep strategy, displays the best EE +performance. On the contrary, typical-cell shows the worst EE +performance due to the absence of both RIS and sleep control. +sleep-only and RIS-only have comparable EE. It is observed +that the former strategy has lower power consumption and +lower throughput, and RIS-only is the opposite (indicated by +Figs. 3 and 4). As a result, these two cases show a close EE. +When the peak traffic load is 8 Mbps, RIS-sleep achieves a +more than doubled EE than other cases. +To better explain how RIS and sleep control are combined, +sleep-only and RIS-sleep are compared in Fig. 6 in terms of +the possibility of keeping SBSs active. During the off-peak +period (from 3:00 to 9:00 in the traffic patterns shown by Fig. +2), most SBSs are shut off to save energy, and the existing +traffic demand is served by MBS. However, after 11:00, sleep- +only has to turn on most SBSs to satisfy the increasing traffic +load, otherwise the MBS will be overloaded and the total +throughput will be greatly affected. By contrast, RIS-sleep is +capable of keeping most SBSs sleep until 17:00, because MBS +can process the increasing traffic demand with a higher SINR +provided by RIS. +Fig. 3. Total power comparison of all BSs against peak traffic load. +Fig. 4. Average throughput per UE in the cell against peak traffic load. +Fig. 5. EE of the BSs against peak traffic load. +Fig. 6. Probability of keeping SBSs active under 8 Mbps peak traffic load. +Apart from the aforementioned discussions, we further +investigate the average SINR of UEs against the number of +RIS reflecting elements in Fig. 7 under different RIS phase +shift resolutions (PSR). A higher PSR generally indicates a +more accurate phase shift design. One can observe that more +5 + +0.7 +Case 1: No RIS + no sleep control +-Case 2: No RIS+ sleep control +0.6 +-Case 3: RIS + no sleep control +-Case 4: RIS + sleep control +0.5 +0.4 +0.3 +Cell +0.2 +2 +3 +4 +6 +7 +8 +9 +10 +Peak Traffic load per UE/Mbps7 +-Case 1:No RIS+ no sleep control +Case 2: No RIS+ sleep control +6 +-Case 3: RIS + no sleep control ++-Case 4: RIS + sleep control +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Peak Traffic load per UE/Mbps0.8 +-Case 1: No RIS + no sleep control +0.7 +-Case 2: No RIS+ sleep control +-Case 3: RIS + no sleep control +0.6 +-Case 4: RIS + sleep control +0.5 +0.4 +0.3 +0.2 +0.1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Peak Traffic load per UE/Mbps-Case 2: No RIS+ sleep control +-Case 4: RIS + sleep control +0.8 +SBS +0.4 +0.2 +3 +5 +9 +11 +13 +15 +17 +19 +21 +23 +Time/HourAccepted by 2022 IEEE Globecom conference, ©2022 IEEE +Fig. 7. Average SINR of UEs under different phase shift resolutions. +Fig. 8. Convergence performance analyses. +RIS elements and higher PSR are as expected essentially +useful to improve the SINR of UEs. On the other hand, the +improvements brought by PSR are barely observable from 3 +to 4 bits. +Finally, Fig. 8 presents the convergence performance. It +shows that both intrinsic and extrinsic rewards increase with +more iterations and finally converge, which means that meta +controller and sub-controller are well coordinated to maintain +the overall performance. +VI. CONCLUSION +The reconfigurable intelligent surface is a promising tech- +nology to enable 5G beyond and 6G networks. In this paper, +we combine reconfigurable intelligent surfaces with sleep +control to improve the energy efficiency of heterogeneous +5G radio access networks. We propose a hierarchical rein- +forcement learning-based method to optimize the sleep control +strategy of small base stations. Compared with the standalone +sleep control method, the simulations show a significantly +higher energy efficiency by jointly deploying reconfigurable +intelligent surface and sleep control in a hierarchical learning +framework. In addition, we conclude that (i) sleep control +largely contributes to reducing power consumption and im- +proving energy efficiency; (ii) reconfigurable intelligent sur- +face is beneficial to the average throughput, especially for high +traffic load conditions. In the future, we will investigate the +control strategy of the phase shift of reconfigurable intelligent +surfaces. +ACKNOWLEDGEMENT +This work has been supported by MITACS and Ericsson +Canada, and NSERC Collaborative Research and Training +Experience Program (CREATE) under Grant 497981. +REFERENCES +[1] M. Usama and M. Erol-Kantarci, “A survey on recent trends and open +issues in energy efficiency of 5g,” Sensors, vol. 19, no. 14, pp. 1–23, +Jul. 2019. +[2] J. Wu, Y. Zhang, M. Zukerman, and E. 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Desset, and et.al, “How much energy is needed +to run a wireless network?” IEEE Wireless Commun., vol. 18, no. 5, pp. +40–49, Oct. 2011. +[21] H. Cho, C. Liu, J. Lee, T. Noh, and T. Q. Quek, “Impact of elevated +base stations on the ultra-dense networks,” IEEE Commun. Lett., vol. 22, +no. 6, pp. 1268–1271, Apr. 2018. +6 + +33 +-PSR: 1 bit +-PSR:: 2 bit +-- PSR:: 3 bit +Average SINR [dB] +28 +PSR:: 4 bit +23 +18 +40 +80 +120 +160 +200 +Number of total RiS elements8 +3 +Extrinsic reward +Intrinsic reward +7.5 +Extrinsic reward +2.5 +Intrinsic reward +6.5 +6 +1.5 +0 +100 +200 +300 +400 +500 +Iterations \ No newline at end of file diff --git a/i9E0T4oBgHgl3EQf7AKY/content/tmp_files/load_file.txt b/i9E0T4oBgHgl3EQf7AKY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..428b35e2d07bca2abc8e25d8f383da1479d186c0 --- /dev/null +++ b/i9E0T4oBgHgl3EQf7AKY/content/tmp_files/load_file.txt @@ -0,0 +1,469 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf,len=468 +page_content='Accepted by 2022 IEEE Globecom conference, ©2022 IEEE Hierarchical Reinforcement Learning for RIS-Assisted Energy-Efficient RAN Hao Zhou1, Long Kong1, Medhat Elsayed2, Majid Bavand2, Raimundas Gaigalas3, Steve Furr2, and Melike Erol-Kantarci1, Senior Member, IEEE 1School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, Ontario, Canada 2 Ericsson Canada, Ottawa, Ontario, Canada 3 Ericsson Sweden, Stockholm County, Sweden Emails: {hzhou98, lkong2, melike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='erolkantarci}@uottawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='ca {medhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='elsayed,majid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='bavand,raimundas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='gaigalas,steve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='furr}@ericsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='com Abstract—Reconfigurable intelligent surface (RIS) is emerging as a promising technology to boost the energy efficiency (EE) of 5G beyond and 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Inspired by this potential, in this paper, we investigate the RIS-assisted energy-efficient radio access networks (RAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' In particular, we combine RIS with sleep control techniques, and develop a hierarchical reinforcement learning (HRL) algorithm for network management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' In HRL, the meta-controller decides the on/off status of the small base stations (SBSs) in heterogeneous networks, while the sub-controller can change the transmission power levels of SBSs to save energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The simulations show that the RIS-assisted sleep control can achieve significantly lower energy consumption, higher throughput, and more than doubled energy efficiency than no-RIS conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Index Terms—Reconfigurable Intelligent Surfaces (RIS), Hier- archical Reinforcement Learning (HRL), energy efficiency (EE), radio access network (RAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' INTRODUCTION In line with previous generations of mobile wireless tech- nologies, 5G is currently on the road to mass deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Meanwhile, the energy efficiency of 5G has been a significant research area in academia and industry [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' As stated in [2], one of the widely considered approaches for energy efficiency has been the sleep control technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Sleep control refers to selectively turning radio transceivers or base stations (BSs) to sleep mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Different than 4G, recurring transmission of always-on signals to guarantee network coverage, the 5G new radio (NR) standard allows to deploy the sleep mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' More recently, reconfigurable intelligent surfaces (RISs) are proposed and considered as key enablers for future wireless communications [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' RIS is essentially an electronically oper- ated metasurface controlled by programmable software, which is physically equivalent to digitally controllable scatterers and software defined surface [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' A large number of small, low- cost, and passive artificial “meta-atoms” integrated into the RIS can smartly change the reflection direction towards any desired users by tuning a series of phase shifters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Accordingly, RISs have been designed for various scenarios and applications including 6G, internet of things (IoT), smart cities [5], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The main benefit of RIS lies in its capability of shaping the wireless propagation environments by adjusting the signal reflections [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Through this, the signal quality and connectivity can be substantially improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Furthermore, the energy consumption of RIS is extremely low, which is a favourable property compared to traditional relaying [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' RIS’s capability and low- power consumption features motivate us to investigate the RIS- aided energy-efficient RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Moreover, machine learning has been generally applied for wireless network management for its advantage in han- dling dynamic environment [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' For example, in reinforcement learning (RL), the optimization problem can be transformed to the unified Markov decision process (MDPs), which avoids the complexity of defining a dedicated optimization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' In this paper, the main contribution is that we propose a novel hi- erarchical reinforcement learning (HRL) architecture for RIS- assisted sleep control in heterogeneous networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Compared with conventional RL that includes one standalone agent, HRL defines a meta-controller and a sub-controller, which enables higher exploration efficiency by the hierarchical architecture [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' In particular, we improve the energy efficiency (EE) in two ways: we consider the macro base station (MBS) as the meta-controller to implement the sleep control of small base stations (SBSs) to save energy, and SBSs as sub-controllers to decide its own transmission power level to reduce energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Besides, RIS is deployed to improve the signal propagation environment and increase the channel capac- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Finally, the simulations show that combining RIS with sleep control can achieve lower energy consumption, higher throughput, and more than doubled EE than the standalone sleep control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Machine learning based sleep control The flourishing machine learning techniques offer promising opportunities for network control and management [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The deep Q-network is deployed in [10] for the sleep control of renewable energy-powered BSs, where the SBSs can share their energy by a micro-grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The neural network is applied in [11] to predict both traffic demand and energy production, and then the prediction results are used for sleep control of BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Similarly, [12] deploys deep neural networks to predict the traffic patterns, and actor-critic reinforcement learning is used for dynamic sleep control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Different from aforementioned works, here we apply the HRL algorithm, including a meta- controller for SBS sleep control and sub-controllers for the 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='02771v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='SP] 7 Jan 2023 Accepted by 2022 IEEE Globecom conference, ©2022 IEEE transmission power control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' This hierarchical control strategy allows more efficient exploration of the environment, and mitigates the long convergence issue of conventional RL [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' RIS related researches RIS, being an appealing approach, has become a hot topic for researchers from both wireless communication and signal processing communities [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The machine learning-enabled RIS-assisted wireless communication systems have been under exploration in terms of channel modelling [4], [14], chan- nel estimation, EE [15], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' More specifically, the authors in [4] and [14] applied the unsupervised machine learning tool, namely, the expectation-maximization (EM) algorithm to model the RIS-assisted wireless communication links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' It is also demonstrated that the machine learning methods are able to provide better performance than central limit theorem-based approaches [4] [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Besides, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' in [15] deployed the deep reinforcement learning to improve the EE of the RIS- aided cellular communication systems, but the sleep control is not involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' To the best knowledge of the authors, no work before has ever investigated the EE problem with sleep control and RIS embedded in the cellular communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' NETWORK AND SYSTEM MODEL As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='1, we consider a heterogeneous network that includes one MBS and several SBSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The SBS may switch to the sleep mode when the traffic demand drops, which will reduce the energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' It is assumed the MBS can take over the active user equipment (UEs) that were previously associated with those small cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' On the other hand, high- density buildings in the urban area lead to high penetration loss and lower received signal-to-interference-plus-noise ratio (SINR) for direct transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' To this end, we deploy RIS to reflect the signal from MBS and mitigate the high penetration loss of direct transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' In this work, it is worth noting that we save energy in two ways: (i) the meta-controller decides the on/off status of SBSs when traffic load changes, and (ii) the sub-controller decides the transmission power of active SBSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' This hierarchical architecture enables a higher management efficiency, which will be introduced in detail in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' RIS-Assisted Channel Model It is assumed that UEs can receive the signal from BSs by direct and indirect transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The direct link is considered as non-line-of-sight (NLOS) transmission due to the dense buildings in the urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The baseband equivalent channel between BS-UE is given by: HBk = gB,khB,k, (1) where gB,k is the path loss from BS to UE, and hB,k is a complex Gaussian distributed random vector with 0 mean and unit variance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=', CN (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The indirect link consists of the BS-RIS and RIS-UE links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Given the fact that RIS is designed to be deployed on the top Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' RIS-aided heterogeneous network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' or surface of tall buildings, the BS-RIS link is assumed to be line-of-sight (LOS) transmission: HBR = gBR[h1, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=', hN] ∈ C1×N, (2) where N is the number of RIS elements, gBR is the path loss between BS and RIS, and hN = exp � −2jπdN λ � is the phase difference, herein j = √−1, dN is the distance between BS and RIS element N, λ is the signal wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Then the RIS will reflect the signal to UEs via a phase shift vector θ = [θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=', θN], we define a diagonal matrix accordingly: Θ = diag(β1ejθ1, β2ejθ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=', βNejθN ) ∈ CN×N, (3) where βN is the amplitude reflection coefficient and β ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Considering the complex environment in the UE side, the RIS-UE link is presumed to be NLOS transmission, and we have HRk = gR,k[h′ 1,k, h′ 2,k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=', h′ N,k] ∈ C1×N, (4) where gR,k is the path loss between RIS and UE k, and h′ N,k is the small scale fading of the signal reflected by the RIS element N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Path loss is given by gRk = d−αRk/2 Rk , where dRk is the distance from RIS to UE k, and αRk is the pathloss exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Finally, the channel gain from BS j to UE k is: Gj,k = |HB,k + HRkΘHT BR|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' (5) For the phase shift control, inspired by [6], we assume the channel state informations (CSIs) are perfectly shared between BS and RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The RIS phase shift is calculated by: θN = arg(HBk)−arg(gBRhNgR,kh′ N,k) to give every term in HRkΘHT BR the same phase as HBk, then the total received signal will be strengthened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' For a downlink transmission between BS j and UE k, the transmission rate is: Cj,k =bj,k log2 (1+ � r∈Rj,k pj,raj,k,rGj,k,r bj,kN0 + � j′∈J−j � k′∈Kj′ � r′∈Rj′ pj′,r′aj′,k′,r′Gj′,k′,r′ � � � , (6) 2 Agent Extrinsic reward Meta-controller Goals Critic Intrinsic High-level policies reward Sub-controller Actions Heterogeneous network environment Macro-BS Small Throughput RIS cell capacity RIS ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Sleeping SBS SBS Small Throughput cell capacity Direct transmission RIS assisted transmissionAccepted by 2022 IEEE Globecom conference, ©2022 IEEE where Rj,k is the set of resource blocks (RBs) allocated to UE k by BS j [16], bj,k is the total bandwidth allocated to UE k, N0 is the power spectral density of noise, and pj,r is the transmission power of RB r allocated by BS j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' aj,k,r is a binary indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' aj,k,r = 1 if the RB r is allocated to UE k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' otherwise aj,k,r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Gj,k,r denotes the channel gain between BS j and UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' J−j denotes the set of BSs except BS j, Kj′ is the UE set of BS j′, and Rj′ is the RB set in BS j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' In this work, the RBs are allocated by the proportional fairness method [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Energy Consumption Model The energy consumption model for the BS is: Pin = � P0 + δpPout, 0 < Pout ≤ Pmax, Psleep, Pout = 0, (7) where P0 is the fixed power consumption, δp is the slope of load-dependent power consumption, Pout is the transmission power, Pmax is the maximum transmission power, and Psleep is the constant power consumption in sleep mode [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Problem Formulation The overall objective is to maximize the total EE, achieve the desired SINR for UEs, and prevent the BSs from overload- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Here the overloading means that current traffic demand has exceeded the transmission capability of one BS, and then the attached UEs may experience a long delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The problem formulation is given by: max Pj � j∈J � k∈Kj Wj,k � j∈JPj − φnod, (8) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' (5) (6) (7), (8a) � k∈Kj � r∈Rj aj,k,r ≤ |Rj|, (8b) � k∈Kj aj,k,r ≤ 1, (8c) SINRthr ≤ SINRj,k, (8d) where Wj,k is the throughput of UE k in BS j, Pj is the power consumption of BS j, and nod is the number of BSs that are overloaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' We apply φ as a penalty factor to prevent overloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Equation (8a) is the system operation constraint, equation (8b) indicates the number of available RBs can not exceed |Rj|, equation (8c) means one RB can only be allocated to at most one UE, and equation (8d) is the SINR threshold constraint of UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' On one hand, turning off SBSs can greatly reduce energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' But it will also increase the risk of MBS overloading, since the MBS has to take over the UEs of the sleeping small cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Therefore, to maximize the total objective, we have to intelligently control the on/off status of SBSs to reduce the energy cost and overload risk, and following we will introduce an HRL based architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' HIERARCHICAL REINFORCEMENT LEARNING FOR ENERGY-EFFICIENT RAN A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Hierarchical Reinforcement Learning In traditional RL, the problem is defined by an MDP < S, A, T, R >, where S is the set of states, A is the set of actions, T is the transition probability with T : S × A × S, and R is the reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Then, one standalone agent will interact with the environment to maximize its long-term expected reward [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' By contrast, in HRL, the agent consists of two controllers, namely meta-controller and sub-controller [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Accordingly, the MDP is rewritten by < S, A, T, R, G>, where Gindicates the set of goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Based on current state s ∈ S, the meta- controller will generate high-level goals g ∈ G for sub- controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Then, these goals are transformed to high-level policies by the critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Consequently, the sub-controller chooses low-level actions a ∈ A according to high-level policies, and receives an intrinsic reward rin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Finally, the meta-controller will receive an extrinsic reward rex from the environment, and select new goals g′ for the sub-controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The idea behind the HRL is to introduce hierarchy architecture in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' In particular, the meta-controller will produce high-level policies to guide the low-level action selection of the sub-controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Compared with traditional RL, HRL is considered as a more efficient learning method due to the hierarchical architecture, and by dividing sub-goals it allows better management of multiple functionalities in RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' MDP Definition To transform the problem formulation showed by equation (8) into the HRL notation, the following MDP for sub- controllers and meta-controller are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Each SBS is regarded as a sub-controller, the MDP is defined by: State: The state ssub of SBS j is defined by its traffic load ratio ssub = {dSBS}, which is given by: dSBS = � k∈Kj Dj,k Dmax j , (9) where Kj indicates the set of UEs that are served by SBS j, Dj,k is the traffic demand of UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Dmax j is the max traffic load of SBS j, which is considered as a constant value to normalize the current traffic load of SBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Meanwhile, note that the transmission demand of UEs often shows strong statistical regularity, and we assume the daily traffic load follows the patterns in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Action: Based on ssub, the SBS may change its trans- mission power PSBS to adapt the traffic demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Then, the action is defined by asub = {PSBS}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Intrinsic reward: The intrinsic reward of SBS is: rin = � k∈Kj Wj,k PSBS − φnod, (10) where Wj,k, φ and nod have been defined in equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' rin aims at maximizing its own EE and preventing overloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 3 Accepted by 2022 IEEE Globecom conference, ©2022 IEEE The meta-controller is responsible for the high-level policies for the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The MBS is defined as the meta-controller, and its MDPs are: State: The state of meta-controller consists of the traffic load ratio of SBSs: smeta = {dSBS,j}, j ∈ JSBS, (11) where dSBS,j is the load ratio of SBS j, and JSBS is the set of SBSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Goals for sub-controller: With the traffic load status of the SBSs, MBS can generate high-level policies for the SBSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The goals gmeta are turning on/off the SBSs: gmeta = {qSBS,j}, j ∈ JSBS, (12) where qSBS,j is a binary variable to indicate the on/off status of SBS j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' qSBS,j = 1 means keeping the SBS j active, otherwise qSBS,j = 0 denotes turning off SBS j to save energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Extrinsic reward: The meta-controller focuses more on the overall performance of the whole cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Accordingly, the extrinsic reward is given by the objective of the problem formulation in equation (8): rex = � j∈J � k∈Kj Wj,k � j∈JPj − φnod, (13) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Q-value Update and Goal Selections In this section, we introduce how to update the Q-values of controllers, and the action and goal selection strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The Q-values of meta-controller is updated by: Qnew meta(smeta, gmeta) = Qold meta(smeta, gmeta)+ α(rex + γ max g Qmeta(s′ meta, g) − Qold meta(smeta, gmeta)), (14) where s′ meta denotes the next state, α is the learning rate, and γ is the discount function (0 < α, γ < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Qold meta and Qnew meta denote old and new Q-values for meta-controller, which means the accumulated reward brought by state-goal pair (smeta, gmeta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Then we use the ϵ-greedy policy for goal selection: π(smeta) = � arg max g Q(smeta, g), rand > ϵ, random goal selection, rand ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' (15) where rand is a random number between 0 and 1, and ϵ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' ϵ-greedy policy can balance the exploration and exploitation of goals to maximize the long-term reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Similarly, for the sub-controller, the Q-values are updated: Qnew sub (ssub, gmeta, asub) = Qold sub(ssub, gmeta, asub)+ α(rin + γ max a Qsub(s′ sub, g′ meta, a) − Qold sub(ssub, gmeta, asub)), (16) where s′ sub is the next state, g′ meta is the next goal generated by meta-controller, Qnew sub and Qold sub are new and old Q-values for sub-controller, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' We still use ϵ-greedy policy for the action selection of sub-controller: π(ssub) = � arg max a Q(ssub, gmeta, a), rand > ϵ, random action selection, rand ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' (17) The HRL based sleep and transmission power control is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Algorithm 1 HRL algorithm for SBS sleep and power control h 1: Initialize: Wireless network and HRL parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 2: for episode=1 to Total do 3: for MBS do 4: With probability ϵ choose goals randomly, otherwise select gmeta by arg max g Q(smeta, g) (Shown by equation (15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 5: for Each active SBS do 6: With probability ϵ choose asub randomly, other- wise select asub by arg max a Q(ssub, gmeta, a) (Shown by equation (17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 7: Calculating intrinsic reward rin, updating state ssub and Q-values by equation (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 8: end for 9: MBS calculates extrinsic reward rex, updating state smeta and Q-values by equation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 10: end for 11: end for 12: Output: Optimal SBS sleep and transmission power con- trol strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' PERFORMANCE EVALUATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Simulation Settings In the simulations, we consider a dense urban environment in the MATLAB simulation platform, where there are 4 SBSs and 4 RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The coverage radius of MBS and SBS are 400m and 80m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The cell includes 50 randomly distributed UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The fixed power consumption of MBS and SBS are 130W and 75W, and the load-dependent power con- sumption slope is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='6 for MBS and SBS, respectively [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' We assume a deep sleep mode at the SBS with 0 power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Each RIS has 10 reflecting elements with 3 bits Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Daily traffic load pattern of residential area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='2 0 1 3 5 7 9 11 13 15 17 19 21 23 Time/HourAccepted by 2022 IEEE Globecom conference, ©2022 IEEE phase shift resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' We assume the RIS power consumption is very low and it is not included in the power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The path loss exponent for LOS and NLOS are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='5, respectively [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The available bandwidth for each BS is bR = 20 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The traffic pattern is presumed to follow Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 2, which is a typical residential area traffic pattern [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The initial learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='95, and we decay the learning rate after every several episodes for a stable learning performance, and the discount factor is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' The simulation is repeated for 10 runs in MATLAB, and we present the average results with 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Simulation Results In this section, we include 4 cases: (1) no RIS and no sleep control (typical-cell), (2) sleep control without RIS (sleep-only), (3) RIS without sleep control (RIS-only), and (4) combining RIS with sleep control (RIS-sleep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' We apply conventional Q-learning for case (1) to (3), and HRL for our proposed case (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 3 to 5 first present the total power consumption of the BSs, average throughput per UEs, and EE against peak traffic load for the 4 cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' One can observe that: (i) typical-cell, as a benchmark here, presents the highest power consumption and lowest EE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' (ii) comparison between typical- cell and sleep-only in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 3 demonstrates that sleep control can significantly reduce the power consumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' (iii) the EE results of typical-cell and RIS-only in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 4 shows that RIS is highly beneficial to the average throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' More specifically, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 4, when the traffic load is lower than 4 Mbps, the existing channel capacity is already huge enough to serve the UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' However, when the peak traffic load becomes higher than 5 Mbps, RIS-only and RIS-sleep show a higher throughput than other two cases, which can be explained by RIS’s capability to improve the SINR of UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' When it comes to the EE metric, as shown by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 5, the case 4, namely RIS-sleep strategy, displays the best EE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' On the contrary, typical-cell shows the worst EE performance due to the absence of both RIS and sleep control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' sleep-only and RIS-only have comparable EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' It is observed that the former strategy has lower power consumption and lower throughput, and RIS-only is the opposite (indicated by Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' As a result, these two cases show a close EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' When the peak traffic load is 8 Mbps, RIS-sleep achieves a more than doubled EE than other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' To better explain how RIS and sleep control are combined, sleep-only and RIS-sleep are compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 6 in terms of the possibility of keeping SBSs active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' During the off-peak period (from 3:00 to 9:00 in the traffic patterns shown by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 2), most SBSs are shut off to save energy, and the existing traffic demand is served by MBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' However, after 11:00, sleep- only has to turn on most SBSs to satisfy the increasing traffic load, otherwise the MBS will be overloaded and the total throughput will be greatly affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' By contrast, RIS-sleep is capable of keeping most SBSs sleep until 17:00, because MBS can process the increasing traffic demand with a higher SINR provided by RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Total power comparison of all BSs against peak traffic load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Average throughput per UE in the cell against peak traffic load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' EE of the BSs against peak traffic load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Probability of keeping SBSs active under 8 Mbps peak traffic load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Apart from the aforementioned discussions, we further investigate the average SINR of UEs against the number of RIS reflecting elements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 7 under different RIS phase shift resolutions (PSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' A higher PSR generally indicates a more accurate phase shift design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' One can observe that more 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='7 Case 1: No RIS + no sleep control Case 2: No RIS+ sleep control 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='6 Case 3: RIS + no sleep control Case 4: RIS + sleep control 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='3 Cell 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='2 2 3 4 6 7 8 9 10 Peak Traffic load per UE/Mbps7 Case 1:No RIS+ no sleep control Case 2: No RIS+ sleep control 6 Case 3: RIS + no sleep control +-Case 4: RIS + sleep control 0 1 2 3 4 5 6 7 8 9 10 Peak Traffic load per UE/Mbps0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='8 Case 1: No RIS + no sleep control 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='7 Case 2: No RIS+ sleep control Case 3: RIS + no sleep control 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='6 Case 4: RIS + sleep control 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='1 0 1 2 3 4 5 6 7 8 9 10 Peak Traffic load per UE/Mbps-Case 2: No RIS+ sleep control Case 4: RIS + sleep control 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='8 SBS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='2 3 5 9 11 13 15 17 19 21 23 Time/HourAccepted by 2022 IEEE Globecom conference, ©2022 IEEE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Average SINR of UEs under different phase shift resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Convergence performance analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' RIS elements and higher PSR are as expected essentially useful to improve the SINR of UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' On the other hand, the improvements brought by PSR are barely observable from 3 to 4 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' 8 presents the convergence performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' It shows that both intrinsic and extrinsic rewards increase with more iterations and finally converge, which means that meta controller and sub-controller are well coordinated to maintain the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' CONCLUSION The reconfigurable intelligent surface is a promising tech- nology to enable 5G beyond and 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' In this paper, we combine reconfigurable intelligent surfaces with sleep control to improve the energy efficiency of heterogeneous 5G radio access networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' We propose a hierarchical rein- forcement learning-based method to optimize the sleep control strategy of small base stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' Compared with the standalone sleep control method, the simulations show a significantly higher energy efficiency by jointly deploying reconfigurable intelligent surface and sleep control in a hierarchical learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' In addition, we conclude that (i) sleep control largely contributes to reducing power consumption and im- proving energy efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' (ii) reconfigurable intelligent sur- face is beneficial to the average throughput, especially for high traffic load conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' In the future, we will investigate the control strategy of the phase shift of reconfigurable intelligent surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' ACKNOWLEDGEMENT This work has been supported by MITACS and Ericsson Canada, and NSERC Collaborative Research and Training Experience Program (CREATE) under Grant 497981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content=' REFERENCES [1] M.' metadata={'source': 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+page_content='5 Intrinsic reward 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='5 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} +page_content='5 0 100 200 300 400 500 Iterations' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E0T4oBgHgl3EQf7AKY/content/2301.02771v1.pdf'} diff --git a/kdE5T4oBgHgl3EQfGw4w/content/tmp_files/2301.05433v1.pdf.txt b/kdE5T4oBgHgl3EQfGw4w/content/tmp_files/2301.05433v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f76224fbfb9c974e95aa0974d872303ab2c23d3 --- /dev/null +++ b/kdE5T4oBgHgl3EQfGw4w/content/tmp_files/2301.05433v1.pdf.txt @@ -0,0 +1,730 @@ +Trends in Explainable AI (XAI) Literature +ALON JACOVI, Bar Ilan University, Israel +The XAI literature is decentralized, both in terminology and in publication venues, but recent years saw the community converge around +keywords that make it possible to more reliably discover papers automatically. We use keyword search using the SemanticScholar +API and manual curation to collect a well-formatted and reasonably comprehensive set of 5199 XAI papers, available at https: +//github.com/alonjacovi/XAI-Scholar. We use this collection to clarify and visualize trends about the size and scope of the literature, +citation trends, cross-field trends, and collaboration trends. Overall, XAI is becoming increasingly multidisciplinary, with relative +growth in papers belonging to increasingly diverse (non-CS) scientific fields, increasing cross-field collaborative authorship, increasing +cross-field citation activity. The collection can additionally be used as a paper discovery engine, by retrieving XAI literature which is +cited according to specific constraints (for example, papers that are influential outside of their field, or influential to non-XAI research). +1 +INTRODUCTION +This is a report on the collection methodology and analysis of XAI-Scholar, a set of XAI1 papers collected as of December +31st 2022. All data and reproduction code are available at https://github.com/alonjacovi/XAI-Scholar. +In recent years the “explainable AI” body of research has started to reach a size that makes it (1) difficult to grasp +with manual surveying; (2) large enough that it’s possible to see overall empirical and statistical trends. The goal of +this report is to collect a large and well-formatted set of XAI papers to make this empirical analysis possible. The +report mostly observes cross-field and multi-disciplinary trends in XAI. We invite others to use the collection for other +purposes as well. +Challenges. +XAI research has several properties that make it difficult to observe in its entirety, compared to many +other adjacent fields in Computer Science: +(1) It is multidisciplinary, with non-negligible communities in many different fields that do not often interact or +share venues. +(2) The terminology that papers use to self-identify as XAI research is not unique to XAI (E.g., “xai” and “Xai Xai” +are names with multiple senses which appear in research), and this terminology is much more recent than the +actual history of XAI. +(3) The most prevalent definitions of “XAI research” papers often include papers that don’t self-identify as XAI, as +long as they research how to explain AI technology. +Findings. +Below is a brief partial summary of findings from the analysis. Most trends are situated around Computer +Science (CS), being the primary field of study for XAI research. +(1) XAI research has had its biggest “expansion” growth spikes outside of Computer Science in 2016, 2018 and 2021. +(2) There is clear growth over time in the relative proportion of papers by authors that traditionally publish in two +or more distinct fields of study. +(3) CS has different citing relationships with different XAI fields. For example, XAI-CS cites XAI-Psychology more +often than the vice versa, but the relationship flips for XAI-CS and XAI-Medicine. This “direction” of influence +shows which fields often inform which fields in the current literature. +1By XAI we refer to a relatively inclusive definition for research articles that discuss the development, implementation, or practice, of explana- +tions/interpretations in “AI” systems (even if those articles don’t refer to their work as explanations, or to their systems as AI systems, as long as they are +treated as such by others). This definition is aligned with what we’ve observed in various curated lists, workshops and journal issues of XAI literature. +1 +arXiv:2301.05433v1 [cs.AI] 13 Jan 2023 + +(4) There is a difference across XAI fields by how often they inform non-XAI research, the highest proportion +being in XAI-Biology, XAI-Engineering and XAI-Law—while the lowest proportion being in XAI-Psychology, +XAI-Business and XAI-Philosophy, whose influence more often carries to other XAI literature. +(5) Citation behavior across papers is significantly different between fields. For example, the top-cited Philosophy +papers cited by XAI-Philosophy are significantly different from those cited by XAI-CS, and so on. Unsurprisingly, +papers outside of a field tend to focus on a smaller variety of papers in that field, but the papers that “break out +of” the traditional boundaries of their field are not always the most cited papers in that field. +(6) The collection can serve as a paper discovery engine by observing which XAI papers, for example, are the most +influential to papers outside of their field, or outside of XAI; or which non-XAI papers of a particular field are the +most informative to another field. Tables with examples of these selections are shown at the end of this report, +and more are available in the accompanying github repository. +2 +DATA COLLECTION +2.1 +Methodology +The collection phase involved five steps. +Step 1: Keyword-based search (3101 papers total). +We derive a set of keywords in a (manual) iterative process to +maximize recall while not compromising near-perfect precision. The keywords are matched against both the title and +abstract together. Due to problems with filtering papers with 1 keyword match, we only collect papers that match 2 +keywords or more. A random sample of 100 papers yielded 99% precision. +The keywords are: xai, (xai), hcxai, explainability, interpretability, explainable ai, explainable artificial intelligence, +interpretable ml, interpretable machine learning, interpretable model, feature attribution, feature importance, global +explanation, local explanation, local interpretation, global interpretation, model explanation, model interpretation, saliency, +counterfactual explanation. +Step 2: Manually curated collections (+ 766; 3867 papers total). +We collected the titles of papers from various +curated XAI collections [1–4, 6–14] and matched them using the SemanticScholar API with fuzzy matching. +Step 3: Citation tree expansion with manual filtering (+ 648; 4515 papers total). +We took the 2000 most cited +papers by the set of papers collected in the previous steps, and manually selected XAI papers from them. +Step 4: Citation tree expansion with automatic filtering (+ 709; 5224 papers total). +We used the citations and +references of all collected papers and filtered them via the 2-keyword-match method from step 1. We repeated this until +no new papers were added. +Step 5: Manual quality check (- 25; 5119 papers total). +Finally, we heuristically found 25 incorrectly-attributed +papers in the set, which we removed from the collection. +2 + +2.2 +Collection Details +Computer +Science +Medicine +Mathematics +Biology +Engineering Psychology +Physics +Business Environmental +Science +Economics +Philosophy +Sociology +Materials +Science +Political +Science +Geography +Law +Agricultural +And +Food +Sciences +Geology +History +Chemistry +Art +Education +0 +1000 +2000 +3000 +4000 +paper count +Medicine +Mathematics +Biology +Engineering +Psychology +Physics +Business +Environmental +Science +Economics +Philosophy +Sociology +Materials +Science +Political +Science +Geography +Law +Agricultural +And +Food +Sciences +Geology +History +Chemistry +Art +Education +0 +50 +100 +150 +200 +250 +300 +350 +paper count +Fig. 1. Paper counts by field of study. The bottom figure shows distribution across fields, sans Computer Science, for readability. +The collection has 5199 papers. Each paper in the collection contains, per the SemanticScholar API: +(1) The SemanticScholar ID and URL +(2) Title +(3) Abstract +(4) Authors +(5) Number of citations +(6) Number of references +(7) Year +(8) Venue +(9) Field of study +(10) SemanticScholar’s “tldr” summary +(11) References +(12) Citations +(13) SemanticScholar’s embedding vector +See Figure 1 for field of study distributions. +3 + +Limitations. +Any insight derived from the collection should account for margin of error based on these limitations: +(1) The data in the collection is noisy as can be expected from a large-scale research database. This includes missing +fields, inconsistent venue names, some incorrect details, and so on. While this is the minority, it is not negligible. +(2) It’s likely that the collection still contains a very small amount of non-XAI papers, despite our efforts. +(3) The collection methodology is biased towards CS, influential papers and papers that self-identify as XAI with the +keywords that we used. Of course, it’s likely that many XAI papers were missed, in particular less-cited papers +and papers which use different terminology. +(4) The collection was retrieved over a period of time in December 2022. Due to delay in proceedings release for +some venues, it may be necessary to consider 2022 a partial year. +3 +GROWTH TRENDS +Figure 2 shows yearly growth in three settings. First, XAI generally shows relative yearly growth—but this growth is +largely controlled by Computer Science (which shows the same growth trend). +Controlling for non-CS papers reveals different growth trends. For example, XAI-Medicine shows exponential +growth, in particular with large relative growth in 2016, 2018 and 2021. These trends also hold when controlling for +non-Medicine and non-CS papers (bottom plot). Overall, it appears that XAI has had the biggest growth into non-central +fields in 2016, 2018 and 2021. +4 +COLLABORATION TRENDS +We define the field of study for an author as the field of study for a majority of their papers (retrieved separately via the +SemanticScholar API). Given this definition, we can define a “collaboration” as the existence of two authors for the +same paper assigned two different fields of study. +Figure 3 shows the most common field pairings. As CS controls the scale, we show a graph that omits CS for a +perspective on multi-field research between non-CS fields. +Figure 4 shows yearly absolute and relative growth for papers with at at least two different fields, via their authors, +out of all yearly papers. If we consider 2016 as an outlier, the trends show a clear relative growth in cross-field XAI +research. +5 +CITATION TRENDS +This section investigates the interaction between the citation graph in XAI-Scholar and the fields of study variable. +5.1 +Computer Science +Figure 5 shows the distribution of citations by field by XAI-CS literature. Unsurprisingly XAI-CS appears to be +significantly informed by XAI-Mathematics, but XAI-Medicine, XAI-Psychology, XAI-Engineering and XAI-Business +also have a non-negligible presence. +Figure 6 shows the citation relationship between XAI-CS and other XAI fields as a directed weighted graph. Every +pair of edges is normalized to 100% to show which of the two XAI fields is informed more by the other. We can observe +that XAI-CS more often cites (vs. is cited by) XAI-Psychology and XAI-Mathematics, while the same is not true for the +other fields (XAI-Engineering and XAI-Engineering being roughly equal, and the rest overwhelmingly citing XAI-CS). +4 + +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +2022 +0 +200 +400 +600 +800 +1000 +1200 +1400 +paper count +XAI growth +2014 +2016 +2017 +2018 +2019 +2020 +2021 +2022 +0 +20 +40 +60 +80 +100 +120 +140 +paper count +XAI growth in Medicine +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +2022 +0 +20 +40 +60 +80 +100 +paper count +XAI growth outside of CS/Medicine +Fig. 2. Yearly growth trends. Papers before 2013 were omitted for readability. +5.2 +Citation Trends Outside of Computer Science +As before we can omit XAI-CS in order to probe into relationships between other fields. Figure 7 shows a cross-field +citation graph where all outgoing edges from a node are normalized to sum to 100%, to show which XAI fields are most +cited by a particular XAI field. +5.3 +Citation Trends Between XAI and non-XAI +Figure 8 show the ranking of XAI fields by the percentage of non-XAI citations out of all citations to that XAI field. +This plot shows which fields most often inform non-XAI literature. For example, XAI-Biology is relatively often cited +by non-XAI literature, and the opposite is true for XAI-Philosophy, which seems to be comparatively more often cited +by XAI. +5 + +Fig. 3. Cross-field collaboration as a weighted undirected graph for all field pairs (left) and excluding CS (right). Edge weights show +the percentage of collaborations for a field pair out of all papers with any collaboration. Low-magnitude edges were omitted. +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +2022 +0 +200 +400 +600 +800 +1000 +1200 +1400 +# papers +Yearly # of papers with more than one different author field of study out of all papers +Total +With collab +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +2022 +0 +5 +10 +15 +20 +25 +30 +% of papers with a collaboration +Yearly % of papers with more than one different author field of study +Fig. 4. Absolute and relative plots of yearly papers with authors of at least two fields of study, out of all papers. +6 +PAPER-LEVEL CITATION TRENDS AND PAPER DISCOVERY +Finally we can look at citation behavior at the level of individual papers. For example, when XAI-CS cites Philosophy, +are they citing a wide variety of papers, or a select minority of papers? Table 1 shows the top cited papers in Philosophy +by XAI-CS, which control 32% of the citations between these fields. More generally, Figure 9 shows the entropy across +paper citations for every field as cited by XAI-CS. The field with the most extreme distribution is Law, which as shown +in Table 2, is due to one paper—“Accountable Algorithms”—being overwhelmingly the most cited paper in Law by +XAI-CS. Table 3 and Table 4 show additional examples for top papers in Psychology and CS as cited by XAI-CS. +Other discovery constraints include: +(1) Which papers in a particular field are the most cited by XAI papers of that field? (e.g., Tables 5 and 6) +(2) What are the XAI papers that are the most cited by papers outside of their field? (e.g., Tables 7 to 9) +(3) What XAI papers in a particular field are most cited in that field? +(4) And so on. +6 + +Psychology +Engineering +7.53 +6.26 +Environmental Science +5.29 +Mathematics +7.82 +Computer Science +47.62 +8.64 +Medicine +10.21 +Physics +6.04 +BiologyChemistry4.06 +Environmental Science +10.43 +23.48 +Engineering +4.64 +4.93 +6.96 +Medicine +Geology +3.77 +5.51 +4.35 +8.99 +Materials S +8.12 +Sclence +Mathematics +Physics +PsychologyMathematics +Medicine +Psychology +Engineering +Business +0 +10 +20 +30 +40 +50 +60 +% citations by Computer Science +% of citations by XAI-Computer Science by field (top 5) out of all cross-field citations by XAI-CS +Fig. 5. The top-5 XAI fields cited by XAI-CS. +7 +CONCLUSIONS +XAI research is converging around specific terminology at scale that makes it possible to observe trends empirically. +While the retrieval process has some limitations that result in a margin of error, it’s possible to account for them on +some level (for example, by acknowledging that the retrieval is biased towards Computer Science and seeing trends that +overcome this bias in the opposite direction). The analysis in this work mostly focused on the field of study variable, +though it is possible to look at many other trends given this collection, as has been explored for other fields [5]. +ACKNOWLEDGMENTS +Thanks to Matan Eyal, Mor Geva, Avi Caciularu, Yoav Goldberg, Yonatan Bitton and Rotem Tsabary, for discussions +and brainstorming. +% cited +Title +6.4% +Contrastive Explanation +5.6% +Causality: Models, Reasoning and Inference +3.5% +The Book of Why: The New Science of Cause and Effect +3.5% +Studies in the Logic of Explanation +3.1% +Causality +3.1% +The philosophical basis of algorithmic recourse +2.0% +Scientific Explanation and the Causal Structure of the World +1.9% +Scientific Explanation +1.7% +Knowledge-Based Causal Attribution : The Abnormal Conditions Focus Model +1.2% +Combining explanation and argumentation in dialogue +Table 1. Top cited papers in Philosophy by XAI-CS. +7 + +Fig. 6. Citation relationship between XAI-CS and other XAI fields. Every pair of edges is normalized to 100% to show which of the +two fields is cited more by the other. +REFERENCES +[1] Anyifei. 2019. All-about-XAI. (2019). https://github.com/feifeife/All-about-XAI +[2] Hubert Baniecki. 2022. Adversarial Explainable AI. (2022). https://github.com/hbaniecki/adversarial-explainable-ai +[3] Przemysław Biecek. 2022. Interesting resources related to XAI (Explainable Artificial Intelligence). (2022). https://github.com/pbiecek/xai_resources +[4] Marina Danilevsky, Kun Qian, Ranit Aharonov, Yannis Katsis, Ban Kawas, and Prithviraj Sen. 2020. A Survey of the State of Explainable AI for +Natural Language Processing. AACL-IJCNLP 2020 (2020). https://xainlp2020.github.io/xainlp/table +[5] Morgan Frank, Dashun Wang, Manuel Cebrian, and Iyad Rahwan. 2019. The evolution of citation graphs in artificial intelligence research. Nature +Machine Intelligence 1 (02 2019), 79–85. https://doi.org/10.1038/s42256-019-0024-5 +[6] Michal Lopuszynski. 2020. Awesome Interpretable Machine Learning. (2020). https://github.com/lopusz/awesome-interpretable-machine-learning +[7] Anh M. Nguyen. 2022. Papers on Explainable Artificial Intelligence. (2022). https://github.com/anguyen8/XAI-papers +[8] Kevin McAreavey. 2022. CHAI-XAI. (2022). https://github.com/kevinmcareavey/chai-xai +[9] Sina Mohseni. 2020. Awesome-XAI-Evaluation. (2020). https://github.com/SinaMohseni/Awesome-XAI-Evaluation +[10] Sina Mohseni, Niloofar Zarei, and Eric D Ragan. 2018. A Multidisciplinary Survey and Framework for Design and Evaluation of Explainable AI +Systems. arXiv preprint arXiv:1811.11839 (2018). +[11] Benedek Rozemberczki. 2022. Awesome Explainable Graph Reasoning. (2022). https://github.com/AstraZeneca/awesome-explainable-graph- +reasoning +[12] Yongjie Wang. 2020. Awesome-explainable-AI. (2020). https://github.com/wangyongjie-ntu/Awesome-explainable-AI +[13] Sam Zabdiel. 2022. XAI. (2022). https://github.com/samzabdiel/XAI +[14] Rehman Zafar. 2022. Interesting resources related to XAI (Explainable Artificial Intelligence). (2022). https://github.com/rehmanzafar/xai-iml-sota +8 + +Biology +Economics +95 +5 +Mathematics +25 +20 +75 +59 +Psychology +82 +Computer Science +41 +18 +43 +47 +57 +12 +88 +53 +Engineering +PhysicsFig. 7. A graph of the top-3 cited XAI fields for every XAI field (omitting fields whose outgoing citations are less than 20). All outgoing +edges from a node are normalized to sum to 100%. +% cited +Title +44.4% +Accountable Algorithms +7.4% +Stability +7.4% +The Role of Explanation in Algorithmic Trust +3.7% +HYPO’S legacy: introduction to the virtual special issue +3.7% +THE UNIVERSITY OF CHICAGO LAW REVIEW +3.7% +The Philadelphia predictive policing experiment +3.7% +Race , Prediction , and Discretion +1.9% +Antidiscriminatory Algorithms +1.9% +Of, for, and by the people: the legal lacuna of synthetic persons +1.9% +FORECASTING THE FUTURE OF PREDICTIVE CRIME MAPPING +Table 2. Top cited papers in Law by XAI-CS. +9 + +Biology +36 +Engineering +Medicine +12 +38 +LawBiology +Engineering +Law +Physics +Economics +Political +Science +Medicine +Mathematics +Computer +Science +Psychology +Business +Philosophy +0 +20 +40 +60 +80 +% citations by non-XAI +% of citations for each XAI subfield by non-XAI out of all citations for that subfield +Fig. 8. The percentage of non-XAI citations (out of all citations) per XAI field. Fields with less than 100 outgoing citations were +omitted. +Computer +Science +Medicine +Psychology Mathematics Engineering +Biology +Business +Physics +Sociology Environmental +Science +Economics +Political +Science +Materials +Science +Geography +Chemistry +Philosophy +Geology +Art +Education +History +Law +0 +2 +4 +6 +8 +entropy +Entropy for the distribution of citations across papers in every field by XAI-Computer Science +Fig. 9. Plot of the entropies for every distribution of citations by XAI-CS across papers, by cited field. For example, XAI-CS citations +of Law papers are concentrated in fewer papers controlling more of the citations, compared to citations of Biology papers. +% cited +Title +1.0% +The structure and function of explanations +1.0% +The Role of Explanations on Trust and Reliance in Clinical Decision Support Systems +0.8% +Explanation and understanding. +0.7% +False Positives, False Negatives, and False Analyses: A Rejoinder to "Machine Bias: There’s +Software Used across the Country to Predict Future Criminals. and It’s Biased against Blacks" +0.6% +A unified view of gradient-based attribution methods for Deep Neural Networks +0.6% +Conversational Processes and Causal Explanation +0.6% +When Explanations Lie: Why Many Modified BP Attributions Fail +0.5% +Explanation and Abductive Inference +0.4% +Causal Inference in Statistics: A Primer +0.4% +Humans and Automation: Use, Misuse, Disuse, Abuse +Table 3. Top cited papers in Psychology by XAI-CS. +10 + +% cited +Title +1.4% +“Why Should I Trust You?”: Explaining the Predictions of Any Classifier +1.0% +A Unified Approach to Interpreting Model Predictions +0.6% +Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps +0.6% +Grad-CAM: Visual Explanations from Deep Networks via Gradient-Based Localization +0.6% +Axiomatic Attribution for Deep Networks +0.5% +Visualizing and Understanding Convolutional Networks +0.5% +Towards A Rigorous Science of Interpretable Machine Learning +0.5% +On Pixel-Wise Explanations for Non-Linear Classifier Decisions by Layer-Wise Relevance Propagation +0.4% +Explanation in Artificial Intelligence: Insights from the Social Sciences +0.4% +Learning Important Features Through Propagating Activation Differences +Table 4. Top cited papers in CS by XAI-CS. +% cited +Title +1.4% +Greedy function approximation: A gradient boosting machine. +1.1% +Regression Shrinkage and Selection via the Lasso +1.0% +European Union Regulations on Algorithmic Decision-Making and a "Right to Explanation" +1.0% +Interpretability Beyond Feature Attribution: Quantitative Testing with Concept Activation Vectors (TCAV) +0.8% +Peeking Inside the Black Box: Visualizing Statistical Learning With Plots of Individual Conditional Expectation +0.8% +An Efficient Explanation of Individual Classifications using Game Theory +0.7% +Understanding Black-box Predictions via Influence Functions +0.7% +PREDICTIVE LEARNING VIA RULE ENSEMBLES +0.6% +Equality of Opportunity in Supervised Learning +0.6% +Generalized Functional ANOVA Diagnostics for High-Dimensional Functions of Dependent Variables +Table 5. Top cited papers in Mathematics by XAI-Mathematics. +% cited +Title +1.7% +Explanation and understanding. +1.0% +How the Mind Explains Behavior: Folk Explanations, Meaning, and Social Interaction +1.0% +Mental Models and Causal Explanation: Judgements of Probable Cause and Explanatory Relevance +1.0% +Functional explanation and the function of explanation +1.0% +The structure and function of explanations +1.0% +The misunderstood limits of folk science: an illusion of explanatory depth +1.0% +Explanatory coherence in social explanations : a parallel distributed processing account +1.0% +Explanatory coherence +0.7% +Attribution theory in social psychology +0.7% +A unified view of gradient-based attribution methods for Deep Neural Networks +Table 6. Top cited papers in Psychology by XAI-Psychology. +11 + +% cited +Title +1.9% +Detection of Influential Observation in Linear Regression +1.8% +Conditional variable importance for random forests +1.7% +To Explain or to Predict +1.7% +Bias in random forest variable importance measures: Illustrations, sources and a solution +1.5% +Illuminating the “black box”: a randomization approach for understanding variable contributions in artificial neural +networks +1.4% +From local explanations to global understanding with explainable AI for trees +1.4% +On the interpretation of weight vectors of linear models in multivariate neuroimaging +1.4% +An accurate comparison of methods for quantifying variable importance in artificial neural networks using simulated +data +1.3% +Permutation importance: a corrected feature importance measure +1.2% +How the machine ‘thinks’: Understanding opacity in machine learning algorithms +Table 7. Top cited papers in XAI-CS by papers outside of CS. +% cited +Title +51.5% +The Right to Explanation, Explained +37.2% +The Shapley value +4.6% +Economic complexity unfolded: Interpretable model for the productive structure of economies +2.5% +Predicting , explaining , and understanding risk of long-term unemployment +1.3% +Explainable AI Models of Stock Crashes: A Machine-Learning Explanation of the Covid March 2020 Equity Meltdown +1.3% +Paving the way towards counterfactual generation in argumentative conversational agents +0.4% +Explainable AI (XAI) Models Applied to Planning in Financial Markets +0.4% +Sell Me the Blackbox! Why eXplainable Artificial Intelligence (XAI) May Hurt Customers +0.4% +An Investigation of the Impact of COVID-19 Non-Pharmaceutical Interventions and Economic Support Policies on +Foreign Exchange Markets with Explainable AI Techniques +0.4% +Frontiers in environmental science a study on China coal price forecasting based on CEEMDAN-GWO-CatBoost hybrid +forecasting model under carbon neutral target +Table 8. Top cited papers in XAI-Economics by papers outside of Economics. +% cited +Title +28.9% +Development and interpretation of a pathomics-based model for the prediction of microsatellite instability in Colorectal Cancer +20.5% +Discovering epistatic feature interactions from neural network models of regulatory DNA sequences +8.4% +Amino Acid k-mer Feature Extraction for Quantitative Antimicrobial Resistance (AMR) Prediction by Machine Learning and +Model Interpretation for Biological Insights +6.3% +Brain age prediction of healthy subjects on anatomic MRI with deep learning : going beyond with an “explainable AI” mindset +5.8% +Reverse-engineering Recurrent Neural Network solutions to a hierarchical inference task for mice +5.8% +Inferring Sequence-Structure Preferences of RNA-Binding Proteins with Convolutional Residual Networks +4.2% +Automated detection of glaucoma with interpretable machine learning using clinical data and multi-modal retinal images +3.2% +BayeSuites: An open web framework for massive Bayesian networks focused on neuroscience +3.2% +Analysis of SARS-CoV-2 RNA-Sequences by Interpretable Machine Learning Models +2.1% +Explainable AI reveals key changes in skin microbiome associated with menopause, smoking, aging and skin hydration +Table 9. Top cited papers in XAI-Biology by papers outside of Biology. +12 + diff --git a/kdE5T4oBgHgl3EQfGw4w/content/tmp_files/load_file.txt b/kdE5T4oBgHgl3EQfGw4w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c11f47b19992104062ca56404c3bc1a5242f8ca7 --- /dev/null +++ b/kdE5T4oBgHgl3EQfGw4w/content/tmp_files/load_file.txt @@ -0,0 +1,431 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf,len=430 +page_content='Trends in Explainable AI (XAI) Literature ALON JACOVI, Bar Ilan University, Israel The XAI literature is decentralized, both in terminology and in publication venues, but recent years saw the community converge around keywords that make it possible to more reliably discover papers automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' We use keyword search using the SemanticScholar API and manual curation to collect a well-formatted and reasonably comprehensive set of 5199 XAI papers, available at https: //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='com/alonjacovi/XAI-Scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' We use this collection to clarify and visualize trends about the size and scope of the literature, citation trends, cross-field trends, and collaboration trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Overall, XAI is becoming increasingly multidisciplinary, with relative growth in papers belonging to increasingly diverse (non-CS) scientific fields, increasing cross-field collaborative authorship, increasing cross-field citation activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' The collection can additionally be used as a paper discovery engine, by retrieving XAI literature which is cited according to specific constraints (for example, papers that are influential outside of their field, or influential to non-XAI research).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 1 INTRODUCTION This is a report on the collection methodology and analysis of XAI-Scholar, a set of XAI1 papers collected as of December 31st 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' All data and reproduction code are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='com/alonjacovi/XAI-Scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' In recent years the “explainable AI” body of research has started to reach a size that makes it (1) difficult to grasp with manual surveying;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (2) large enough that it’s possible to see overall empirical and statistical trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' The goal of this report is to collect a large and well-formatted set of XAI papers to make this empirical analysis possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' The report mostly observes cross-field and multi-disciplinary trends in XAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' We invite others to use the collection for other purposes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' XAI research has several properties that make it difficult to observe in its entirety, compared to many other adjacent fields in Computer Science: (1) It is multidisciplinary, with non-negligible communities in many different fields that do not often interact or share venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (2) The terminology that papers use to self-identify as XAI research is not unique to XAI (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=', “xai” and “Xai Xai” are names with multiple senses which appear in research), and this terminology is much more recent than the actual history of XAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (3) The most prevalent definitions of “XAI research” papers often include papers that don’t self-identify as XAI, as long as they research how to explain AI technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Below is a brief partial summary of findings from the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Most trends are situated around Computer Science (CS), being the primary field of study for XAI research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (1) XAI research has had its biggest “expansion” growth spikes outside of Computer Science in 2016, 2018 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (2) There is clear growth over time in the relative proportion of papers by authors that traditionally publish in two or more distinct fields of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (3) CS has different citing relationships with different XAI fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' For example, XAI-CS cites XAI-Psychology more often than the vice versa, but the relationship flips for XAI-CS and XAI-Medicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' This “direction” of influence shows which fields often inform which fields in the current literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 1By XAI we refer to a relatively inclusive definition for research articles that discuss the development, implementation, or practice, of explana- tions/interpretations in “AI” systems (even if those articles don’t refer to their work as explanations, or to their systems as AI systems, as long as they are treated as such by others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' This definition is aligned with what we’ve observed in various curated lists, workshops and journal issues of XAI literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='05433v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='AI] 13 Jan 2023 (4) There is a difference across XAI fields by how often they inform non-XAI research, the highest proportion being in XAI-Biology, XAI-Engineering and XAI-Law—while the lowest proportion being in XAI-Psychology, XAI-Business and XAI-Philosophy, whose influence more often carries to other XAI literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (5) Citation behavior across papers is significantly different between fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' For example, the top-cited Philosophy papers cited by XAI-Philosophy are significantly different from those cited by XAI-CS, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Unsurprisingly, papers outside of a field tend to focus on a smaller variety of papers in that field, but the papers that “break out of” the traditional boundaries of their field are not always the most cited papers in that field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (6) The collection can serve as a paper discovery engine by observing which XAI papers, for example, are the most influential to papers outside of their field, or outside of XAI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' or which non-XAI papers of a particular field are the most informative to another field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Tables with examples of these selections are shown at the end of this report, and more are available in the accompanying github repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 2 DATA COLLECTION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='1 Methodology The collection phase involved five steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Step 1: Keyword-based search (3101 papers total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' We derive a set of keywords in a (manual) iterative process to maximize recall while not compromising near-perfect precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' The keywords are matched against both the title and abstract together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Due to problems with filtering papers with 1 keyword match, we only collect papers that match 2 keywords or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' A random sample of 100 papers yielded 99% precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' The keywords are: xai, (xai), hcxai, explainability, interpretability, explainable ai, explainable artificial intelligence, interpretable ml, interpretable machine learning, interpretable model, feature attribution, feature importance, global explanation, local explanation, local interpretation, global interpretation, model explanation, model interpretation, saliency, counterfactual explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Step 2: Manually curated collections (+ 766;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 3867 papers total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' We collected the titles of papers from various curated XAI collections [1–4, 6–14] and matched them using the SemanticScholar API with fuzzy matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Step 3: Citation tree expansion with manual filtering (+ 648;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 4515 papers total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' We took the 2000 most cited papers by the set of papers collected in the previous steps, and manually selected XAI papers from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Step 4: Citation tree expansion with automatic filtering (+ 709;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 5224 papers total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' We used the citations and references of all collected papers and filtered them via the 2-keyword-match method from step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' We repeated this until no new papers were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Step 5: Manual quality check (- 25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 5119 papers total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Finally, we heuristically found 25 incorrectly-attributed papers in the set, which we removed from the collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='Collection Details ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='Computer ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Paper counts by field of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' The bottom figure shows distribution across fields, sans Computer Science, for readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' The collection has 5199 papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Each paper in the collection contains, per the SemanticScholar API: (1) The SemanticScholar ID and URL (2) Title (3) Abstract (4) Authors (5) Number of citations (6) Number of references (7) Year (8) Venue (9) Field of study (10) SemanticScholar’s “tldr” summary (11) References (12) Citations (13) SemanticScholar’s embedding vector See Figure 1 for field of study distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 3 Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Any insight derived from the collection should account for margin of error based on these limitations: (1) The data in the collection is noisy as can be expected from a large-scale research database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' This includes missing fields, inconsistent venue names, some incorrect details, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' While this is the minority, it is not negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (2) It’s likely that the collection still contains a very small amount of non-XAI papers, despite our efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (3) The collection methodology is biased towards CS, influential papers and papers that self-identify as XAI with the keywords that we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Of course, it’s likely that many XAI papers were missed, in particular less-cited papers and papers which use different terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (4) The collection was retrieved over a period of time in December 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Due to delay in proceedings release for some venues, it may be necessary to consider 2022 a partial year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 3 GROWTH TRENDS Figure 2 shows yearly growth in three settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' First, XAI generally shows relative yearly growth—but this growth is largely controlled by Computer Science (which shows the same growth trend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Controlling for non-CS papers reveals different growth trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' For example, XAI-Medicine shows exponential growth, in particular with large relative growth in 2016, 2018 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' These trends also hold when controlling for non-Medicine and non-CS papers (bottom plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Overall, it appears that XAI has had the biggest growth into non-central fields in 2016, 2018 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 4 COLLABORATION TRENDS We define the field of study for an author as the field of study for a majority of their papers (retrieved separately via the SemanticScholar API).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Given this definition, we can define a “collaboration” as the existence of two authors for the same paper assigned two different fields of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Figure 3 shows the most common field pairings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' As CS controls the scale, we show a graph that omits CS for a perspective on multi-field research between non-CS fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Figure 4 shows yearly absolute and relative growth for papers with at at least two different fields, via their authors, out of all yearly papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' If we consider 2016 as an outlier, the trends show a clear relative growth in cross-field XAI research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 5 CITATION TRENDS This section investigates the interaction between the citation graph in XAI-Scholar and the fields of study variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='1 Computer Science Figure 5 shows the distribution of citations by field by XAI-CS literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Unsurprisingly XAI-CS appears to be significantly informed by XAI-Mathematics, but XAI-Medicine, XAI-Psychology, XAI-Engineering and XAI-Business also have a non-negligible presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Figure 6 shows the citation relationship between XAI-CS and other XAI fields as a directed weighted graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Every pair of edges is normalized to 100% to show which of the two XAI fields is informed more by the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' We can observe that XAI-CS more often cites (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' is cited by) XAI-Psychology and XAI-Mathematics, while the same is not true for the other fields (XAI-Engineering and XAI-Engineering being roughly equal, and the rest overwhelmingly citing XAI-CS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 4 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 0 200 400 600 800 1000 1200 1400 paper count XAI growth 2014 2016 2017 2018 2019 2020 2021 2022 0 20 40 60 80 100 120 140 paper count XAI growth in Medicine 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 0 20 40 60 80 100 paper count XAI growth outside of CS/Medicine Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Yearly growth trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Papers before 2013 were omitted for readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='2 Citation Trends Outside of Computer Science As before we can omit XAI-CS in order to probe into relationships between other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Figure 7 shows a cross-field citation graph where all outgoing edges from a node are normalized to sum to 100%, to show which XAI fields are most cited by a particular XAI field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='3 Citation Trends Between XAI and non-XAI Figure 8 show the ranking of XAI fields by the percentage of non-XAI citations out of all citations to that XAI field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' This plot shows which fields most often inform non-XAI literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' For example, XAI-Biology is relatively often cited by non-XAI literature, and the opposite is true for XAI-Philosophy, which seems to be comparatively more often cited by XAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Cross-field collaboration as a weighted undirected graph for all field pairs (left) and excluding CS (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Edge weights show the percentage of collaborations for a field pair out of all papers with any collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Low-magnitude edges were omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 2014 2015 2016 2017 2018 2019 2020 2021 2022 0 200 400 600 800 1000 1200 1400 # papers Yearly # of papers with more than one different author field of study out of all papers Total With collab 2014 2015 2016 2017 2018 2019 2020 2021 2022 0 5 10 15 20 25 30 % of papers with a collaboration Yearly % of papers with more than one different author field of study Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Absolute and relative plots of yearly papers with authors of at least two fields of study, out of all papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 6 PAPER-LEVEL CITATION TRENDS AND PAPER DISCOVERY Finally we can look at citation behavior at the level of individual papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' For example, when XAI-CS cites Philosophy, are they citing a wide variety of papers, or a select minority of papers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Table 1 shows the top cited papers in Philosophy by XAI-CS, which control 32% of the citations between these fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' More generally, Figure 9 shows the entropy across paper citations for every field as cited by XAI-CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' The field with the most extreme distribution is Law, which as shown in Table 2, is due to one paper—“Accountable Algorithms”—being overwhelmingly the most cited paper in Law by XAI-CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Table 3 and Table 4 show additional examples for top papers in Psychology and CS as cited by XAI-CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Other discovery constraints include: (1) Which papers in a particular field are the most cited by XAI papers of that field?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=', Tables 5 and 6) (2) What are the XAI papers that are the most cited by papers outside of their field?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=', Tables 7 to 9) (3) What XAI papers in a particular field are most cited in that field?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (4) And so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 6 Psychology Engineering 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='53 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='26 Environmental Science 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='29 Mathematics 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='82 Computer Science 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='62 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='64 Medicine 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='21 Physics 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='04 BiologyChemistry4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='06 Environmental Science 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='43 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='48 Engineering 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='93 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='96 Medicine Geology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='77 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='51 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='35 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='99 Materials S 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='12 Sclence Mathematics Physics PsychologyMathematics Medicine Psychology Engineering Business 0 10 20 30 40 50 60 % citations by Computer Science % of citations by XAI-Computer Science by field (top 5) out of all cross-field citations by XAI-CS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' The top-5 XAI fields cited by XAI-CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 7 CONCLUSIONS XAI research is converging around specific terminology at scale that makes it possible to observe trends empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' While the retrieval process has some limitations that result in a margin of error, it’s possible to account for them on some level (for example, by acknowledging that the retrieval is biased towards Computer Science and seeing trends that overcome this bias in the opposite direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' The analysis in this work mostly focused on the field of study variable, though it is possible to look at many other trends given this collection, as has been explored for other fields [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' ACKNOWLEDGMENTS Thanks to Matan Eyal, Mor Geva, Avi Caciularu, Yoav Goldberg, Yonatan Bitton and Rotem Tsabary, for discussions and brainstorming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' % cited Title 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='4% Contrastive Explanation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='6% Causality: Models, Reasoning and Inference 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='5% The Book of Why: The New Science of Cause and Effect 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='5% Studies in the Logic of Explanation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='1% Causality 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='1% The philosophical basis of algorithmic recourse 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='0% Scientific Explanation and the Causal Structure of the World 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='9% Scientific Explanation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='7% Knowledge-Based Causal Attribution : The Abnormal Conditions Focus Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='2% Combining explanation and argumentation in dialogue Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Top cited papers in Philosophy by XAI-CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Citation relationship between XAI-CS and other XAI fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Every pair of edges is normalized to 100% to show which of the two fields is cited more by the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' REFERENCES [1] Anyifei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' All-about-XAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='com/feifeife/All-about-XAI [2] Hubert Baniecki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Adversarial Explainable AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='com/hbaniecki/adversarial-explainable-ai [3] Przemysław Biecek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Interesting resources related to XAI (Explainable Artificial Intelligence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='com/pbiecek/xai_resources [4] Marina Danilevsky, Kun Qian, Ranit Aharonov, Yannis Katsis, Ban Kawas, and Prithviraj Sen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' A Survey of the State of Explainable AI for Natural Language Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='com/anguyen8/XAI-papers [8] Kevin McAreavey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' CHAI-XAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} 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Zarei, and Eric D Ragan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' A Multidisciplinary Survey and Framework for Design and Evaluation of Explainable AI Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' arXiv preprint arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='11839 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' [11] Benedek Rozemberczki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 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Title 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='4% Accountable Algorithms 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='4% Stability 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='4% The Role of Explanation in Algorithmic Trust 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='7% HYPO’S legacy: introduction to the virtual special issue 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='7% THE UNIVERSITY OF CHICAGO LAW REVIEW 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='7% The Philadelphia predictive policing experiment 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='7% Race , Prediction , and Discretion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='9% Antidiscriminatory Algorithms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='9% Of, for, and by the people: the legal lacuna of synthetic persons 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='9% FORECASTING THE FUTURE OF PREDICTIVE CRIME MAPPING Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Top cited papers in Law by XAI-CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 9 Biology 36 Engineering Medicine 12 38 LawBiology Engineering Law Physics Economics Political Science Medicine Mathematics Computer Science Psychology Business Philosophy 0 20 40 60 80 % citations by non-XAI % of citations for each XAI subfield by non-XAI out of all citations for that subfield Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' The percentage of non-XAI citations (out of all citations) per XAI field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Fields with less than 100 outgoing citations were omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Computer Science Medicine Psychology Mathematics Engineering Biology Business Physics Sociology Environmental Science Economics Political Science Materials Science Geography Chemistry Philosophy Geology Art Education History Law 0 2 4 6 8 entropy Entropy for the distribution of citations across papers in every field by XAI-Computer Science Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' Plot of the entropies for every distribution of citations by XAI-CS across papers, by cited field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' For example, XAI-CS citations of Law 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content=' % cited Title 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='9% Development and interpretation of a pathomics-based model for the prediction of microsatellite instability in Colorectal Cancer 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='5% Discovering epistatic feature interactions from neural network models of regulatory DNA sequences 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} +page_content='4% Amino Acid k-mer Feature Extraction for Quantitative Antimicrobial Resistance (AMR) Prediction by Machine Learning and Model Interpretation for Biological Insights 6.' metadata={'source': 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+page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE5T4oBgHgl3EQfGw4w/content/2301.05433v1.pdf'} diff --git a/ldFLT4oBgHgl3EQfdy92/content/tmp_files/2301.12088v1.pdf.txt b/ldFLT4oBgHgl3EQfdy92/content/tmp_files/2301.12088v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1cc892e971086791efd8f8b4a93e4b961f0d494 --- /dev/null +++ b/ldFLT4oBgHgl3EQfdy92/content/tmp_files/2301.12088v1.pdf.txt @@ -0,0 +1,2121 @@ +arXiv:2301.12088v1 [cs.NI] 28 Jan 2023 +Wireless and Service Allocation for Mobile +Computation Offloading with Task Deadlines +Hong Chen∗, Terence D. Todd∗, Dongmei Zhao∗ and George Karakostas† +∗Department of Electrical and Computer Engineering +†Department of Computing and Software +McMaster University +Hamilton, Ontario, CANADA +Email: {chenh151,todd,dzhao,karakos}@mcmaster.ca +Abstract—In mobile computation offloading (MCO), mobile +devices (MDs) can choose to either execute tasks locally or +to have them executed on a remote edge server (ES). This +paper addresses the problem of assigning both the wireless +communication bandwidth needed, along with the ES capacity +that is used for the task execution, so that task completion time +constraints are satisfied. The objective is to obtain these alloca- +tions so that the average power consumption of the mobile devices +is minimized, subject to a cost budget constraint. The paper +includes contributions for both soft and hard task completion +deadline constraints. The problems are first formulated as mixed +integer nonlinear programs (MINLPs). Approximate solutions +are then obtained by decomposing the problems into a collection +of convex subproblems that can be efficiently solved. Results are +presented that demonstrate the quality of the proposed solutions, +which can achieve near optimum performance over a wide range +of system parameters. +Index Terms—Edge computing, mobile computation offloading, +soft and hard task completion deadlines, cost budget constraints, +power efficiency. +I. INTRODUCTION +Mobile computation offloading (MCO) can be used to +improve mobile device (MD) performance by running compu- +tational tasks on a remote cloud server rather than executing +them locally [1]–[3]. Since the energy needed for task exe- +cution is incurred by the cloud server, a reduction in mobile +device energy consumption can often be obtained [4]–[10]. +During MCO, wireless communications is used by the MD +to communicate with the cloud server. This interaction incurs +MD energy use that would not otherwise exist if the task were +executed at the MD. MCO also incurs added latency due to +the time needed for the MD to interact with the cloud server +[11], [12]. An edge server (ES) located close to the network +base stations is typically used to reduce this delay by providing +high interconnection bandwidth between the base station (BS) +and the ES [13]. +The question of whether a given task should be offloaded +has been studied extensively [14]–[23]. It is clear from this +work that in order to obtain good performance, the offloading +decisions should incorporate both the limited edge server +computational capacity [21]–[23], and the temporal evolution +This work has been submitted to the IEEE for possible publication. +Copyright may be transferred without notice, after which this version may +no longer be accessible. +of the system during the computation offload. This includes the +queueing behaviour experienced by offloaded tasks awaiting +execution at the ES [18]–[20]. Prior work has also considered +the question of how best to configure system resources so +that MCO is best accommodated [14], [18]–[20], [24], [25]. +These are the issues that are considered in our paper and +involve the tradeoffs between wireless communication and +edge server capacity assignment and how these affect the delay +performance experienced by the MDs. +The wireless and execution capacity assignment problem +in MCO can be informally stated as follows. A network +leaseholder (NL) purchases both wireless channel capacity +and edge server execution services, subject to a cost budget +constraint. The leased resources are then used to provide MCO +to a large set of mobile devices [26]. When an MD generates +a task for execution, there is an associated deadline, which +gives the time by which task execution should be completed +with a high degree of certainty [27]. The objective is to find a +joint wireless and ES resource assignment that minimizes the +mean MD power consumption subject to the budget constraint +and constraints on the task completion times. Note that this +problem is different than that of network slice creation [28]. In +this case, the NL simply purchases services from the network +owner (NO), who prices the cost of unit wireless channel and +computational resources. Due to the edge server placement, +we consider the case where the dominant latencies are that of +wireless access and edge server execution [13]. +The paper is novel in that it includes formulations for both +soft and hard task completion time deadlines. In the soft dead- +line case, the wireless and edge server capacity assignments +are designed so that the probability of task completion time +deadline violation is upper bounded. In the hard deadline case, +task execution deadlines must always be respected, which is +accomplished by including concurrent local execution (CLE) +[29] into the problem formulation. In CLE, local execution of +the task may be initiated while offloading is ongoing, so that +the task completion time deadline is always met. +The inclusion of task deadline constraints significantly in- +creases the difficulty of the problem compared to that of prior +work with no completion time requirements or that uses a +mean delay criterion [30], [31]. In order to obtain solutions to +the problem, a queuing model is used to obtain the delay distri- +bution experienced by tasks that are offloaded to the ES [31], + +2 +[32]. This model is incorporated into the resulting optimization +problems, which are formulated as mixed integer nonlinear +programming problems (MINLPs) that are computationally +hard to solve exactly. Approximate solutions are obtained by +decomposing the non-convex non-linear formulation into a +collection of convex subproblems that can be solved efficiently, +and then picking the best of these solutions. +A variety of results are presented that characterize the +tradeoffs between task deadline violation, average MD power +consumption and the cost budget. Our results show the quality +of the proposed solutions, which can achieve close-to-optimum +performance for a wide range of system parameters. The +results also show that with CLE, the proposed solution not +only guarantees respecting all hard task completion deadlines, +but does so with only slightly higher MD power consumption +when compared to the soft task completion deadlines solution +with a small deadline violation probability. On the other hand, +we show that there is an apparent trade-off in the case of +soft task completion deadlines between the average power +consumption and the deadline violation probability. Namely, +the average MD power consumption of our solution is signif- +icantly reduced when a higher deadline violation probability +is tolerable. +The main contributions of the paper are summarized below. +• This paper addresses the problem of assigning computa- +tional and wireless channel resources for MCO, subject +to task execution completion time deadlines. The work +is the first that generates joint resource assignments for +both soft and hard task deadlines using very general +system modelling assumptions compared to prior work. +The soft deadline case aims to create assignments so that +the probability of task completion time deadline violation +is upper bounded. In the hard deadline case, the paper is +also unique in that it creates resource assignments where +task completion time deadlines are always satisfied. This +is done by incorporating CLE into the problem formula- +tions. For this reason, this is the first paper that obtains +system resource assignments for MCO that ensure that +task completion time deadlines are always satisfied. +• Modeling both soft and hard job completion time targets +significantly increases the difficulty of the problem com- +pared to prior work with no completion time requirements +or that uses a mean delay criterion [30] [31]. In both +deadline cases, the paper addresses this by incorporating +an ES queueing system into the problem formulation +that models the delay distribution experienced by arriving +tasks. The assignment problem is addressed by inverting +the estimated probability density function (PDF) of the +task completion time and incorporating it into the opti- +mizations. These resource assignments are obtained under +very general modeling assumptions, where the wireless +channels are modeled as arbitrary base station specific +sets of Markov processes and task execution times have +a general probability distribution. +• The problems are first formulated as MINLPs, with +integral decision variables for the number of wireless +channels reserved, and a continuous decision variable for +the portion of ES reserved. Even the relaxations of these +MINLPs are difficult to solve, since they are non-convex. +Hence, instead of following the common practice of solv- +ing the relaxation and rounding the fractional solution, we +observe that the discretization of the continuous variable +and the replacement of the discrete channel variables +by approximate functions of the continuous blocking +probabilities, allows us to break the original non-convex +MINLPs into collections of convex subproblems, that +can be solved efficiently. Our solutions are approximate, +and their accuracy depends on both the discretization +granularity and the approximation functions used for +blocking probabilities. On the other hand, they are based +on very general assumptions, i.e., the existence of convex +upper bound approximations of the inversion of blocking +probabilities. The more restricted the system model is, +the better these approximations are. +The remainder of the paper is organized as follows. In +Section II the prior work most related to our paper is reviewed. +The system model and problem formulation is then described +in Section III. In Section III-A, the general design problem is +first considered assuming soft task completion time deadlines, +where the probability of deadline violation is bounded. Follow- +ing this, in Section III-B a formulation is described when task +completion times are subject to hard deadlines. The problem +formulations in both cases are non-convex and difficult to deal +with directly using conventional optimization approaches. In +Section IV, approximation solutions are proposed where the +original problems are decomposed into convex subproblems +that can be efficiently solved. Both the soft and hard deadline +cases are considered in Sections IV-A and IV-B. Section V +then introduces some common system assumptions used in the +remainder of the paper when solving the optimizations. Both +the soft and hard deadline cases are then treated in detail in +Sections V-A and V-B. In Section VI simulation results that +demonstrate the proposed designs are given. Both the single +class and multiple classes of tasks cases are considered in +Sections VI-A and VI-B. Finally, we present our conclusions +of the work in Section VII. +II. RELATED WORK +A large amount of prior MCO work considers the problem +based on system state inputs sampled at task generation times, +i.e., the models assume that the system is static throughout +the offload period [14], [15], [17]–[25], [33], [34]. As in our +paper, task offloading decisions become more complex when +the MD interacts with the network over wireless channels +that may change randomly during the offload. Reference [32] +studies a distributed computation offloading problem with +delay constraints using stochastic communication channels but +does not take into account the energy consumption incurred +during task offloading. The work in [30] uses a Markov +decision process that analyzes the mean task delay and the +average system throughput. Unlike our paper, a throughput +maximization problem is formulated with constraints on the +average task delay, rather than using the delay distribution. In +[31], task offloading is modeled as a game using a network +of queues to obtain the end-to-end delay. The problem is + +3 +transformed into one with a generalized Nash equilibrium +solution that captures the conflicting interests in resource +allocation among mobile network operators and computing +resource providers. In references [30] and [31] the average +delay is considered rather than the stringent types of soft and +hard delay constraints considered in our paper. Reference [35] +considers task offloading with statistical QoS guarantees (i.e., +tasks are allowed to complete before a given deadline with +a probability above a given threshold) to maximize the MD +energy efficiency. The energy efficiency is defined as the ratio +of the overall executed (transmitted) bits of tasks to the total +energy consumption of the MDs. Statistical computation and +transmission models are introduced to quantify the correlation +between the statistical quality of service (QoS) guarantee and +task offloading process. Unlike the models used in [31] and +[35], our paper uses a task offloading and resource allocation +formulation that uses very general system model assumptions, +including base station specific sets of Markov processes for +channel modelling. +Reducing both mobile energy consumption and task exe- +cution time is a common objective in mobile computation +offloading. The work in [36] investigates a latency minimiza- +tion problem in a multi-user time-division multiple access +system with joint communication and computation resource +allocation. Our paper, instead, uses a soft task deadline crite- +rion based on modelling the distributions of both upload and +execution time delays. Hard completion time constraints are +considered in references [17], [21]–[23], [32], [33]. However, +unlike our work, they consider the hard completion time re- +quirement as a constraint in the problem formulation. For this +reason, if the provided network resources or the MD transmit +power are insufficient, the hard completion time constraints +may not be satisfied. In our work, we avoid this infeasibility +by applying CLE that ensures that hard completion time +constraints are always satisfied. A benefit from integrating +CLE into the problem formulation is that we no longer +require the hard completion time constraints in the problem +formulation. The objective in [21] is to minimize the energy +consumption of the entire system, and in references [22] and +[33], the objective is to minimize the total energy consumption +of all MDs. Instead of satisfying delay constraints, the work in +[24], [25], [34] optimize a utility function that is a weighted +sum of task completion time and energy consumption. Unlike +the above work, two different kinds of delay constraints are +introduced in our paper, i.e., soft deadlines captured by the +statistics of the completion time of the tasks and hard deadlines +that are always satisfied by CLE. +Prior work has considered the optimization of wireless +network and computational server resources to improve MCO +performance [14], [18]–[20], [24], [25]. In particular, offload- +ing decisions and base station associations are optimized with +transmission power and channel assignments in a cellular +network to minimize the total energy consumption of all +MDs, subject to task’s latency constraints [17]. Reference [21] +studies the problem of task offloading and channel resource +allocation for ultra-dense networks and minimizes the total en- +ergy consumption of the system with a limited delay tolerance. +The work in [22] studies MCO by considering application +latency fairness and minimizes MD energy consumption by +jointly optimizing the offloading ratio, channel assignments, +and channel time allocations. Reference [23] investigates the +power minimization problem for meeting the service delay +requirements in multi-cell multi-user mobile edge computing +networks. Channel assignment and power allocation problems +are considered jointly. The work in [26] studies the joint +resource management of link scheduling, channel assignment +and power control for device-to-device communication as- +sisted multi-tier fog computing with the objective of maximiz- +ing the network operator profit under deadline requirements. +It considers the service charge collected from all end users, +total expense in renting third-party fog nodes, and the en- +ergy cost of the ES. All of this work [17], [21]–[23], [26] +optimizes radio resources and offloading decisions without +considering edge server computational capability. The work +in [24] investigates relay-assisted computation offloading to +minimize the weighted sum of task execution delay and the +energy consumption by jointly optimizing the offloading ratio, +bandwidth allocation, processor speeds, and transmit power. +Table I summarizes the work described above that is most +related to our paper, and compares it to this paper on five key +properties: +Joint channel and computation resource assignment: +The column denotes work where both channel and +computation resource assignments are jointly generated. +Our work differs from the rest in that we assign aggregate +channel resources from the network operator to each +base station so that it can support its associated mobile +device population, i.e., we do not allocate channel and +computation resources of each BS and ES to individual +MDs. +Soft task deadlines: The work selected in this column con- +siders some form of soft (i.e., statistical) task deadlines. +However, the models we use in this paper are quite differ- +ent with more general underlying assumptions. Since our +soft deadline model aims to set bounds on the probability +of task deadline violation, we model the complete delay +distribution experienced by executed tasks. This includes +the base station channel delay (which is modeled by base +station specific Markov processes) and the queueing delay +experienced at the ES, where execution times can have a +general distribution. +Hard task deadlines: Although there is other work selected +in this column, a significant difference exists compared +with our paper, which we have already discussed above. +Namely, our work can always satisfy all hard task +deadlines by incorporating the CLE mechanism into the +modeled system. The related work, instead, considers +the existence of hard deadlines as a problem constraint +that may result in problem infeasibility, which can never +happen in our case. +Resource expense: This column denotes work where the re- +sources provided to the MDs are charged by a third-party +(e.g., network operator). The work selected considers +computational resource expense but not on the wireless +base station side. A network profit maximization problem + +4 +TABLE I: Related Work Summary +References +Joint channel and +computation resource +assignment +Soft task +deadlines +Hard task +deadlines +Resource +expense +Temporal +evolution +[17] [21] [22] [23] +✓ +[24] [36] +✓ +[26] +✓ +✓ +[30] [31] +✓ +✓ +[32] +✓ +✓ +[35] +✓ +✓ +✓ +Our paper +✓ +✓ +✓ +✓ +✓ +is studied where an expense budget is not considered, +unlike the case in our work. +Temporal evolution: Temporal evolution means that the of- +fload periods may include stochastic changes to the +wireless channels and the ES, so that this information +must be modeled in the problem formulation, as in our +paper. The randomness modeled in the selected work has +different underlying assumptions compared to our paper. +III. SYSTEM MODEL AND PROBLEM FORMULATION +As shown in Fig. 1, we consider a network that consists of +N BSs that are owned and operated by a NO. The set of BSs +is denoted by N = {1, 2, . . ., N} and indexed by n ∈ N. +The network also contains an ES. Tasks generated by an MD +can be offloaded through the wireless network and executed +on the ES. +The NO permits a NL to rent wireless communication and +ES computational capacity that the NL can use for mobile +computation offloading for its MDs. When this is done, for +each BS n, there are up to Kn available channels that can +be selected by the NL. The cost of renting a channel from +BS n is set by the NO to αn. When a channel is included in +the agreement, the NO agrees to provision its network so that +sufficient resources are available to allow the traffic generated +on the channel to be carried to the ES with an acceptable +delay with a high degree of certainty. Since the ES is located +at the edge of the network, we focus on the dominant sources +of delay, i.e., wireless access at the BSs and task execution at +the ES [13]. +In order to use the computing resources at the ES, the NL +must also lease CPU resources at the ES. The cost (based on +the number of CPU cycles per second) for leasing on the CPU +resource is denoted by β. The maximum available CPU speed +for rental is f C CPU cycles per second. +When an agreement is made between the NO and NL, xn is +defined as the number of channels from BS n that are included, +and y ∈ [0, 1] is defined as the fraction of maximum CPU +speed at the ES that is included, i.e., the CPU speed available +for the NL will be yf C. It is assumed that the NL has a +cost budget, denoted by Bmax. Accordingly, the total rent must +satisfy the following constraint: +�N +n=1 αnxn + βyf C ≤ Bmax. +(1) +There are J classes of tasks generated by the MDs, which +may need to be offloaded to the ES. Let J = {1, 2, . . ., J} +be the set of task classes. The class j of a task is defined +Edge Server (ES) + + +Fig. 1: System Model +by parameters sj, qj, and dj, where sj is the input data size +in bits, qj is the computation load in number of CPU cycles, +and dj is the deadline of the task in seconds. In what follows, +˜dj = ⌊dj/τ⌋ is the task deadline rounded down to time slots +of the same duration τ as the wireless transmission time slots +(see below). The probability of a task generated by an MD +belonging to class j is denoted by P C +j ; we assume that this +probability is known, e.g., by observing the past history of +offloading requests. +Our objective is to create a NO/NL contract for MCO. +In MCO, tasks generated by an MD can be executed either +locally (at the MD itself) or offloaded through the network and +executed on the ES. We focus on two goals, each depending +on how hard the task deadline constraint is. Our first goal is to +accomplish this so that the mean mobile power consumption +is minimized subject to the cost budget constraint and such +that the probability that task execution deadline violation is +bounded, i.e., the deadline constraints can be violated, albeit +rarely. Our second goal is to create a power-efficient, budget- +respecting assignment which respects all task deadlines, i.e., +deadline constraints are hard; for that purpose we will employ +CLE [29]. +We model the wireless channels between the MDs and the + +5 +BSs as discrete-time Markov processes. It is assumed that there +are In channel models for BS n, which are a function of +the radio propagation environment that the MDs experience +at that BS. In = {1, 2, . . ., In} is the set of all wireless +channel models in BS n. For each of the channel models, +the Markovian transition probabilities are defined in the usual +way, i.e., given the channel state in the current time slot, there +is a probability associated to its transition to another state in +the next time slot. The time slot duration is defined to be +τ seconds. A class j task, offloaded to BS n by the MD, +encounters channel model k with probability P G +n,j,k; as with +task generation probabilities P C +j +above, we assume that this +probability is also known, e.g., by observing the past history +of offloading requests. +To obtain the design, the decision to offload the execution +of a task is made using a local execute on blocking (LEB) +mechanism as follows. When an MD in BS n generates a +class j task, the MD offloads the task if at least one of the +xn channels is available for immediate use. Otherwise, the +MD executes the task locally. When a channel is available, +the MD begins the offload by uploading the sj task bits +needed for execution on the ES. The LEB mechanism is +useful in that either local execution or remote offloading is +initiated immediately at task release time, which may be +advantageous when task deadlines are tight. It also provides a +simple mechanism for assessing when the current level of local +congestion is high, which would suggest that local execution +is beneficial. +Tasks arrive at BS n according to a stationary process with +average arrival rate λn tasks per second. According to the LEB +mechanism, a new task is blocked from BS channel access if +all the xn channels are busy with uploading other tasks. We +denote the task blocking probability at BS n by PBn(xn), +which is a function of xn. For the sake of notation simplicity, +we use PBn in the rest of the paper. Let pL be the power needed +in the MD to process tasks. When a class j task is blocked +from offloading and executed locally, the local execution time +is given as Lj = qj/f, where f is the MD’s execution speed in +number of CPU cycles per time slot1. Define ¯L as the average +local execution time of tasks. Since the task blocking is caused +by channel access, which is the same for all task classes, we +have ¯L = �J +j=1 P C +j Lj. The average energy consumption for +executing a task locally is given by pL ¯L. Consider all the tasks +that are generated in BS n and blocked from offloading in one +second, then the mean energy for executing these tasks locally +is +EL +n(xn) = PBnλnpL ¯L, +(2) +which is the average power consumption of the MDs. +The wireless upload transmission time tW +n,j,k of a jth class +task in BS n when the wireless channel model is k, is +measured in time slots. The mean wireless upload transmission +time ¯tW +n,j,k for jth class tasks in BS n according to channel +model k can be calculated, since Pr[tW +n,j,k = l] can be com- +1Lj is normally measured in CPU cycles, but in order to apply CLE and +to simplify the system, we round it up to a multiple of τ. +puted for all l from channel model k. Moreover, the mean +wireless transmission time ¯tW +n for BS n is +¯tW +n = +J +� +j=1 +In +� +k=1 +P C +j P G +n,j,k¯tW +n,j,k. +(3) +Under the stated assumptions, the aggregate mean task +arrival rate λ at the ES is given by +λ = �N +n=1 (1 − PBn)λn. +(4) +As is normally the case for stability in a single server queueing +system, the following constraint must always be satisfied, +λ < µC, +(5) +where µC denotes the mean service rate at the ES, i.e, µC = +yf C/ �J +j=1 P C +j qj. As will become clear later, we can relax +this constraint to λ ≤ yf C/ �J +j=1 P C +j qj without affecting our +proposed solutions. +Let tC +n,j,k be the delay (including both queueing and exe- +cution time) experienced by a jth class task from BS n at +the ES, under wireless channel model k. It takes continuous +values, and Pr[tC +n,j,k ≤ t], for any t ≥ 0, is a function of λ +and µC. In what follows, ˜tC +n,j,k is the discretization of tC +n,j,k, +measured in time slots; its distribution is calculated by +Pr[˜tC +n,j,k = b] = Pr[tC +n,j,k ≤ bτ] − Pr[tC +n,j,k ≤ (b − 1)τ] (6) +for any number of time slots b ≥ 0. Table II lists the related +notation and their associated meanings. +A. Problem Formulation with Soft Deadlines +We consider the distribution of total delay for an offloaded +task, which is the sum of the data upload delay tW +n,j,k and +the task execution at ES delay tC +n,j,k, for BS n, task class +j, and channel model k. Note that both delays are random +variables. As mentioned earlier, the data transmission delay +from the BS to the ES is negligible. In addition, in this paper +we consider the case of a very small amount of data returned +once the execution is completed, and, therefore, we consider +only uploading delays between MD and BS. +Following common practice (e.g., [27]) in modelling soft +deadlines along the lines of QoS requirements, a jth class +task in BS n under wireless channel model k, must have a +total delay satisfying +Pr[tW +n,j,k + tC +n,j,k ≤ dj] ≥ 1 − εj, +(7) +where 0 < εj ≤ 1 is the (given) tolerated probability the +completion time of a class j task exceeds its deadline.2 Note +that tW +n,j,k takes discrete values (number of time slots), tC +n,j,k +takes discrete values (number of CPU cycle periods), while dj +is continuous (in seconds), so (7) assumes that all quantities +are first converted to secs. Its LHS is a function of xn, y. +The joint probability distribution of total delay is +Pr[tW +n,j,k + tC +n,j,k ≤ dj] = +2The case εj = 0 corresponds to the case of hard deadlines, and will be +dealt with in the next section. + +6 +TABLE II: Summary of Notation +Notation +Definition +Units +N +Set of BSs, |N | = N +J +Set of task classes, |J | = J +In +Set of channel models of BS n, |In| = In +Kn +Number of available channels in BS n +fC +Maximum available ES capacity +CPU cycles/sec +αn +Unit price of wireless channels from BS n +$ per channel +β +Unit price of ES capacity +$ per bps +xn +Number of channels from BS n +y +Fraction of maximum ES capacity +Bmax +Cost budget +$ +sj +Data size of a task in class j +bits +qj +Computation load of a task in class j +CPU cycles +dj +Deadline of a task in class j +sec +˜dj +Discretized deadline of a task in class j +Time slots +P C +j +Probability of a task belonging to class j +P G +n,j,k +Probability of a class j task in BS n with +channel model k +PBn +Blocking probability in BS n +µC +Mean service rate at the ES +Tasks/sec +λn +Average task arrival rate in BS n +Tasks/sec +λ +Aggregate average task arrival rate at ES +Tasks/sec +τ +Time slot +sec +tW +n,j,k +Wireless transmission time of a jth class +task in BS n with channel model k +Time slots +¯tW +n,j,k +Mean wireless transmission time of a jth +class task in BS n with channel model k +Time slots +¯tW +n +Mean task uploading transmission time in +BS n +Time slots +tC +n,j,k +Execution time at ES for class j tasks from +BS n with channel model k +sec +˜tC +n,j,k +Discretized value of tC +n,j,k +Time slots +tL +j +Latest feasible starting time for local exe- +cution +Time slots +εj +Tolerable probability a class j task exceeds +deadline +pL +Local energy consumption per time slot +Joules +pT +Wireless transmission energy per time slot +Joules +ET +n +Average MD power consumption for up- +loading tasks in BS n +Watts +EC +n +Average MD power consumption for up- +loading and executing tasks in BS n +Watts +lmax +� +l=1 +Pr[tW +n,j,k = l] Pr[tC +n,j,k ≤ dj − lτ], +(8) +where lmax = ⌊(dj −qj/yf C)/τ⌋ is the maximum value that l +can take, since qj/yf C is the execution time at the ES without +queueing. +The average power consumption of MDs in BS n to upload +tasks that are granted channels for offloading is +ET +n (xn) = (1 − PBn)λnpT¯tW +n , +(9) +where pT is the transmission energy per time slot used by +the MD for uploading the task bits. Therefore, the expected +average power consumption of the MDs for uploading and +executing tasks arriving at BS n is EL +n(xn) + ET +n (xn). +Our objective is to create an allocation that minimizes +EL +n(xn) + ET +n (xn) under the cost budget and deadline con- +straints (1) and (7). The problem can be formulated as follows: +min +x,y +N +� +n=1 +[EL +n(xn) + ET +n (xn)] s.t. +(10) +N +� +n=1 +αnxn + βf Cy ≤ Bmax +(11) +Pr[tW +n,j,k + tC +n,j,k ≤ dj] ≥ 1 − εj, +∀n, j, k +(12) +(f C/ +J +� +j=1 +P C +j qj)y ≥ λ +(13) +xn ∈ {0, 1, . . ., Kn}, +∀n ∈ N +(14) +0 ≤ y ≤ 1. +(15) +Constraints (11) and (12) are constraints (1) and (7). Con- +straint (13) is the (relaxed) queue stability requirement for ES; +it is equivalent to (5), since equality leads to infinite mean +queueing delay, which is never optimal. The optimization +problem (10)-(15) is a mixed integer nonlinear programming +(MINLP) problem. Constraint (14) ensures that the number +of channels assigned does not exceed the maximum number +available in each BS. Even the fractional relaxation of MINLP +problem (10)-(15) is non-convex, due to its objective and con- +straints (12), and, as a result, it is computationally inefficient +to solve it exactly. Hence we are going to propose approximate +solutions for it. +B. Problem Formulation with Hard Deadlines +For the case of hard deadline constraints, i.e., when the task +deadline must be respected, we employ CLE [29]. In CLE, +local execution of the task may be initiated while offloading +is ongoing, so that the task deadline is always met, even if +offloading fails to finish in time due to the stochastic nature +of the wireless channels. Guaranteeing task completion before +its deadline may incur additional costs (due to potentially +simultaneous local and remote execution of the same task). +When CLE is employed, and in order to ensure that the +local execution of a task from class j finishes by its deadline, +the latest feasible starting time for local execution is +tL +j = ˜dj − Lj + 1. +(16) +The expected wireless transmission power is still given +by (9). However, due to the overlap of offloading and local +execution because of CLE, there is an extra mean power +consumption due to a (potential) overlap with local execution. +This expected overlap power consumption is +EO +n,j,k(xn, y) = (1 − PBn)λn +· +˜dj +� +t=tL +j +t−⌈ +qj +yfCτ ⌉ +� +l=1 +Pr[tW +n,j,k = l] Pr[˜tC +n,j,k = t − l] · pL(t − tL +j + 1), +(17) + +7 +where t is the number of time slots needed to complete the +offloaded task, and (t − tL +j + 1) is the offloading and local +execution overlap. Note that Pr[˜tC +n,j,k = t − l] is a function of +xn and y. +In case the task offloading goes beyond the finish of the +local execution of a task, there is an extra power consumption +incurred, whose expected value is +EB +n,j,k(xn, y) = (1 − PBn)λn +· ++∞ +� +t= ˜dj+1 +t−⌈ +qj +yfCτ ⌉ +� +l=1 +Pr[tW +n,j,k = l] Pr[˜tC +n,j,k = t − l]pLLj. +(18) +Hence, the expected power consumption of MDs for offloaded +tasks in BS n in one second is +EC +n (xn, y) = ET +n (xn) ++ +J +� +j=1 +In +� +k=1 +P C +j P G +n,j,k[EO +n,j,k(xn, y) + EB +n,j,k(xn, y)], +(19) +and the expected power consumption of MDs for tasks arriving +at BS n in one second is EL +n(xn) + EC +n (xn, y). +As before, our objective is to minimize the total expected +power consumption of the MDs for uploading and executing +the tasks that are generated in one second, but now subject +to hard deadline constraints. The problem is formulated as +follows: +min +x,y +N +� +n=1 +[EL +n(xn) + EC +n (xn, y)] s.t. +(20) +N +� +n=1 +αnxn + βf Cy ≤ Bmax +(21) +(f C/ +J +� +j=1 +P C +j qj)y ≥ λ +(22) +xn ∈ {0, 1, . . ., Kn}, +∀n ∈ N +(23) +0 ≤ y ≤ 1. +(24) +IV. GENERAL APPROXIMATE ALLOCATION SOLUTIONS +In this section, we propose approximate solutions for opti- +mization problems (10)-(15) and (20)-(24), by decomposing +them into convex optimization subproblems which can be +solved efficiently. +A. Approximate Solution for Soft Deadlines +In this subsection, we propose an approximate solution for +the optimization problem (10)-(15) by decomposing it into +several convex subproblems that can be solved efficiently, +solve them, and then keep the best solution. More specifically, +we discretize variable y ∈ [0, 1] by breaking [0, 1] into +Y equal segments, so that y takes values ya = a/Y , for +a = 0, 1, . . ., Y . With y fixed, we show that the relaxation of +(10)-(15) can be approximated by a convex optimization prob- +lem, which can be solved in polynomial time. The resulting +(fractional) xn’s are then rounded to integer values (and this is +another source of suboptimality for our solution method). After +solving the resulting Y +1 problems, we output the minimum +solution x∗, y∗. Obviously, the quality of the approximation +depends on the discretization parameter Y . +We consider the relaxed version of problem (10)-(15), i.e., +constraint (14) has been replaced by xn ≥ 0, ∀n. With y +fixed, we show that the non-convex problem (10)-(15) can be +transformed into an equivalent convex optimization problem +with the PBn’s as the decision variables. First, we concentrate +on constraints (12), (13). Note that Pr[tW +n,j,k + tC +n,j,k ≤ dj] is +a monotonically decreasing function of the aggregate mean +task arrival rate λ. Hence, by binary search in the range +[0, yf C/ �J +j=1 P C +j qj], we can approximate within any desired +accuracy the maximum possible value of λ that satisfies +constraints (12) for all n, j, k. Let λ∗ be this maximum value +(note that λ∗ < µC, so stability is ensured). Using (4), +constraints (12), (13) can be replaced by constraint +N +� +n=1 +(1 − PBn)λn ≤ λ∗. +(25) +Next, we note that the blocking probability PBn is mono- +tonically decreasing in xn; let P min +Bn be the blocking probability +when xn = Kn. Then constraints (14) can be replaced by the +equivalent constraints +P min +Bn ≤ PBn ≤ 1, ∀n ∈ N. +(26) +Constraint (11) is the only remaining constraint with an +explicit dependence on the xn’s. Since PBn is a function of +xn, one could potentially use its inverse to replace xn with a +function of PBn. However, such an inversion function may not +exist explicitly (and even if it does, it may be non-convex). In +its stead, we can use a convex upper bound approximation F +of the inversion of blocking probability, so that +xn ≤ F(PBn), ∀n ∈ N. +(27) +Hence, the new convex optimization problem that approxi- +mates the original one when y is fixed, is the following: +min +PB +N +� +n=1 +[EL +n(PBn) + ET +n (PBn)] s.t. +(28) +N +� +n=1 +αnF(PBn) ≤ Bmax − βf Cy +(29) +N +� +n=1 +(1 − PBn)λn ≤ λ∗ +(30) +P min +Bn ≤ PBn ≤ 1, +∀n ∈ N. +(31) +After solving (28)-(31) and obtaining the PBn’s, we can +compute the largest integral x∗ +n which achieves a blocking +probability equal to or bigger than PBn, for all n ∈ N. +Complexity Analysis: Algorithm GCASD (cf. Algorithm 1) +codifies the solution method described above. Problem (28)- +(31) is convex, and can be solved in time O(L), for a poly- +nomial L. Line 6 takes time O(N log Kmax) (recall that there +are N BSs, and Kmax is the largest Kn). Hence Algorithm +1 has a running time of O(Y (L + log µC +ǫ + N log Kmax)), +where Y is the granularity of y, and O(log µC +ǫ ) is the binary + +8 +Algorithm 1 General Case Approximation for Soft Deadlines +(GCASD) +Require: λn, pT, pL, αn, Kn, β, f C, Y, sj, dj, qj, P C +j , P G +n,j,k, +PDFs of tW, tC +1: cost∗ = ∞ +2: for all a = 0, . . . , Y do +3: +y = a/Y +4: +Obtain λ∗, the upper bound of λ, by binary search in +[0, µC] +5: +[PB, cost] = [solution, objective] of (28)-(31) +6: +xint = max integral x with blocking probabilities ≥ PB +7: +if cost < cost∗ then +8: +x∗ = xint; y∗ = y; cost∗ = cost +9: +end if +10: end for +11: return x∗, y∗ +search cost of line 4 of the algorithm, in order to get a λ∗ +within ǫ of the optimal. +B. Approximate Solution for Hard Deadlines +In this subsection, we use a similar approach in order to +solve (20)-(24). Here we decompose the original problem into +several subproblems by discretizing both variable y as before, +and λ. Then, for every possible (fixed) pair (y, λ), the non- +convex problem (20)-(24) can be transformed into a convex +optimization problem with PBn as its decision variables, which +can be solved in polynomial time. By calculating the pair +(y∗, λ∗) whose subproblem achieves minimum average power +consumption, integer values x∗ +n for the original optimization +problem are obtained from P ∗ +Bn. +In more detail, we discretize y +∈ +[0, 1] by break- +ing +[0, 1] +into +Y +equal +segments, +and +then +we +dis- +cretize λ +∈ +[0, yf C/ �J +j=1 P C +j qj] by breaking interval +[0, yf C/ �J +j=1 P C +j qj] into Λ equal segments. At iteration +(m, i) of this discretization, y = y(m) and λ = λ(i) are fixed. +Then Pr[˜tC +n,j,k = t − l] can be calculated directly for any t and +l, and the original optimization problem (20)-(24) becomes +min +x +N +� +n=1 +[EL +n(xn) + EC +n (xn)] s.t. +(32) +N +� +n=1 +αnxn ≤ Bmax − βf Cy(m) +(33) +N +� +n=1 +(1 − PBn)λn ≤ λ(i) +(34) +xn ∈ {0, 1, . . ., Kn}, +∀n ∈ N +(35) +This is still a non-convex non-linear integer program, which +cannot be solved efficiently. As in Section III, and by using +(26)-(27), it becomes +min +PB +N +� +n=1 +[EL +n(PBn)+EC +n (PBn)] s.t. +(36) +N +� +n=1 +αnF(PBn) ≤ Bmax − βf Cy(m) +(37) +N +� +n=1 +(1 − PBn)λn ≤ λ(i) +(38) +P min +Bn ≤ PBn ≤ 1, +∀n ∈ N. +(39) +Problem (36)-(39) is a convex program and can be solved +efficiently. Hence, we can obtain the optimal blocking prob- +abilities P ∗ +Bn, corresponding to a pair (y(m), λ(i)). We can +compute the largest integral x∗ +n which achieves blocking +probabilities no smaller than P ∗ +Bn, for all n ∈ N, by using +binary search based on the fact that the PBn’s are decreasing +functions of the xn’s. After collecting the solutions for all +iterations (m, i), we output the minimum cost one x∗, y∗. +Algorithm 2 General Case Approximation for Hard Deadlines +(GCAHD) +Require: λn, pT, pL, αn, Kn, β, f C, Y, sj, dj, qj, Λ, P C +j , P G +n,j,k, +PDFs of tW, tC +1: cost∗ = ∞, y = 0, λ = 0 +2: while y ≤ 1 do +3: +while λ ≤ yf C/ �J +j=1 P C +j qj do +4: +[PB, cost] = [solution, objective] of (36)-(39) +5: +xint = max integral x with blocking probabilities +≥ PB +6: +if cost < cost∗ then +7: +x∗ = xint; y∗ = y; cost∗ = cost +8: +end if +9: +λ = λ + +yf C/ �J +j=1 P C +j qj +Λ +10: +end while +11: +y = y + 1 +Y +12: end while +13: return x∗, y∗ +Complexity Analysis: Algorithm GCAHD (cf. Algorithm 2) +codifies the solution method described above. Problem (36)- +(39) is convex, and can be solved (line 4) in time O(L), for +a polynomial L. Line 5 takes time O(N log Kmax) (recall +that there are N BSs, and Kmax is the largest Kn). Hence +Algorithm 2 has a running time of O(Y Λ(L+ N log Kmax)), +where Y and Λ are the granularity of y and λ respectively. +V. TASK ARRIVAL AND OFFLOADING ASSUMPTIONS +In the remainder of this paper, we assume that tasks arrive +from the MDs at BS n according to a Poisson process with +mean arrival rate λn. The Poisson process assumption is +commonly made in this type of situation, since the number +of mobile devices in a given coverage area is typically quite +large, each contributing to a small fraction of the total load +[37]. In this case, we can invoke the insensitivity property +of the Erlang B formula, to compute the probability of +blocking at each BS [38]. Note that, typically, the Erlang +B result is derived using the M/M/N/N Markovian queue, +which assumes exponentially distributed channel upload (i.e., +service) times [39]. Due to insensitivity, the result holds for + +9 +any service time distribution with the same mean. Therefore, +the blocking probability for a task arriving at BS n is +PBn = +� λn +µW +n +�xn 1 +xn! +� xn +� +r=0 +� λn +µW +n +�r 1 +r! +�−1 +(40) +where µW +n +denotes the mean service rate, which can be +calculated by µW +n = 1/¯tW +n . Function (40) is convex in xn +[40]. +Note that due to the Poisson process task arrival assump- +tion, the channel state sampled by arriving tasks is given +by the steady-state equilibrium probability distribution of the +Markovian channel at that MD. This follows from the PASTA +rule [41]. +We assume that the aggregate task arrival process at ES is +Poisson [42], and, therefore, arriving tasks sample the asymp- +totic equilibrium state distribution of ES. This approximation +is justified due to the mixing of arrivals at ES from BSs +operating independently. In this case, ES can be modeled +as an M/G/1 queue, whose waiting time is given by the +random variable wC. Given λ and knowledge of the data +upload distribution, the distribution of wC can be obtained +by numerical inversion of the probability generating function +of system waiting time for M/G/1 [37]. In this case, the +execution time of a task at the ES depends only on which +class it belongs to, i.e., tC +n,j,k = tC +j , for all n and k, and +tC +j = wC + qj/yf C. Thus, Pr[tW +n,j,k + tC +j ≤ dj] can be easily +obtained. +When applying algorithms GCASD (Algorithm 1) and GC- +AHD (Algorithm 2) in this case, the upper bound F used in +problem (28)-(31) and (36)-(39) becomes [43]: +xn ≤ λn +µW +n +(1 − PBn) + +1 +PBn +, ∀n. +(41) +A. Approximation with Soft Deadlines +In this case, problem (28)-(31) becomes: +min +PB +N +� +n=1 +[EL +n(PBn) + ET +n (PBn)] s.t. +(42) +N +� +n=1 +αn( λn +µW +n +(1 − PBn) + +1 +PBn +) ≤ Bmax−βf Cy +(43) +N +� +n=1 +(1 − PBn)λn ≤ λ∗ +(44) +P min +Bn ≤ PBn ≤ 1, +∀n ∈ N. +(45) +Problem (42)-(45) is convex, and can be solved in polynomial +time. Hence Algorithm 1 can be implemented efficiently. +B. Approximation with Hard Deadlines +In this case, problem (36)-(39) becomes +min +PB +N +� +n=1 +[EL +n(PBn) + EC +n (PBn)] s.t. +(46) +N +� +n=1 +αn( λn +µW +n +(1 − PBn) + +1 +PBn +) ≤ Bmax−βf Cy(m) +(47) +N +� +n=1 +(1 − PBn)λn ≤ λ(i) +(48) +P min +Bn ≤ PBn ≤ 1, +∀n ∈ N. +(49) +Problem (46)-(49) is convex, and can be solved efficiently. +VI. SIMULATION RESULTS +In this section, we present simulation results to demon- +strate the performance of our proposed algorithms GCASD +(Algorithm 1) and GCAHD (Algorithm 2). We adopt the +two-state Gilbert-Elliot channel model [44], i.e., the channel +states change by following a Markov chain with two states, +“Good” (G) and “Bad” (B). This model is commonly used +to characterize the effects of burst noise in wireless channels, +where the channel can abruptly transition between good and +bad conditions [45]. The Gilbert-Elliot channel is a difficult +one for computation offloading algorithms to deal with com- +pared to those where there is much more correlation in the +channel quality as the offloading progresses. Let Bg and Bb, +respectively, be the data transmission rate when the channel +is in the G and B states. We consider that all channels have +the same Bg and Bb values but differ in their state transition +probabilities that result in different propagation models. The +transition probabilities for propagation model k in BS n are +denoted as P GG +n,k , P GB +n,k , P BG +n,k , and P BB +n,k . In each time slot, +the channel state Markov chain transitions in accordance with +these probabilities. Denote πG +n,k and πB +n,k, respectively, as the +stationary probabilities of a channel in BS n for propagation +model k being in the G and B states. Two sets of simulations +are performed with set 1 for single class of tasks and set +2 for multiple classes of tasks. Default parameters used in +the simulations are summarized in Table III. The parameter +settings that we use were taken from the references [23], +[26] and [32]. These references summarize parameter settings +for various types of applications including those that are +inherently delay sensitive, such as gaming, face recognition +and healthcare use. We intentionally use a wide range of +parameter values based on the referenced ranges so that we +can make conclusions that apply in general settings. +A. Simulation set 1: single class of tasks +In this subsection, we will assume that all the tasks gen- +erated at the MDs have the same data size s and same +computation load q, i.e., sj = s and qj = q for all j. When the +channel is in the G state, the transmission rate of the wireless +channel allows a task to be uploaded within one time slot; +while when the channel is in the B state, the data transmission +rate is zero. Since there is only one class of the tasks, subscript +j can be dropped from the notation. +Let tW +n,k be the time needed for uploading a task in BS n +with channel model k. The probability that one task in BS n +with channel model k can be uploaded in l time slots is given +as follows +Pr[tW +n,k = l] = + + + +πG +n,k, +when l = 1 +πB +n,kP BB +n,k +l−2P BG +n,k , +when l ≥ 2 +(50) + +10 +The mean wireless transmission time of a task in BS n +uploaded through a channel with propagation model k can +be calculated as follows +¯tW +n,k = +∞ +� +l=1 +l Pr[tW +n,k = l] = 1 + +P GB +n,k +P BG +n,k +2 + P GB +n,k P BG +n,k +. +(51) +Based on this, the mean wireless transmission time of the +tasks in BS n is ¯tW +n += �In +k=1 P G +n,k¯tW +n,k, where P G +n,k is the +probability that a task in BS n is uploaded through a channel +with propagation model k. +With a single class of tasks, the ES server becomes an +M/D/1 queueing system, tC +n,j,k = tC for all n, j and k, +and the distribution of delay is given by [46] +Pr[tC ≤ ˆt] = +� +1 − λ +µC +� ⌊ˆtµC⌋ +� +z=0 +[λ( z +µC − ˆt)] +z +z! +e +−λ( +z +µC −ˆt) +(52) +where µC = yf C/q. +For comparison, we also run a discrete event simulation +(DES) of the system using the xn’s and y solutions obtained +from the proposed algorithms to validate our model assump- +tions, and these solutions are denoted as DESSD and DESHD, +respectively, for the soft deadline (SD) and hard deadline (HD) +cases. In addition, we simulate a DES-based OPT scheme for +each proposed algorithm as follows. For GCASD, we first +obtain all the possible combinations of xn’s under constraint +(14); for a given combination of xn’s, we can obtain the +solution of y based on (11) and (15), and then check if +constraint (13) is satisfied based on the current set of xn’s +and y. If not, we go to the next set of xn’s and repeat this +procedure. If it is satisfied, we use this set of xn’s and y to run +the DES for the system, and then check if (12) is satisfied. If +not, we proceed to the next combination of xn’s and repeat the +above procedure. If the constraints are satisfied, we save the +obtained average power. After going through all the possible +combinations of xn’s, we obtain the minimum average power +and the corresponding xn’s and y. For GCAHD, we first obtain +all the possible combinations of xn’s under constraint (23); for +a given combination of xn’s, we can obtain the solution of y +based on (21) and (24), and then check if constraint (22) is +satisfied based on the current set of xn’s and y. If not, we go to +the next set of xn’s and repeat this procedure. If it is satisfied, +we use this set of xn’s and y to run the DES for the system. +Then, we save the obtained mean power consumption. After +going through all the possible combinations of xn’s, we obtain +the minimum average power and the corresponding xn’s and +y. +In the simulation, we consider a cellular network consisting +of 3 BSs. There are two propagation models at each BS with +transition probabilities P GG +n,1 = 0.9, P GG +n,2 = 0.7, P BB +n,1 = 0.1, +and P BB +n,2 = 0.3 for n = 1, 2, 3. The probabilities of the differ- +ent channel models in BS 1 are P G +1,1 = 0.8 and P G +1,2 = 0.2; and +those in BSs 2 and 3 are P G +2,1 = 0.5, P G +2,2 = 0.5, P G +3,1 = 0.2, +and P G +3,2 = 0.8. +Figs. 2(a) and 2(b) show the average power consumption of +MDs versus Bmax for the SD and HD cases, respectively. In +Fig. 2(a), when the tolerable violation of latency ε is 1%, +TABLE III: Default Parameters +Parameter +Value in set 1 +Value in set 2 +τ +1 s +pL +250 mW +pT +2.5 mW +λn +11, 13, 15 tasks/s +Kn +15, 15, 20 +αn +1, 1, 1 $ +β +0.3 × 10−6 $ +0.25 × 10−6 $ +fC +75M cycles/s +200M cycles/s +f +1M cycles/s +2M cycles/s +Bmax +140 $ +90 $ +Bg, Bb +2M, 0 bits per time slot +5M, 1M bits per time slot +sj +2M bits +5M, 10M, 15M bits +dj +4 s +6, 11, 16 s +qj +3M CPU cycles +10M, 20M, 30M CPU cycles +the average power consumption of MDs is a constant for +all the solutions. This is because all the tasks are executed +locally regardless of the cost budget, since the tight delay +constraints cannot be satisfied if a task is offloaded. When +ε is 3% or 5%, some tasks are allowed to be offloaded, and +the average power consumption of the MDs decreases with +Bmax for all the solutions. This happens since, when the +cost budget is small, the optimization is constrained by the +cost budget, which limits the number of offloaded tasks; and +with the increase of Bmax, more channel and ES resource is +available, leading to more MDs offloading their tasks. When +Bmax is large, the budget constraint is loose, and the task +offloading completion is mainly affected by the changing +wireless transmission conditions. Fig. 2(a) also shows that the +average MD power consumption decreases with ε for all the +solutions, since larger ε makes it easier to meet the latency +constraint through offloading, which results in more offloaded +tasks and saves power in the MDs. +By comparing the average MD power consumption for +ε = 3% and ε = 5% in Fig. 2(a), it is seen that the +gap is small when the cost budget is small. The gap then +increases as the cost budget increases, and finally becomes +constant when the cost budget is sufficiently large. When +the cost budget is low, the number of channels is small, +which forces most tasks to be executed locally, regardless of +the value of ε. As the cost budget increases, more channels +are available, and the offloading decisions are determined by +both ε and the available channel resources. When the cost +budget is sufficiently high, the offloading decisions are mainly +determined by the value of ε. The figure also shows that +the average MD power consumption using GCASD is almost +the same as using DESSD, which validates the model and +approximations used in designing GCASD. The performance +of GCASD is also close to DESSD-based OPT, which further +shows good performance of the former. +By comparing Figs. 2(b) and 2(a), it can be seen that the +average MD power consumption for the HD case is slightly +higher than that for the SD case with ε = 3% and much + +11 +�� +�� +�� +��� +��� +��������������� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +Average power��������������W� +������ ǫ��� +������ ǫ��� +�����������������ǫ��� +������ ǫ��� +������ ǫ��� +�����������������ǫ��� +������ ǫ��� +������ ǫ��� +�����������������ǫ��� +(a) Soft deadlines +�� +�� +�� +��� +��� +��������������� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +�� +����������������������������� +����� +����� +��������������� +(b) Hard deadlines +Fig. 2: Average power consumption versus cost budget (Single class of tasks) +�� +�� +�� +�� +�� +�� +������������������������������������� +� +�� +�� +�� +�� +�� +�� +�� +�� +Average power��������������W� +������ ǫ��� +������ ǫ��� +�����������������ǫ��� +������ ǫ��� +������ ǫ��� +�����������������ǫ��� +������ ǫ��� +������ ǫ��� +�����������������ǫ��� +(a) Soft deadlines +�� +�� +�� +�� +�� +�� +������������������������������������� +� +�� +�� +�� +�� +�� +�� +�� +�� +Average power��������������W� +����� +����� +��������������� +(b) Hard deadlines +Fig. 3: Average power consumption versus mean arrival rate (Single class of tasks) +lower than that for the SD case with ε = 1%. For the +SD case, when ε = 1%, the tight (soft) delay constraint +forces all the tasks to be executed locally, which results in +the highest average power consumption of the MDs; and the +power consumption decreases as ε increases and more tasks +are allowed to be offloaded. Without having to use CLE, the +SD solutions result in lower average MD power consumption +than the corresponding HD solutions. However, this is at a +price that up to ε of the tasks do not meet their completion +deadlines. On the other hand, using CLE in the GCAHD only +incur slightly higher power consumption of the MDs compared +to GCASD when ε = 3% For the HD case, the total average +power consumption of the MDs decreases with Bmax when +Bmax is small and becomes a constant when Bmax becomes +larger for all schemes, which is the same as that of the SD +case with ε = 3% and 5%. +Figs. 3(a) and 3(b) show the average power consumption +versus λn (same for all BSs) for the SD and HD cases, +respectively. The figures show that the power consumption +increases linearly with λn for all schemes, since both the +local execution power and the uploading transmission power +are proportional to the mean task arrival rate. The average + +12 +�� +�� +�� +�� +��� +��� +��� +��� +�������������������������������������� +� +�� +�� +�� +�� +�� +Average power��������������W� +������ ǫ��� +������ ǫ��� +�����������������ǫ��� +������ ǫ��� +������ ǫ��� +�����������������ǫ��� +������ ǫ��� +������ ǫ��� +�����������������ǫ��� +(a) Soft deadlines +�� +�� +�� +�� +��� +��� +��� +��� +�������������������������������������� +� +�� +�� +�� +�� +�� +����������������������������� +����� +����� +��������������� +(b) Hard deadlines +Fig. 4: Average power consumption versus available ES capacity (Single class of tasks) +MD power consumption using GCAHD is close to that using +GCASD with ε = 3% but much lower than that using GCASD +with ε = 1%. This demonstrates that the use of CLE in +GCAHD is minimized, while always ensuring the HD of the +tasks. Fig. 3(a) shows that the performance of GCASD is very +close to DESSD and DESSD-based OPT; and Fig. 3(b) shows +that the performance of GCAHD is very close to DESHD and +DESHD-based OPT. These observations are consistent with +the ones from Figs. 2(a) and 2(b). This further demonstrates +the good performance of GCASD and GCAHD and validates +the model and approximations used in designing the proposed +algorithms. +Figs. 4(a) and 4(b) show the average power consumption +of the MDs versus f C, which is the ES capacity, for the SD +and HD cases, respectively. For the SD case with ε = 1%, +all tasks are executed locally; and when ε = 3% and 5%, +offloading is possible for some tasks, and the number of +tasks that can be offloaded increases with the ES capacity, +resulting in lower power consumption of the MDs. As the ES +capacity is sufficiently high, the average power consumption +of MDs becomes a constant, since the offloading decisions +are determined by the cost budget which limits the number +of wireless channels for uploading tasks. Note that the slight +increase in average power consumption when f C is between +60 and 80 is caused by the discretization errors of variable y in +algorithms 1 and 2. Increasing the Y values in the algorithms +helps reduce the discretization errors but significantly increase +the amount of time for running the simulations. Comparing +the average power consumption of the HD and the SD cases +shown in Figs. 4(a) and 4(b), we have consistent observations +as in previous figures. +B. Simulation set 2: multiple classes of tasks +In this subsection, tasks have multiple classes. The two- +state Gilbert-Elliot channels are considered. Let Bg and Bb, +respectively, be the data transmission rates when a channel +is in the G and B states. Given the channel state transision +probabilities, the distribution of wireless transmission time +tW +n,j,k for uploading a class j task in BS n through a channel +with propagation model k can be calculated from [29]. +At the ES, the system of serving the uploaded tasks becomes +an M/G/1 queueing system. Let B be a random variable +representing the execution time of the tasks. We have Pr[B = +qj +yf C ] = P C +j , then the probability density function of B can be +written as +fB(˜b) = +J +� +j=1 +Pr +� +B = +qj +yf C +� +δ +� +˜b − qj +yf C +� += +J +� +j=1 +P C +j δ +� +˜b − qj +yf C +� +, +(53) +and the Laplace-Stieltjes transform of fB(˜b) is given by +g(s) = +J +� +j=1 +P C +j e +− +qj +yfC s. +(54) +The Laplace-Stieltjes transform of the probability density +function of queuing time wC is given by the Pollaczek- +Khinchine transform [37] as +W ∗(s) = +(1 − λ¯b)s +s − λ(1 − g(s)), +(55) +where ¯b is the mean of B. The distribution of wC can be +obtained by numerical inversion of (55). +In the simulation, we consider a cellular network consisting +of 3 BSs, 3 task classes, and 2 channel propagation models. + +13 +The channel state transition probabilities are P GG +n,1 += 0.9, +P BB +n,1 = 0.1, P GG +n,2 = 0.6, and P BB +n,1 = 0.4 for n = 1, 2, 3. The +probabilities of accessing channels with different propagation +models in BS 1 are P G +1,1 = 0.8 and P G +1,2 = 0.2; those in BSs 2 +and 3 are P G +2,1 = 0.5, P G +2,2 = 0.5, P G +3,1 = 0.2, and P G +3,2 = 0.8. +The probabilities of a task belonging to different classes are +P C +1 = 0.6, P C +2 = 0.3, and P C +3 = 0.1. +Figs. 5(a) and 5(b) show the average power consumption of +MDs versus Bmax for the SD and HD cases, respectively. In +Fig. 5(a), when ε is 0.5%, all the tasks are executed locally +regardless of the cost budget, since offloading cannot satisfy +the tight delay constraints. When ε is 1% or 6%, the average +power consumption of MDs decreases with Bmax and then +becomes a constant. By comparing the power consumption +of the MD in the SD and HD cases, we can see that the +average power consumption of MDs for the HD case is slightly +higher than that for the SD case with ε = 1% and much +lower than that for the SD case with ε = 0.5%. Figs. 6(a) and +6(b) show the total average power consumption of the MDs +versus f C. All the results show that our GCASD and GCAHD +solutions achieve the average power consumption performance +that is very close to DES-based OPT, and the observations in +the multi-class simulations are consistent with the single-class +simulations. +VII. CONCLUSIONS +This paper has studied joint wireless network and task +service allocation for mobile computation offloading. The +objective is to minimize the total average power consumption +of MDs for completing the arriving tasks, while satisfying +the delay constraints of tasks and the cost budget of the +network customer. The formulations presented included both +soft and hard task completion time deadlines. The designs +were formulated as MINLPs and approximate solutions were +obtained by decomposing the formulations into convex sub- +problems. Simulation results were presented that characterize +the performance of the system and show various tradeoffs +between task deadline violation, average mobile device power +consumption and the cost budget. Results were presented that +demonstrate the quality of the proposed solutions, which can +achieve close-to-optimum performance over a wide range of +system parameters. The optimum allocation were obtained +by doing exhaustive search-based discrete event simulations +for assigning the wireless channels from each BSs and ES +capacity. +REFERENCES +[1] T. H. 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Franx, “A simple solution for the m/d/1 waiting time distribution,” +Operations Research Letters, vol. 29, no. 5, pp. 221–229, 2001. + diff --git a/ldFLT4oBgHgl3EQfdy92/content/tmp_files/load_file.txt b/ldFLT4oBgHgl3EQfdy92/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7edace4004972fe983fd1b0e279b0302713755e5 --- /dev/null +++ b/ldFLT4oBgHgl3EQfdy92/content/tmp_files/load_file.txt @@ -0,0 +1,1107 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf,len=1106 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='12088v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='NI] 28 Jan 2023 Wireless and Service Allocation for Mobile Computation Offloading with Task Deadlines Hong Chen∗, Terence D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Todd∗, Dongmei Zhao∗ and George Karakostas† ∗Department of Electrical and Computer Engineering †Department of Computing and Software McMaster University Hamilton, Ontario, CANADA Email: {chenh151,todd,dzhao,karakos}@mcmaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='ca Abstract—In mobile computation offloading (MCO), mobile devices (MDs) can choose to either execute tasks locally or to have them executed on a remote edge server (ES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This paper addresses the problem of assigning both the wireless communication bandwidth needed, along with the ES capacity that is used for the task execution, so that task completion time constraints are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The objective is to obtain these alloca- tions so that the average power consumption of the mobile devices is minimized, subject to a cost budget constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The paper includes contributions for both soft and hard task completion deadline constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The problems are first formulated as mixed integer nonlinear programs (MINLPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Approximate solutions are then obtained by decomposing the problems into a collection of convex subproblems that can be efficiently solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Results are presented that demonstrate the quality of the proposed solutions, which can achieve near optimum performance over a wide range of system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Index Terms—Edge computing, mobile computation offloading, soft and hard task completion deadlines, cost budget constraints, power efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' INTRODUCTION Mobile computation offloading (MCO) can be used to improve mobile device (MD) performance by running compu- tational tasks on a remote cloud server rather than executing them locally [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Since the energy needed for task exe- cution is incurred by the cloud server, a reduction in mobile device energy consumption can often be obtained [4]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' During MCO, wireless communications is used by the MD to communicate with the cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This interaction incurs MD energy use that would not otherwise exist if the task were executed at the MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' MCO also incurs added latency due to the time needed for the MD to interact with the cloud server [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' An edge server (ES) located close to the network base stations is typically used to reduce this delay by providing high interconnection bandwidth between the base station (BS) and the ES [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The question of whether a given task should be offloaded has been studied extensively [14]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' It is clear from this work that in order to obtain good performance, the offloading decisions should incorporate both the limited edge server computational capacity [21]–[23], and the temporal evolution This work has been submitted to the IEEE for possible publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Copyright may be transferred without notice, after which this version may no longer be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' of the system during the computation offload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This includes the queueing behaviour experienced by offloaded tasks awaiting execution at the ES [18]–[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Prior work has also considered the question of how best to configure system resources so that MCO is best accommodated [14], [18]–[20], [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' These are the issues that are considered in our paper and involve the tradeoffs between wireless communication and edge server capacity assignment and how these affect the delay performance experienced by the MDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The wireless and execution capacity assignment problem in MCO can be informally stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' A network leaseholder (NL) purchases both wireless channel capacity and edge server execution services, subject to a cost budget constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The leased resources are then used to provide MCO to a large set of mobile devices [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When an MD generates a task for execution, there is an associated deadline, which gives the time by which task execution should be completed with a high degree of certainty [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The objective is to find a joint wireless and ES resource assignment that minimizes the mean MD power consumption subject to the budget constraint and constraints on the task completion times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Note that this problem is different than that of network slice creation [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In this case, the NL simply purchases services from the network owner (NO), who prices the cost of unit wireless channel and computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Due to the edge server placement, we consider the case where the dominant latencies are that of wireless access and edge server execution [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The paper is novel in that it includes formulations for both soft and hard task completion time deadlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In the soft dead- line case, the wireless and edge server capacity assignments are designed so that the probability of task completion time deadline violation is upper bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In the hard deadline case, task execution deadlines must always be respected, which is accomplished by including concurrent local execution (CLE) [29] into the problem formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In CLE, local execution of the task may be initiated while offloading is ongoing, so that the task completion time deadline is always met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The inclusion of task deadline constraints significantly in- creases the difficulty of the problem compared to that of prior work with no completion time requirements or that uses a mean delay criterion [30], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In order to obtain solutions to the problem, a queuing model is used to obtain the delay distri- bution experienced by tasks that are offloaded to the ES [31], 2 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This model is incorporated into the resulting optimization problems, which are formulated as mixed integer nonlinear programming problems (MINLPs) that are computationally hard to solve exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Approximate solutions are obtained by decomposing the non-convex non-linear formulation into a collection of convex subproblems that can be solved efficiently, and then picking the best of these solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' A variety of results are presented that characterize the tradeoffs between task deadline violation, average MD power consumption and the cost budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Our results show the quality of the proposed solutions, which can achieve close-to-optimum performance for a wide range of system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The results also show that with CLE, the proposed solution not only guarantees respecting all hard task completion deadlines, but does so with only slightly higher MD power consumption when compared to the soft task completion deadlines solution with a small deadline violation probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' On the other hand, we show that there is an apparent trade-off in the case of soft task completion deadlines between the average power consumption and the deadline violation probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Namely, the average MD power consumption of our solution is signif- icantly reduced when a higher deadline violation probability is tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The main contributions of the paper are summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This paper addresses the problem of assigning computa- tional and wireless channel resources for MCO, subject to task execution completion time deadlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The work is the first that generates joint resource assignments for both soft and hard task deadlines using very general system modelling assumptions compared to prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The soft deadline case aims to create assignments so that the probability of task completion time deadline violation is upper bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In the hard deadline case, the paper is also unique in that it creates resource assignments where task completion time deadlines are always satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This is done by incorporating CLE into the problem formula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' For this reason, this is the first paper that obtains system resource assignments for MCO that ensure that task completion time deadlines are always satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Modeling both soft and hard job completion time targets significantly increases the difficulty of the problem com- pared to prior work with no completion time requirements or that uses a mean delay criterion [30] [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In both deadline cases, the paper addresses this by incorporating an ES queueing system into the problem formulation that models the delay distribution experienced by arriving tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The assignment problem is addressed by inverting the estimated probability density function (PDF) of the task completion time and incorporating it into the opti- mizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' These resource assignments are obtained under very general modeling assumptions, where the wireless channels are modeled as arbitrary base station specific sets of Markov processes and task execution times have a general probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The problems are first formulated as MINLPs, with integral decision variables for the number of wireless channels reserved, and a continuous decision variable for the portion of ES reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Even the relaxations of these MINLPs are difficult to solve, since they are non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Hence, instead of following the common practice of solv- ing the relaxation and rounding the fractional solution, we observe that the discretization of the continuous variable and the replacement of the discrete channel variables by approximate functions of the continuous blocking probabilities, allows us to break the original non-convex MINLPs into collections of convex subproblems, that can be solved efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Our solutions are approximate, and their accuracy depends on both the discretization granularity and the approximation functions used for blocking probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' On the other hand, they are based on very general assumptions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', the existence of convex upper bound approximations of the inversion of blocking probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The more restricted the system model is, the better these approximations are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In Section II the prior work most related to our paper is reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The system model and problem formulation is then described in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In Section III-A, the general design problem is first considered assuming soft task completion time deadlines, where the probability of deadline violation is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Follow- ing this, in Section III-B a formulation is described when task completion times are subject to hard deadlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The problem formulations in both cases are non-convex and difficult to deal with directly using conventional optimization approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In Section IV, approximation solutions are proposed where the original problems are decomposed into convex subproblems that can be efficiently solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Both the soft and hard deadline cases are considered in Sections IV-A and IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Section V then introduces some common system assumptions used in the remainder of the paper when solving the optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Both the soft and hard deadline cases are then treated in detail in Sections V-A and V-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In Section VI simulation results that demonstrate the proposed designs are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Both the single class and multiple classes of tasks cases are considered in Sections VI-A and VI-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Finally, we present our conclusions of the work in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' RELATED WORK A large amount of prior MCO work considers the problem based on system state inputs sampled at task generation times, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', the models assume that the system is static throughout the offload period [14], [15], [17]–[25], [33], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' As in our paper, task offloading decisions become more complex when the MD interacts with the network over wireless channels that may change randomly during the offload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Reference [32] studies a distributed computation offloading problem with delay constraints using stochastic communication channels but does not take into account the energy consumption incurred during task offloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The work in [30] uses a Markov decision process that analyzes the mean task delay and the average system throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Unlike our paper, a throughput maximization problem is formulated with constraints on the average task delay, rather than using the delay distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In [31], task offloading is modeled as a game using a network of queues to obtain the end-to-end delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The problem is 3 transformed into one with a generalized Nash equilibrium solution that captures the conflicting interests in resource allocation among mobile network operators and computing resource providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In references [30] and [31] the average delay is considered rather than the stringent types of soft and hard delay constraints considered in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Reference [35] considers task offloading with statistical QoS guarantees (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', tasks are allowed to complete before a given deadline with a probability above a given threshold) to maximize the MD energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The energy efficiency is defined as the ratio of the overall executed (transmitted) bits of tasks to the total energy consumption of the MDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Statistical computation and transmission models are introduced to quantify the correlation between the statistical quality of service (QoS) guarantee and task offloading process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Unlike the models used in [31] and [35], our paper uses a task offloading and resource allocation formulation that uses very general system model assumptions, including base station specific sets of Markov processes for channel modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Reducing both mobile energy consumption and task exe- cution time is a common objective in mobile computation offloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The work in [36] investigates a latency minimiza- tion problem in a multi-user time-division multiple access system with joint communication and computation resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Our paper, instead, uses a soft task deadline crite- rion based on modelling the distributions of both upload and execution time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Hard completion time constraints are considered in references [17], [21]–[23], [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' However, unlike our work, they consider the hard completion time re- quirement as a constraint in the problem formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' For this reason, if the provided network resources or the MD transmit power are insufficient, the hard completion time constraints may not be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In our work, we avoid this infeasibility by applying CLE that ensures that hard completion time constraints are always satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' A benefit from integrating CLE into the problem formulation is that we no longer require the hard completion time constraints in the problem formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The objective in [21] is to minimize the energy consumption of the entire system, and in references [22] and [33], the objective is to minimize the total energy consumption of all MDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Instead of satisfying delay constraints, the work in [24], [25], [34] optimize a utility function that is a weighted sum of task completion time and energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Unlike the above work, two different kinds of delay constraints are introduced in our paper, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', soft deadlines captured by the statistics of the completion time of the tasks and hard deadlines that are always satisfied by CLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Prior work has considered the optimization of wireless network and computational server resources to improve MCO performance [14], [18]–[20], [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In particular, offload- ing decisions and base station associations are optimized with transmission power and channel assignments in a cellular network to minimize the total energy consumption of all MDs, subject to task’s latency constraints [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Reference [21] studies the problem of task offloading and channel resource allocation for ultra-dense networks and minimizes the total en- ergy consumption of the system with a limited delay tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The work in [22] studies MCO by considering application latency fairness and minimizes MD energy consumption by jointly optimizing the offloading ratio, channel assignments, and channel time allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Reference [23] investigates the power minimization problem for meeting the service delay requirements in multi-cell multi-user mobile edge computing networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Channel assignment and power allocation problems are considered jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The work in [26] studies the joint resource management of link scheduling, channel assignment and power control for device-to-device communication as- sisted multi-tier fog computing with the objective of maximiz- ing the network operator profit under deadline requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' It considers the service charge collected from all end users, total expense in renting third-party fog nodes, and the en- ergy cost of the ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' All of this work [17], [21]–[23], [26] optimizes radio resources and offloading decisions without considering edge server computational capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The work in [24] investigates relay-assisted computation offloading to minimize the weighted sum of task execution delay and the energy consumption by jointly optimizing the offloading ratio, bandwidth allocation, processor speeds, and transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Table I summarizes the work described above that is most related to our paper, and compares it to this paper on five key properties: Joint channel and computation resource assignment: The column denotes work where both channel and computation resource assignments are jointly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Our work differs from the rest in that we assign aggregate channel resources from the network operator to each base station so that it can support its associated mobile device population, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', we do not allocate channel and computation resources of each BS and ES to individual MDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Soft task deadlines: The work selected in this column con- siders some form of soft (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', statistical) task deadlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' However, the models we use in this paper are quite differ- ent with more general underlying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Since our soft deadline model aims to set bounds on the probability of task deadline violation, we model the complete delay distribution experienced by executed tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This includes the base station channel delay (which is modeled by base station specific Markov processes) and the queueing delay experienced at the ES, where execution times can have a general distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Hard task deadlines: Although there is other work selected in this column, a significant difference exists compared with our paper, which we have already discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Namely, our work can always satisfy all hard task deadlines by incorporating the CLE mechanism into the modeled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The related work, instead, considers the existence of hard deadlines as a problem constraint that may result in problem infeasibility, which can never happen in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Resource expense: This column denotes work where the re- sources provided to the MDs are charged by a third-party (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', network operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The work selected considers computational resource expense but not on the wireless base station side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' A network profit maximization problem 4 TABLE I: Related Work Summary References Joint channel and computation resource assignment Soft task deadlines Hard task deadlines Resource expense Temporal evolution [17] [21] [22] [23] ✓ [24] [36] ✓ [26] ✓ ✓ [30] [31] ✓ ✓ [32] ✓ ✓ [35] ✓ ✓ ✓ Our paper ✓ ✓ ✓ ✓ ✓ is studied where an expense budget is not considered, unlike the case in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Temporal evolution: Temporal evolution means that the of- fload periods may include stochastic changes to the wireless channels and the ES, so that this information must be modeled in the problem formulation, as in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The randomness modeled in the selected work has different underlying assumptions compared to our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' SYSTEM MODEL AND PROBLEM FORMULATION As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 1, we consider a network that consists of N BSs that are owned and operated by a NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The set of BSs is denoted by N = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', N} and indexed by n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The network also contains an ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Tasks generated by an MD can be offloaded through the wireless network and executed on the ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The NO permits a NL to rent wireless communication and ES computational capacity that the NL can use for mobile computation offloading for its MDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When this is done, for each BS n, there are up to Kn available channels that can be selected by the NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The cost of renting a channel from BS n is set by the NO to αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When a channel is included in the agreement, the NO agrees to provision its network so that sufficient resources are available to allow the traffic generated on the channel to be carried to the ES with an acceptable delay with a high degree of certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Since the ES is located at the edge of the network, we focus on the dominant sources of delay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', wireless access at the BSs and task execution at the ES [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In order to use the computing resources at the ES, the NL must also lease CPU resources at the ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The cost (based on the number of CPU cycles per second) for leasing on the CPU resource is denoted by β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The maximum available CPU speed for rental is f C CPU cycles per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When an agreement is made between the NO and NL, xn is defined as the number of channels from BS n that are included, and y ∈ [0, 1] is defined as the fraction of maximum CPU speed at the ES that is included, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', the CPU speed available for the NL will be yf C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' It is assumed that the NL has a cost budget, denoted by Bmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Accordingly, the total rent must satisfy the following constraint: �N n=1 αnxn + βyf C ≤ Bmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (1) There are J classes of tasks generated by the MDs, which may need to be offloaded to the ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Let J = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', J} be the set of task classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The class j of a task is defined Edge Server (ES) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 1: System Model by parameters sj, qj, and dj, where sj is the input data size in bits, qj is the computation load in number of CPU cycles, and dj is the deadline of the task in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In what follows, ˜dj = ⌊dj/τ⌋ is the task deadline rounded down to time slots of the same duration τ as the wireless transmission time slots (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The probability of a task generated by an MD belonging to class j is denoted by P C j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' we assume that this probability is known, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', by observing the past history of offloading requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Our objective is to create a NO/NL contract for MCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In MCO, tasks generated by an MD can be executed either locally (at the MD itself) or offloaded through the network and executed on the ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' We focus on two goals, each depending on how hard the task deadline constraint is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Our first goal is to accomplish this so that the mean mobile power consumption is minimized subject to the cost budget constraint and such that the probability that task execution deadline violation is bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', the deadline constraints can be violated, albeit rarely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Our second goal is to create a power-efficient, budget- respecting assignment which respects all task deadlines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', deadline constraints are hard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' for that purpose we will employ CLE [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' We model the wireless channels between the MDs and the 5 BSs as discrete-time Markov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' It is assumed that there are In channel models for BS n, which are a function of the radio propagation environment that the MDs experience at that BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', In} is the set of all wireless channel models in BS n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' For each of the channel models, the Markovian transition probabilities are defined in the usual way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', given the channel state in the current time slot, there is a probability associated to its transition to another state in the next time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The time slot duration is defined to be τ seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' A class j task, offloaded to BS n by the MD, encounters channel model k with probability P G n,j,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' as with task generation probabilities P C j above, we assume that this probability is also known, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', by observing the past history of offloading requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' To obtain the design, the decision to offload the execution of a task is made using a local execute on blocking (LEB) mechanism as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When an MD in BS n generates a class j task, the MD offloads the task if at least one of the xn channels is available for immediate use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Otherwise, the MD executes the task locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When a channel is available, the MD begins the offload by uploading the sj task bits needed for execution on the ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The LEB mechanism is useful in that either local execution or remote offloading is initiated immediately at task release time, which may be advantageous when task deadlines are tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' It also provides a simple mechanism for assessing when the current level of local congestion is high, which would suggest that local execution is beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Tasks arrive at BS n according to a stationary process with average arrival rate λn tasks per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' According to the LEB mechanism, a new task is blocked from BS channel access if all the xn channels are busy with uploading other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' We denote the task blocking probability at BS n by PBn(xn), which is a function of xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' For the sake of notation simplicity, we use PBn in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Let pL be the power needed in the MD to process tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When a class j task is blocked from offloading and executed locally, the local execution time is given as Lj = qj/f, where f is the MD’s execution speed in number of CPU cycles per time slot1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Define ¯L as the average local execution time of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Since the task blocking is caused by channel access, which is the same for all task classes, we have ¯L = �J j=1 P C j Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The average energy consumption for executing a task locally is given by pL ¯L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Consider all the tasks that are generated in BS n and blocked from offloading in one second, then the mean energy for executing these tasks locally is EL n(xn) = PBnλnpL ¯L, (2) which is the average power consumption of the MDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The wireless upload transmission time tW n,j,k of a jth class task in BS n when the wireless channel model is k, is measured in time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The mean wireless upload transmission time ¯tW n,j,k for jth class tasks in BS n according to channel model k can be calculated, since Pr[tW n,j,k = l] can be com- 1Lj is normally measured in CPU cycles, but in order to apply CLE and to simplify the system, we round it up to a multiple of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' puted for all l from channel model k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Moreover, the mean wireless transmission time ¯tW n for BS n is ¯tW n = J � j=1 In � k=1 P C j P G n,j,k¯tW n,j,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (3) Under the stated assumptions, the aggregate mean task arrival rate λ at the ES is given by λ = �N n=1 (1 − PBn)λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (4) As is normally the case for stability in a single server queueing system, the following constraint must always be satisfied, λ < µC, (5) where µC denotes the mean service rate at the ES, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e, µC = yf C/ �J j=1 P C j qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' As will become clear later, we can relax this constraint to λ ≤ yf C/ �J j=1 P C j qj without affecting our proposed solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Let tC n,j,k be the delay (including both queueing and exe- cution time) experienced by a jth class task from BS n at the ES, under wireless channel model k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' It takes continuous values, and Pr[tC n,j,k ≤ t], for any t ≥ 0, is a function of λ and µC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In what follows, ˜tC n,j,k is the discretization of tC n,j,k, measured in time slots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' its distribution is calculated by Pr[˜tC n,j,k = b] = Pr[tC n,j,k ≤ bτ] − Pr[tC n,j,k ≤ (b − 1)τ] (6) for any number of time slots b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Table II lists the related notation and their associated meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Problem Formulation with Soft Deadlines We consider the distribution of total delay for an offloaded task, which is the sum of the data upload delay tW n,j,k and the task execution at ES delay tC n,j,k, for BS n, task class j, and channel model k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Note that both delays are random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' As mentioned earlier, the data transmission delay from the BS to the ES is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In addition, in this paper we consider the case of a very small amount of data returned once the execution is completed, and, therefore, we consider only uploading delays between MD and BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Following common practice (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', [27]) in modelling soft deadlines along the lines of QoS requirements, a jth class task in BS n under wireless channel model k, must have a total delay satisfying Pr[tW n,j,k + tC n,j,k ≤ dj] ≥ 1 − εj, (7) where 0 < εj ≤ 1 is the (given) tolerated probability the completion time of a class j task exceeds its deadline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='2 Note that tW n,j,k takes discrete values (number of time slots), tC n,j,k takes discrete values (number of CPU cycle periods), while dj is continuous (in seconds), so (7) assumes that all quantities are first converted to secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Its LHS is a function of xn, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The joint probability distribution of total delay is Pr[tW n,j,k + tC n,j,k ≤ dj] = 2The case εj = 0 corresponds to the case of hard deadlines, and will be dealt with in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 6 TABLE II: Summary of Notation Notation Definition Units N Set of BSs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' |N | = N J Set of task classes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' |J | = J In Set of channel models of BS n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' |In| = In ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Number of available channels in BS n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='fC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Maximum available ES capacity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='CPU cycles/sec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='αn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Unit price of wireless channels from BS n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='$ per channel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Unit price of ES capacity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='$ per bps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='xn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Number of channels from BS n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Fraction of maximum ES capacity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Bmax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Cost budget ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='$ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='sj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Data size of a task in class j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='bits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='qj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Computation load of a task in class j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='CPU cycles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='dj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Deadline of a task in class j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='sec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='˜dj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Discretized deadline of a task in class j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Time slots ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='P C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Probability of a task belonging to class j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='P G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='k Probability of a class j task in BS n with channel model k PBn Blocking probability in BS n µC Mean service rate at the ES Tasks/sec λn Average task arrival rate in BS n Tasks/sec λ Aggregate average task arrival rate at ES Tasks/sec τ Time slot sec tW n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='k Wireless transmission time of a jth class task in BS n with channel model k Time slots ¯tW n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='k Mean wireless transmission time of a jth class task in BS n with channel model k Time slots ¯tW n Mean task uploading transmission time in BS n Time slots tC n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='k Execution time at ES for class j tasks from BS n with channel model k sec ˜tC n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='k Discretized value of tC n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='k Time slots tL j Latest feasible starting time for local exe- cution Time slots εj Tolerable probability a class j task exceeds deadline pL Local energy consumption per time slot Joules pT Wireless transmission energy per time slot Joules ET n Average MD power consumption for up- loading tasks in BS n Watts EC n Average MD power consumption for up- loading and executing tasks in BS n Watts lmax � l=1 Pr[tW n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='k = l] Pr[tC n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='k ≤ dj − lτ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (8) where lmax = ⌊(dj −qj/yf C)/τ⌋ is the maximum value that l can take,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' since qj/yf C is the execution time at the ES without queueing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The average power consumption of MDs in BS n to upload tasks that are granted channels for offloading is ET n (xn) = (1 − PBn)λnpT¯tW n , (9) where pT is the transmission energy per time slot used by the MD for uploading the task bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Therefore, the expected average power consumption of the MDs for uploading and executing tasks arriving at BS n is EL n(xn) + ET n (xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Our objective is to create an allocation that minimizes EL n(xn) + ET n (xn) under the cost budget and deadline con- straints (1) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The problem can be formulated as follows: min x,y N � n=1 [EL n(xn) + ET n (xn)] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (10) N � n=1 αnxn + βf Cy ≤ Bmax (11) Pr[tW n,j,k + tC n,j,k ≤ dj] ≥ 1 − εj, ∀n, j, k (12) (f C/ J � j=1 P C j qj)y ≥ λ (13) xn ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', Kn}, ∀n ∈ N (14) 0 ≤ y ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (15) Constraints (11) and (12) are constraints (1) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Con- straint (13) is the (relaxed) queue stability requirement for ES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' it is equivalent to (5), since equality leads to infinite mean queueing delay, which is never optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The optimization problem (10)-(15) is a mixed integer nonlinear programming (MINLP) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Constraint (14) ensures that the number of channels assigned does not exceed the maximum number available in each BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Even the fractional relaxation of MINLP problem (10)-(15) is non-convex, due to its objective and con- straints (12), and, as a result, it is computationally inefficient to solve it exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Hence we are going to propose approximate solutions for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Problem Formulation with Hard Deadlines For the case of hard deadline constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', when the task deadline must be respected, we employ CLE [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In CLE, local execution of the task may be initiated while offloading is ongoing, so that the task deadline is always met, even if offloading fails to finish in time due to the stochastic nature of the wireless channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Guaranteeing task completion before its deadline may incur additional costs (due to potentially simultaneous local and remote execution of the same task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When CLE is employed, and in order to ensure that the local execution of a task from class j finishes by its deadline, the latest feasible starting time for local execution is tL j = ˜dj − Lj + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (16) The expected wireless transmission power is still given by (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' However, due to the overlap of offloading and local execution because of CLE, there is an extra mean power consumption due to a (potential) overlap with local execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This expected overlap power consumption is EO n,j,k(xn, y) = (1 − PBn)λn ˜dj � t=tL j t−⌈ qj yfCτ ⌉ � l=1 Pr[tW n,j,k = l] Pr[˜tC n,j,k = t − l] · pL(t − tL j + 1), (17) 7 where t is the number of time slots needed to complete the offloaded task, and (t − tL j + 1) is the offloading and local execution overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Note that Pr[˜tC n,j,k = t − l] is a function of xn and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In case the task offloading goes beyond the finish of the local execution of a task, there is an extra power consumption incurred, whose expected value is EB n,j,k(xn, y) = (1 − PBn)λn +∞ � t= ˜dj+1 t−⌈ qj yfCτ ⌉ � l=1 Pr[tW n,j,k = l] Pr[˜tC n,j,k = t − l]pLLj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (18) Hence, the expected power consumption of MDs for offloaded tasks in BS n in one second is EC n (xn, y) = ET n (xn) + J � j=1 In � k=1 P C j P G n,j,k[EO n,j,k(xn, y) + EB n,j,k(xn, y)], (19) and the expected power consumption of MDs for tasks arriving at BS n in one second is EL n(xn) + EC n (xn, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' As before, our objective is to minimize the total expected power consumption of the MDs for uploading and executing the tasks that are generated in one second, but now subject to hard deadline constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The problem is formulated as follows: min x,y N � n=1 [EL n(xn) + EC n (xn, y)] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (20) N � n=1 αnxn + βf Cy ≤ Bmax (21) (f C/ J � j=1 P C j qj)y ≥ λ (22) xn ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', Kn}, ∀n ∈ N (23) 0 ≤ y ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (24) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' GENERAL APPROXIMATE ALLOCATION SOLUTIONS In this section, we propose approximate solutions for opti- mization problems (10)-(15) and (20)-(24), by decomposing them into convex optimization subproblems which can be solved efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Approximate Solution for Soft Deadlines In this subsection, we propose an approximate solution for the optimization problem (10)-(15) by decomposing it into several convex subproblems that can be solved efficiently, solve them, and then keep the best solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' More specifically, we discretize variable y ∈ [0, 1] by breaking [0, 1] into Y equal segments, so that y takes values ya = a/Y , for a = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' With y fixed, we show that the relaxation of (10)-(15) can be approximated by a convex optimization prob- lem, which can be solved in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The resulting (fractional) xn’s are then rounded to integer values (and this is another source of suboptimality for our solution method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' After solving the resulting Y +1 problems, we output the minimum solution x∗, y∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Obviously, the quality of the approximation depends on the discretization parameter Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' We consider the relaxed version of problem (10)-(15), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', constraint (14) has been replaced by xn ≥ 0, ∀n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' With y fixed, we show that the non-convex problem (10)-(15) can be transformed into an equivalent convex optimization problem with the PBn’s as the decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' First, we concentrate on constraints (12), (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Note that Pr[tW n,j,k + tC n,j,k ≤ dj] is a monotonically decreasing function of the aggregate mean task arrival rate λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Hence, by binary search in the range [0, yf C/ �J j=1 P C j qj], we can approximate within any desired accuracy the maximum possible value of λ that satisfies constraints (12) for all n, j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Let λ∗ be this maximum value (note that λ∗ < µC, so stability is ensured).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Using (4), constraints (12), (13) can be replaced by constraint N � n=1 (1 − PBn)λn ≤ λ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (25) Next, we note that the blocking probability PBn is mono- tonically decreasing in xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' let P min Bn be the blocking probability when xn = Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Then constraints (14) can be replaced by the equivalent constraints P min Bn ≤ PBn ≤ 1, ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (26) Constraint (11) is the only remaining constraint with an explicit dependence on the xn’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Since PBn is a function of xn, one could potentially use its inverse to replace xn with a function of PBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' However, such an inversion function may not exist explicitly (and even if it does, it may be non-convex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In its stead, we can use a convex upper bound approximation F of the inversion of blocking probability, so that xn ≤ F(PBn), ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (27) Hence, the new convex optimization problem that approxi- mates the original one when y is fixed, is the following: min PB N � n=1 [EL n(PBn) + ET n (PBn)] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (28) N � n=1 αnF(PBn) ≤ Bmax − βf Cy (29) N � n=1 (1 − PBn)λn ≤ λ∗ (30) P min Bn ≤ PBn ≤ 1, ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (31) After solving (28)-(31) and obtaining the PBn’s, we can compute the largest integral x∗ n which achieves a blocking probability equal to or bigger than PBn, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Complexity Analysis: Algorithm GCASD (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Algorithm 1) codifies the solution method described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Problem (28)- (31) is convex, and can be solved in time O(L), for a poly- nomial L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Line 6 takes time O(N log Kmax) (recall that there are N BSs, and Kmax is the largest Kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Hence Algorithm 1 has a running time of O(Y (L + log µC ǫ + N log Kmax)), where Y is the granularity of y, and O(log µC ǫ ) is the binary 8 Algorithm 1 General Case Approximation for Soft Deadlines (GCASD) Require: λn, pT, pL, αn, Kn, β, f C, Y, sj, dj, qj, P C j , P G n,j,k, PDFs of tW, tC 1: cost∗ = ∞ 2: for all a = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' , Y do 3: y = a/Y 4: Obtain λ∗, the upper bound of λ, by binary search in [0, µC] 5: [PB, cost] = [solution, objective] of (28)-(31) 6: xint = max integral x with blocking probabilities ≥ PB 7: if cost < cost∗ then 8: x∗ = xint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' y∗ = y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' cost∗ = cost 9: end if 10: end for 11: return x∗, y∗ search cost of line 4 of the algorithm, in order to get a λ∗ within ǫ of the optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Approximate Solution for Hard Deadlines In this subsection, we use a similar approach in order to solve (20)-(24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Here we decompose the original problem into several subproblems by discretizing both variable y as before, and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Then, for every possible (fixed) pair (y, λ), the non- convex problem (20)-(24) can be transformed into a convex optimization problem with PBn as its decision variables, which can be solved in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' By calculating the pair (y∗, λ∗) whose subproblem achieves minimum average power consumption, integer values x∗ n for the original optimization problem are obtained from P ∗ Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In more detail, we discretize y ∈ [0, 1] by break- ing [0, 1] into Y equal segments, and then we dis- cretize λ ∈ [0, yf C/ �J j=1 P C j qj] by breaking interval [0, yf C/ �J j=1 P C j qj] into Λ equal segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' At iteration (m, i) of this discretization, y = y(m) and λ = λ(i) are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Then Pr[˜tC n,j,k = t − l] can be calculated directly for any t and l, and the original optimization problem (20)-(24) becomes min x N � n=1 [EL n(xn) + EC n (xn)] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (32) N � n=1 αnxn ≤ Bmax − βf Cy(m) (33) N � n=1 (1 − PBn)λn ≤ λ(i) (34) xn ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', Kn}, ∀n ∈ N (35) This is still a non-convex non-linear integer program, which cannot be solved efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' As in Section III, and by using (26)-(27), it becomes min PB N � n=1 [EL n(PBn)+EC n (PBn)] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (36) N � n=1 αnF(PBn) ≤ Bmax − βf Cy(m) (37) N � n=1 (1 − PBn)λn ≤ λ(i) (38) P min Bn ≤ PBn ≤ 1, ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (39) Problem (36)-(39) is a convex program and can be solved efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Hence, we can obtain the optimal blocking prob- abilities P ∗ Bn, corresponding to a pair (y(m), λ(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' We can compute the largest integral x∗ n which achieves blocking probabilities no smaller than P ∗ Bn, for all n ∈ N, by using binary search based on the fact that the PBn’s are decreasing functions of the xn’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' After collecting the solutions for all iterations (m, i), we output the minimum cost one x∗, y∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Algorithm 2 General Case Approximation for Hard Deadlines (GCAHD) Require: λn, pT, pL, αn, Kn, β, f C, Y, sj, dj, qj, Λ, P C j , P G n,j,k, PDFs of tW, tC 1: cost∗ = ∞, y = 0, λ = 0 2: while y ≤ 1 do 3: while λ ≤ yf C/ �J j=1 P C j qj do 4: [PB, cost] = [solution, objective] of (36)-(39) 5: xint = max integral x with blocking probabilities ≥ PB 6: if cost < cost∗ then 7: x∗ = xint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' y∗ = y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' cost∗ = cost 8: end if 9: λ = λ + yf C/ �J j=1 P C j qj Λ 10: end while 11: y = y + 1 Y 12: end while 13: return x∗, y∗ Complexity Analysis: Algorithm GCAHD (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Algorithm 2) codifies the solution method described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Problem (36)- (39) is convex, and can be solved (line 4) in time O(L), for a polynomial L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Line 5 takes time O(N log Kmax) (recall that there are N BSs, and Kmax is the largest Kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Hence Algorithm 2 has a running time of O(Y Λ(L+ N log Kmax)), where Y and Λ are the granularity of y and λ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' TASK ARRIVAL AND OFFLOADING ASSUMPTIONS In the remainder of this paper, we assume that tasks arrive from the MDs at BS n according to a Poisson process with mean arrival rate λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The Poisson process assumption is commonly made in this type of situation, since the number of mobile devices in a given coverage area is typically quite large, each contributing to a small fraction of the total load [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In this case, we can invoke the insensitivity property of the Erlang B formula, to compute the probability of blocking at each BS [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Note that, typically, the Erlang B result is derived using the M/M/N/N Markovian queue, which assumes exponentially distributed channel upload (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', service) times [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Due to insensitivity, the result holds for 9 any service time distribution with the same mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Therefore, the blocking probability for a task arriving at BS n is PBn = � λn µW n �xn 1 xn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' � xn � r=0 � λn µW n �r 1 r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' �−1 (40) where µW n denotes the mean service rate, which can be calculated by µW n = 1/¯tW n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Function (40) is convex in xn [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Note that due to the Poisson process task arrival assump- tion, the channel state sampled by arriving tasks is given by the steady-state equilibrium probability distribution of the Markovian channel at that MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This follows from the PASTA rule [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' We assume that the aggregate task arrival process at ES is Poisson [42], and, therefore, arriving tasks sample the asymp- totic equilibrium state distribution of ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This approximation is justified due to the mixing of arrivals at ES from BSs operating independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In this case, ES can be modeled as an M/G/1 queue, whose waiting time is given by the random variable wC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Given λ and knowledge of the data upload distribution, the distribution of wC can be obtained by numerical inversion of the probability generating function of system waiting time for M/G/1 [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In this case, the execution time of a task at the ES depends only on which class it belongs to, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', tC n,j,k = tC j , for all n and k, and tC j = wC + qj/yf C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Thus, Pr[tW n,j,k + tC j ≤ dj] can be easily obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When applying algorithms GCASD (Algorithm 1) and GC- AHD (Algorithm 2) in this case, the upper bound F used in problem (28)-(31) and (36)-(39) becomes [43]: xn ≤ λn µW n (1 − PBn) + 1 PBn , ∀n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (41) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Approximation with Soft Deadlines In this case, problem (28)-(31) becomes: min PB N � n=1 [EL n(PBn) + ET n (PBn)] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (42) N � n=1 αn( λn µW n (1 − PBn) + 1 PBn ) ≤ Bmax−βf Cy (43) N � n=1 (1 − PBn)λn ≤ λ∗ (44) P min Bn ≤ PBn ≤ 1, ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (45) Problem (42)-(45) is convex, and can be solved in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Hence Algorithm 1 can be implemented efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Approximation with Hard Deadlines In this case, problem (36)-(39) becomes min PB N � n=1 [EL n(PBn) + EC n (PBn)] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (46) N � n=1 αn( λn µW n (1 − PBn) + 1 PBn ) ≤ Bmax−βf Cy(m) (47) N � n=1 (1 − PBn)λn ≤ λ(i) (48) P min Bn ≤ PBn ≤ 1, ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (49) Problem (46)-(49) is convex, and can be solved efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' SIMULATION RESULTS In this section, we present simulation results to demon- strate the performance of our proposed algorithms GCASD (Algorithm 1) and GCAHD (Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' We adopt the two-state Gilbert-Elliot channel model [44], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', the channel states change by following a Markov chain with two states, “Good” (G) and “Bad” (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This model is commonly used to characterize the effects of burst noise in wireless channels, where the channel can abruptly transition between good and bad conditions [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The Gilbert-Elliot channel is a difficult one for computation offloading algorithms to deal with com- pared to those where there is much more correlation in the channel quality as the offloading progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Let Bg and Bb, respectively, be the data transmission rate when the channel is in the G and B states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' We consider that all channels have the same Bg and Bb values but differ in their state transition probabilities that result in different propagation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The transition probabilities for propagation model k in BS n are denoted as P GG n,k , P GB n,k , P BG n,k , and P BB n,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In each time slot, the channel state Markov chain transitions in accordance with these probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Denote πG n,k and πB n,k, respectively, as the stationary probabilities of a channel in BS n for propagation model k being in the G and B states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Two sets of simulations are performed with set 1 for single class of tasks and set 2 for multiple classes of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Default parameters used in the simulations are summarized in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The parameter settings that we use were taken from the references [23], [26] and [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' These references summarize parameter settings for various types of applications including those that are inherently delay sensitive, such as gaming, face recognition and healthcare use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' We intentionally use a wide range of parameter values based on the referenced ranges so that we can make conclusions that apply in general settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Simulation set 1: single class of tasks In this subsection, we will assume that all the tasks gen- erated at the MDs have the same data size s and same computation load q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=', sj = s and qj = q for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When the channel is in the G state, the transmission rate of the wireless channel allows a task to be uploaded within one time slot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' while when the channel is in the B state, the data transmission rate is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Since there is only one class of the tasks, subscript j can be dropped from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Let tW n,k be the time needed for uploading a task in BS n with channel model k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The probability that one task in BS n with channel model k can be uploaded in l time slots is given as follows Pr[tW n,k = l] = \uf8f1 \uf8f2 \uf8f3 πG n,k, when l = 1 πB n,kP BB n,k l−2P BG n,k , when l ≥ 2 (50) 10 The mean wireless transmission time of a task in BS n uploaded through a channel with propagation model k can be calculated as follows ¯tW n,k = ∞ � l=1 l Pr[tW n,k = l] = 1 + P GB n,k P BG n,k 2 + P GB n,k P BG n,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (51) Based on this, the mean wireless transmission time of the tasks in BS n is ¯tW n = �In k=1 P G n,k¯tW n,k, where P G n,k is the probability that a task in BS n is uploaded through a channel with propagation model k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' With a single class of tasks, the ES server becomes an M/D/1 queueing system, tC n,j,k = tC for all n, j and k, and the distribution of delay is given by [46] Pr[tC ≤ ˆt] = � 1 − λ µC � ⌊ˆtµC⌋ � z=0 [λ( z µC − ˆt)] z z!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' e −λ( z µC −ˆt) (52) where µC = yf C/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' For comparison, we also run a discrete event simulation (DES) of the system using the xn’s and y solutions obtained from the proposed algorithms to validate our model assump- tions, and these solutions are denoted as DESSD and DESHD, respectively, for the soft deadline (SD) and hard deadline (HD) cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In addition, we simulate a DES-based OPT scheme for each proposed algorithm as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' For GCASD, we first obtain all the possible combinations of xn’s under constraint (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' for a given combination of xn’s, we can obtain the solution of y based on (11) and (15), and then check if constraint (13) is satisfied based on the current set of xn’s and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' If not, we go to the next set of xn’s and repeat this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' If it is satisfied, we use this set of xn’s and y to run the DES for the system, and then check if (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' If not, we proceed to the next combination of xn’s and repeat the above procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' If the constraints are satisfied, we save the obtained average power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' After going through all the possible combinations of xn’s, we obtain the minimum average power and the corresponding xn’s and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' For GCAHD, we first obtain all the possible combinations of xn’s under constraint (23);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' for a given combination of xn’s, we can obtain the solution of y based on (21) and (24), and then check if constraint (22) is satisfied based on the current set of xn’s and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' If not, we go to the next set of xn’s and repeat this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' If it is satisfied, we use this set of xn’s and y to run the DES for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Then, we save the obtained mean power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' After going through all the possible combinations of xn’s, we obtain the minimum average power and the corresponding xn’s and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In the simulation, we consider a cellular network consisting of 3 BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' There are two propagation models at each BS with transition probabilities P GG n,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='9, P GG n,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='7, P BB n,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='1, and P BB n,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='3 for n = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The probabilities of the differ- ent channel models in BS 1 are P G 1,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='8 and P G 1,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' and those in BSs 2 and 3 are P G 2,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='5, P G 2,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='5, P G 3,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='2, and P G 3,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 2(a) and 2(b) show the average power consumption of MDs versus Bmax for the SD and HD cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 2(a), when the tolerable violation of latency ε is 1%, TABLE III: Default Parameters Parameter Value in set 1 Value in set 2 τ 1 s pL 250 mW pT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='5 mW λn 11, 13, 15 tasks/s Kn 15, 15, 20 αn 1, 1, 1 $ β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='3 × 10−6 $ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='25 × 10−6 $ fC 75M cycles/s 200M cycles/s f 1M cycles/s 2M cycles/s Bmax 140 $ 90 $ Bg, Bb 2M, 0 bits per time slot 5M, 1M bits per time slot sj 2M bits 5M, 10M, 15M bits dj 4 s 6, 11, 16 s qj 3M CPU cycles 10M, 20M, 30M CPU cycles the average power consumption of MDs is a constant for all the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This is because all the tasks are executed locally regardless of the cost budget, since the tight delay constraints cannot be satisfied if a task is offloaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When ε is 3% or 5%, some tasks are allowed to be offloaded, and the average power consumption of the MDs decreases with Bmax for all the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This happens since, when the cost budget is small, the optimization is constrained by the cost budget, which limits the number of offloaded tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' and with the increase of Bmax, more channel and ES resource is available, leading to more MDs offloading their tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When Bmax is large, the budget constraint is loose, and the task offloading completion is mainly affected by the changing wireless transmission conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 2(a) also shows that the average MD power consumption decreases with ε for all the solutions, since larger ε makes it easier to meet the latency constraint through offloading, which results in more offloaded tasks and saves power in the MDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' By comparing the average MD power consumption for ε = 3% and ε = 5% in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 2(a), it is seen that the gap is small when the cost budget is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The gap then increases as the cost budget increases, and finally becomes constant when the cost budget is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When the cost budget is low, the number of channels is small, which forces most tasks to be executed locally, regardless of the value of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' As the cost budget increases, more channels are available, and the offloading decisions are determined by both ε and the available channel resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When the cost budget is sufficiently high, the offloading decisions are mainly determined by the value of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The figure also shows that the average MD power consumption using GCASD is almost the same as using DESSD, which validates the model and approximations used in designing GCASD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The performance of GCASD is also close to DESSD-based OPT, which further shows good performance of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' By comparing Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 2(b) and 2(a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' it can be seen that the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='average MD power consumption for the HD case is slightly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='higher than that for the SD case with ε = 3% and much ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Average power��������������W� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������ ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������ ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�����������������ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������ ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������ ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�����������������ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������ ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������ ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�����������������ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='(a) Soft deadlines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='(b) Hard deadlines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 2: Average power consumption versus cost budget (Single class of tasks) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������������������������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Average power��������������W� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������ ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������ ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�����������������ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������ ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������ ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�����������������ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������ ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������ ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�����������������ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='(a) Soft deadlines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������������������������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Average power��������������W� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='(b) Hard deadlines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 3: Average power consumption versus mean arrival rate (Single class of tasks) lower than that for the SD case with ε = 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' For the SD case, when ε = 1%, the tight (soft) delay constraint forces all the tasks to be executed locally, which results in the highest average power consumption of the MDs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' and the power consumption decreases as ε increases and more tasks are allowed to be offloaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Without having to use CLE, the SD solutions result in lower average MD power consumption than the corresponding HD solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' However, this is at a price that up to ε of the tasks do not meet their completion deadlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' On the other hand, using CLE in the GCAHD only incur slightly higher power consumption of the MDs compared to GCASD when ε = 3% For the HD case, the total average power consumption of the MDs decreases with Bmax when Bmax is small and becomes a constant when Bmax becomes larger for all schemes, which is the same as that of the SD case with ε = 3% and 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 3(a) and 3(b) show the average power consumption versus λn (same for all BSs) for the SD and HD cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The figures show that the power consumption increases linearly with λn for all schemes, since both the local execution power and the uploading transmission power are proportional to the mean task arrival rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The average 12 �� �� �� �� ��� ��� ��� ��� �������������������������������������� � �� �� �� �� �� Average power��������������W� ������ ǫ��� ������ ǫ��� �����������������ǫ��� ������ ǫ��� ������ ǫ��� �����������������ǫ��� ������ ǫ��� ������ ǫ��� �����������������ǫ��� (a) Soft deadlines �� �� �� �� ��� ��� ��� ��� �������������������������������������� � �� �� �� �� �� ����������������������������� ����� ����� ��������������� (b) Hard deadlines Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 4: Average power consumption versus available ES capacity (Single class of tasks) MD power consumption using GCAHD is close to that using GCASD with ε = 3% but much lower than that using GCASD with ε = 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This demonstrates that the use of CLE in GCAHD is minimized, while always ensuring the HD of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 3(a) shows that the performance of GCASD is very close to DESSD and DESSD-based OPT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 3(b) shows that the performance of GCAHD is very close to DESHD and DESHD-based OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' These observations are consistent with the ones from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 2(a) and 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' This further demonstrates the good performance of GCASD and GCAHD and validates the model and approximations used in designing the proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 4(a) and 4(b) show the average power consumption of the MDs versus f C, which is the ES capacity, for the SD and HD cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' For the SD case with ε = 1%, all tasks are executed locally;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' and when ε = 3% and 5%, offloading is possible for some tasks, and the number of tasks that can be offloaded increases with the ES capacity, resulting in lower power consumption of the MDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' As the ES capacity is sufficiently high, the average power consumption of MDs becomes a constant, since the offloading decisions are determined by the cost budget which limits the number of wireless channels for uploading tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Note that the slight increase in average power consumption when f C is between 60 and 80 is caused by the discretization errors of variable y in algorithms 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Increasing the Y values in the algorithms helps reduce the discretization errors but significantly increase the amount of time for running the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Comparing the average power consumption of the HD and the SD cases shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 4(a) and 4(b), we have consistent observations as in previous figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Simulation set 2: multiple classes of tasks In this subsection, tasks have multiple classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The two- state Gilbert-Elliot channels are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Let Bg and Bb, respectively, be the data transmission rates when a channel is in the G and B states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Given the channel state transision probabilities, the distribution of wireless transmission time tW n,j,k for uploading a class j task in BS n through a channel with propagation model k can be calculated from [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' At the ES, the system of serving the uploaded tasks becomes an M/G/1 queueing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Let B be a random variable representing the execution time of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' We have Pr[B = qj yf C ] = P C j , then the probability density function of B can be written as fB(˜b) = J � j=1 Pr � B = qj yf C � δ � ˜b − qj yf C � = J � j=1 P C j δ � ˜b − qj yf C � , (53) and the Laplace-Stieltjes transform of fB(˜b) is given by g(s) = J � j=1 P C j e − qj yfC s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' (54) The Laplace-Stieltjes transform of the probability density function of queuing time wC is given by the Pollaczek- Khinchine transform [37] as W ∗(s) = (1 − λ¯b)s s − λ(1 − g(s)), (55) where ¯b is the mean of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The distribution of wC can be obtained by numerical inversion of (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In the simulation, we consider a cellular network consisting of 3 BSs, 3 task classes, and 2 channel propagation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 13 The channel state transition probabilities are P GG n,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='9, P BB n,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='1, P GG n,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='6, and P BB n,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='4 for n = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The probabilities of accessing channels with different propagation models in BS 1 are P G 1,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='8 and P G 1,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' those in BSs 2 and 3 are P G 2,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='5, P G 2,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='5, P G 3,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='2, and P G 3,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The probabilities of a task belonging to different classes are P C 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='6, P C 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='3, and P C 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 5(a) and 5(b) show the average power consumption of MDs versus Bmax for the SD and HD cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 5(a), when ε is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='5%, all the tasks are executed locally regardless of the cost budget, since offloading cannot satisfy the tight delay constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' When ε is 1% or 6%, the average power consumption of MDs decreases with Bmax and then becomes a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' By comparing the power consumption of the MD in the SD and HD cases, we can see that the average power consumption of MDs for the HD case is slightly higher than that for the SD case with ε = 1% and much lower than that for the SD case with ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 6(a) and 6(b) show the total average power consumption of the MDs versus f C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' All the results show that our GCASD and GCAHD solutions achieve the average power consumption performance that is very close to DES-based OPT, and the observations in the multi-class simulations are consistent with the single-class simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' CONCLUSIONS This paper has studied joint wireless network and task service allocation for mobile computation offloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The objective is to minimize the total average power consumption of MDs for completing the arriving tasks, while satisfying the delay constraints of tasks and the cost budget of the network customer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The formulations presented included both soft and hard task completion time deadlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The designs were formulated as MINLPs and approximate solutions were obtained by decomposing the formulations into convex sub- problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Simulation results were presented that characterize the performance of the system and show various tradeoffs between task deadline violation, average mobile device power consumption and the cost budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' Results were presented that demonstrate the quality of the proposed solutions, which can achieve close-to-optimum performance over a wide range of system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' The optimum allocation were obtained by doing exhaustive search-based discrete event simulations for assigning the wireless channels from each BSs and ES capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' REFERENCES [1] T.' metadata={'source': 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“Multi-user multi-channel computation offloading and resource allocation for mobile edge computing,” in ICC 2020 - 2020 IEEE International Conference on Communications (ICC), 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 14 �� �� ��� ��� ��� ��������������� �� �� �� �� �� �� Average power��������������W� ��������� ǫ����� ��������� ǫ����� ��������������������ǫ����� ��������� ǫ��� ��������� ǫ��� ��������������������ǫ��� ��������� ǫ��� ��������� ǫ��� ��������������������ǫ��� (a) Soft deadlines �� �� ��� ��� ��� ��������������� �� �� �� �� �� �� Average power��������������W� �������� �������� ������������������ (b) Hard deadlines Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content=' 5: Average power consumption versus cost budget (Multiple classes of tasks) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�������������������������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='����������������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��������� ǫ����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��������� ǫ����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��������������������ǫ����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��������� ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��������� ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��������������������ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��������� ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��������� ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��������������������ǫ��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='(a) Soft deadlines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�������������������������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='Average power��������������W� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} +page_content='(b) Hard deadlines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFLT4oBgHgl3EQfdy92/content/2301.12088v1.pdf'} 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a/m9AzT4oBgHgl3EQfN_u0/content/tmp_files/2301.01159v1.pdf.txt b/m9AzT4oBgHgl3EQfN_u0/content/tmp_files/2301.01159v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..360cd0d4dd47a79ff46d7c036e094a22551e922b --- /dev/null +++ b/m9AzT4oBgHgl3EQfN_u0/content/tmp_files/2301.01159v1.pdf.txt @@ -0,0 +1,3604 @@ +Wave propagation in one-dimensional quasiperiodic media +Pierre Amenoagbadji, Sonia Fliss, Patrick Joly +Abstract +This work is devoted to the resolution of the Helmholtz equation −(µ u′)′ −ρ ω2u = f +in a one-dimensional unbounded medium. We assume the coefficients of this equation +to be local perturbations of quasiperiodic functions, namely the traces along a particular +line of higher-dimensional periodic functions. Using the definition of quasiperiodicity, +the problem is lifted onto a higher-dimensional problem with periodic coefficients. The +periodicity of the augmented problem allows us to extend the ideas of the DtN-based +method developed in [10, 19] for the elliptic case. However, the associated mathematical +and numerical analysis of the method are more delicate because the augmented PDE is +degenerate, in the sense that the principal part of its operator is no longer elliptic. We +also study the numerical resolution of this PDE, which relies on the resolution of Dirichlet +cell problems as well as a constrained Riccati equation. +1 +Introduction and motivation +We consider the Helmholtz equation +− d +dx +� +µ du +dx +� +− ρ ω2 u = f +in +R, +(1.1) +where the coefficients µ and ρ have positive upper and lower bounds: +∃ µ±, ρ±, +∀ x ∈ R, +0 < µ− ≤ µ(x) ≤ µ+ +and +0 < ρ− ≤ ρ(x) ≤ ρ+. +(1.2) +The source term f belongs to L2(R) and is assumed to have a compact support: +∃ a > 0, +supp f ⊂ (−a, a). +(1.3) +Equation (1.1) is encountered when one is looking for time-harmonic solutions u(x) eiωt of +the linear wave equation in heterogeneous media. For real frequencies ω, the well-posedness +of this problem is unclear. In fact, on one hand, one expects that the physical solution u, if +it exists, may not belong to H1(R) due to possible wave propagation phenomena and a lack +of decay at infinity. On the other hand, uniqueness of a solution in H1 +loc(R) does not hold in +general. In this case, one needs a so-called a radiation condition that imposes the behaviour +of the solution at infinity. Such a condition can be obtained in practice using the limiting +absorption principle, which consists in (i) adding some absorption – that is some imaginary +part to ω: Im ω, and (ii) studying the limit of the solution u ≡ u(ω) as the absorption tends +to 0. The limiting absorption principle is a classical approach to study time-harmonic wave +propagation problems in unbounded domains; see for instance [1, 9, 31]. More recently, it +has been successfully applied for locally perturbed periodic media [10, 17, 20, 25]. +1 +arXiv:2301.01159v1 [math.AP] 3 Jan 2023 + +In this paper, we will only address the case with absorption, that is +the frequency ω satisfies Im ω > 0. +(1.4) +Under these assumptions, (1.1) admits a unique solution in H1(R) by Lax-Milgram’s theorem. +Moreover, it can be shown (using for instance an argument similar to the one in [7]) that this +solution satisfies a sharp exponential decay property +∃ c, α > 0, +∀ x ∈ R, +|u(x)| ≤ c e−α Im ω|x|. +(1.5) +Exploiting (1.5), a naive numerical method for treating the unboundedness would consist in +truncating the computational domain (with homogeneous Dirichlet boundary conditions for +instance) at a certain distance related to Im ω. However the cost and the accuracy of the +method would deteriorate when Im ω tends to 0. Our objective in this paper is to develop +a numerical method which is robust when Im ω tends to 0, in the particular case of locally +perturbed quasiperiodic media. More precisely, we solve the problem in the bounded domain +(−a, a) (which is independent of Im ω) by constructing transparent boundary conditions of +Dirichlet-to-Neumann type: +± µ du +dx + λ± u = 0 +on +x = ±a, +(1.6) +where λ± are called Dirichlet-to-Neumann (DtN) coefficients. These coefficients are defined +by +λ± = ∓ +� +µ du± +dx +� +(±a), +(1.7) +where u± is the unique solution in H1(±a, ±∞) of +������� +− d +dx +� +µ du± +dx +� +− ρ ω2 u± = 0, +for +±x > a, +u±(±a) = 1. +(1.8) +Knowing λ±, one is then reduced to compute u|(−a,a) by solving the problem +��������� +− d +dx +� +µ dui +dx +� +− ρ ω2 ui = f, +for +x ∈ (−a, a), +� +± µ dui +dx + λ± ui� +(±a) = 0. +(1.9) +The well-posedness of this problem is a direct consequence of the sign property +Im λ± < 0, +which, through a Green’s formula, results itself from the presence of dissipation (1.4) in (1.8). +Then the solution u of (1.1) is given by +∀ x ∈ R, +u(x) = +� +� +� +� +� +� +� +� +� +ui(−a) u−(x), +x < −a, +ui(x), +x ∈ (−a, a), +ui(a) u+(x), +x > a. +(1.10) +In general, the problem is that computing λ±, that is to say solving (1.8), is as difficult as +the original problem. However, this is no longer true when the exterior medium (i.e. outside +(−a, a)) has a certain structure: +2 + +• if the exterior medium is homogeneous (ρ and µ are constant), these coefficients can be +computed explicitly; +• if the exterior medium is periodic (ρ and µ are periodic), several methods for the +computation of these DtN coefficients are developed in [10, 19, 20]; +• if the exterior medium is a weakly random perturbation of a periodic medium, the +coefficients can be approximated via an asymptotic analysis; see [11]. +Our main objective in this paper is to compute the DtN coefficients for a quasiperiodic +exterior medium, in order to develop a numerical method according to (1.8), (1.9), (1.10). +The outline of the rest of the paper is as follows. In Section 2, we introduce the fundamental +notion of quasiperiodic functions (in 1D) and define what is a locally perturbed quasiperiodic +medium in the context of the problem (1.1). Sections 3 and 4 are the most important sections +of the paper. In Section 3, we link the solution of the 1D half-line problem with quasiperiodic +coefficients to the solution of a degenerate directional Helmholtz equation posed in dimension +n, with n > 1 defined as in Section 2. This is the so-called lifting approach whose principle +is presented in Section 3.1. More precisely, in Section 3.3, we characterize the solution of +the 1D quasiperiodic problem as the trace along a (broken) line of a nD problem posed in +a domain with the geometry of a half-waveguide: (0, 1)n−1 × R+. In between, we need to +dedicate the (rather long) Section 3.2 to fix the notations used in the rest of the paper and +present some useful preliminary material about an adapted functional framework for the +rigorous setting of our method. This concerns anisotropic Sobolev spaces with an emphasis +on trace theorems and related Green’s formula. In Section 4, we provide a method for solving +the half-waveguide problem of Section 3.3. In Section 4.1, we describe the structure of the +solution with the help of a propagation operator P and local cell problems. In Section 4.2, we +show that the operator P is characterized as a particular solution of a Riccati equation. In +Section 4.3, we first build a directional DtN operator Λ for the half-waveguide problem, from +which we deduce the DtN coefficients λ± we are looking for (cf. (1.7)). Finally, in Section +4.4, we analyze the Riccati equation from a spectral point of view and in Section 4.5 we +describe the spectrum of P. In Section 5 devoted to numerical results, we restrict ourselves +to n = 2 for the sake of simplicity. The first two subsections are devoted to the discretization +of the cell problems evoked above. We have considered two approaches: one, natural but +naive, consists in using 2D Lagrange finite elements (Section 5.1) while the other, called the +quasi-1D method, is better fitted to the anisotropy of the problem (Section 5.2). In Section +5.3, we explain how we can construct a discrete propagation operator from a discrete Riccati +equation that we choose to solve via a spectral approach, while Section 5.4 simply mimics +Section 4.3 at the discrete level. Section 5.5 is devoted to numerical results. In the first three +subsections, we provide various validations of our method for the half-line problem (Sections +5.5.1 and 5.5.3) and the whole line problem (Section 5.5.2). At last, in Section 5.5.4, we +address the question of the approximation of the spectrum of the propagation operator P by +the one of its discrete approximation. +Particular notation used throughout the paper. +In what follows, +1. the equality modulo 1 is denoted by +∀ y ∈ R, +z = y [1] +⇐⇒ +z ∈ [0, 1) and y − z ∈ Z. +3 + +and for all p, q ∈ N, p < q, we set �p, q� := {j ∈ N, p ≤ j ≤ q}. +2. We introduce Cper(Rn) as the space of continuous functions F : Rn → R that are 1– +periodic with respect to each variable, and C ∞ +0 (O) as the space of smooth functions +that are compactly supported in O ⊂ Rn. +3. For i ∈ �1, n�, we denote by ⃗ei the i-th unit vector from the canonical basis of Rn. For +any element y = (y1, . . . , yn) in Rn, we define ˆy as the vector (y1, . . . , yn−1) ∈ Rn−1, so +that y = (ˆy, yn). For y = (y1, . . . , yn) and z = (z1, . . . , zn), the Euclidean inner product +of y and z is denoted y · z := y1 z1 + · · · yn zn, and the associated norm is |y| := √y · y. +2 +Quasiperiodicity +2.1 +Quasiperiodic functions of one real variable +In this section, we present a brief overview of the main properties of quasiperiodic functions. +We refer to [3, 5, 22] for more complete presentations. Quasiperiodicity is defined as follows. +Definition 2.1. A continuous function f : R → R is said to be quasiperiodic of order n > 1 +if there exist a constant real vector θ = (θ1, . . . , θn), with θi > 0 for all i ∈ �1, n�, and a +continuous function F : Rn → R, 1–periodic with respect to each variable, such that +∀ x ∈ R, +f(x) = F(x θ). +(2.1) +The vector θ is called a cut direction, and F is a periodic extension of f. +A geometrical interpretation of this definition is to see the one-dimensional function f as the +trace of a n-dimensional function F along the line passing through (0, 0) and parallel to the +vector θ. This is illustrated in Figure 1 for n = 2 and θ = (1, +√ +2). +θ +0 +0.4 +0.8 +0 +0.4 +0.8 +0 +Size of periodicity cell +−4 +−2 +0 +2 +4 +−2 +0 +2 +Figure 1: Function F : (y1, y2) �→ cos 2πy1 + cos 2πy2 in its periodicity cell (left), and whose +trace along θ = (1, +√ +2) leads to a quasiperiodic function (right). +Periodic functions are obviously quasiperiodic. Other examples of quasiperiodic functions +are finite sums or products of periodic functions: if f1 and f2 are periodic, then f1 + f2 and +f1f2 can be expressed under the form (2.1). Note that f1 + f2 and f1f2 are not periodic if f1 +and f2 are continuous functions with non-commensurable least periods. For instance, with +4 + +f1(x) = cos 2πx and f2(x) = cos 2π +√ +2x, one easily checks that the sum f1 + f2, represented +in Figure 1, is not periodic since it equals 2 only when x = 0. +In Definition 2.1, it is easy to see that neither the periodic extension nor the cut direction +are uniquely defined. Given (F, θ), it is always possible to lower the value of n, and change +the function F accordingly, so that the coefficients θ1, . . . , θn are linearly independent over +the integers (see [22, Chapter 2]), that is +∀ k ∈ Zn, +k · θ = 0 +⇐⇒ +k = 0. +(2.2) +For n = 2 and θ = (θ1, θ2), the above condition amounts to saying that the ratio θ1/θ2 is +irrational. Due to this observation, vectors that satisfy (2.2) will be abusively referred to as +irrational vectors. A consequence of (2.2) is given by Kronecker’s approximation theorem. +Theorem 2.2 ([16, Theorem 444]). If θ is an irrational vector, then the set θ R + Nn is +dense in Rn. +If θ is an irrational vector, and if F ∈ Cper(Rn) satisfies F(θ R) = 0, then Theorem 2.2 +ensures that F = 0. In other words, under the linear independence assumption, F is uniquely +determined by its restriction on the line θ R. +For n = 2, Theorem 2.2 implies that the broken line +�(x θ1[1], x θ2[1]), x ∈ R +� is dense in the +unit cell (0, 1)2. To illustrate this, Figure 2 represents the set +�(x θ1[1], x θ2[1]), x ∈ (0, M) +� +in the unit cell for different values of M, when (1) θ1/θ2 ∈ Q (see the first row), and when +(2) θ1/θ2 ∈ R \ Q (see the second row for θ = ( +√ +2, 1) and the third one for θ = (π, 1)). For +M large enough, in the first case, this set is reduced to a finite union of segments, whereas in +the second case, it seems to fill the unit cell without ever passing through the same positions. +It is also interesting to see that for θ = ( +√ +2, 1), the unit cell is somehow filled uniformly, +contrary to the case where θ = (π, 1). +Finally, it is worth mentioning that Definition 2.1 extends to higher-dimensional continuous +functions as well. Moreover, the notion of quasiperiodicty can be defined at a discrete level, to +describe the properties of tilings that are cuts and projections of higher-dimensional periodic +tilings. These quasiperiodic tilings have been extensively studied [13, 23, 24, 27], and are +used for modelling quasicrystals [28]. +2.2 +Locally perturbed quasiperiodic media +A locally perturbed quasiperiodic medium is a medium corresponding to functions µ and +ρ that satisfy (1.2) and that are quasiperiodic outside a bounded interval, which can be +supposed to be (−a, a) (see (1.3)) without any loss of generality. More precisely, +µ(x) = +����� +µi(x) +x ∈ (−a, a) +µp(x θ) +x ∈ R \ (−a, a) +and +ρ(x) = +����� +ρi(x) +x ∈ (−a, a) +ρp(x θ) +x ∈ R \ (−a, a), +where the functions µp, ρp belong to Cper(Rn) with n > 1, and θ ∈ Rn is an irrational vector +(see Condition (2.2)). +5 + +0 +1 +0 +1 +θ = (3, 1) +M = 1/3 +0 +1 +M = 2/3 +0 +1 +M = 1 +0 +1 +M ≥ 1 +0 +1 +0 +1 +θ = ( +√ +2, 1) +M = 1 +0 +1 +M = 20 +0 +1 +M = 40 +0 +1 +M = 80 +0 +1 +0 +1 +θ = (π, 1) +M = 1 +0 +1 +M = 20 +0 +1 +M = 40 +0 +1 +M = 80 +Figure 2: Representation of the set +�(x θ1[1], x θ2[1]), x ∈ (0, M) +� in (0, 1)2 for different +values of M, when θ1/θ2 ∈ Q (first row), and when θ1/θ2 ∈ R \ Q (second row for θ = ( +√ +2, 1) +and third row for θ = (π, 1)). +Remark 2.3. (a). Since θ is an irrational vector, Kronecker’s approximation theorem 2.2 +ensures that the functions µp and ρp are entirely determined by their restrictions on the line +R θ. Therefore, µp and ρp satisfy (1.2) with respectively the same bounds as µ and ρ. +(b). The present study can be extended without difficulty to the case where µ (resp. ρ) +coincides with two different quasiperiodic functions in (−∞, −a) and in (a, +∞): +for ± x > ±a, +µ(x) = µ± +p (x θ± ) +and +ρ(x) = ρ± +p (x θ± ), +where µ± +p , ρ± +p belong to Cper(Rn±) with n± > 1, and where θ± ∈ Rn± are irrational vectors. +6 + +3 +The half-line quasiperiodic problem +We now focus on the half-line quasiperiodic problems (1.8). +As these problems are very +similar to each other, it is sufficient to study the half-line problem set on (a, +∞) and suppose +without loss of generality that a = 0. Let µθ := µp(θ ·) and ρθ := ρp(θ ·). Therefore, the +problem we consider in this section is the following: +������� +− d +dx +� +µθ +du+ +θ +dx +� +− ρθ ω2 u+ +θ = 0, +in +R+, +u+ +θ (0) = 1. +(3.1) +Remark 3.1. The function u+ +θ corresponds exactly to the solution u+ of (1.8) that was +introduced in Section 1 for very general media. The reason why this solution is relabeled u+ +θ +is due to the fact that, because we consider here quasiperiodic media, the coefficients µ and ρ +that appear in (1.8) have been replaced by µθ and ρθ. +3.1 +Lifting in a higher-dimensional periodic problem +We wish to exhibit some structure of the solution u+ +θ . As the coefficients µθ and ρθ in (3.1) +are by definition traces of n–dimensional functions along the half-line θ R+, it is natural to +seek u+ +θ as the trace along the same line of a function y ∈ Rn �→ �U+ +θ (y), that is to say: +a. e. x ∈ R, +u+ +θ (x) = �U+ +θ (x θ), +(3.2) +where �U+ +θ shall be characterized as the solution of a n–dimensional PDE (in some sense, an +“augmented” problem in which y is the augmented space variable) with periodic coefficients, +as illustrated in Figure 3. This so-called lifting approach has been used in the homogenization +setting for the analysis of some correctors in presence of periodic halfspaces [14, 15] or periodic +structures separated by an interface [4], as well as for the homogenization of quasicrystals +and Penrose tilings [6, 30]. However, to our knowledge, very little seems to have been done +in other contexts (such as wave propagation), and in particular for numerical analysis and +simulation purposes. +To build a higher-dimensional PDE, one has to exploit the correspondence between the deriva- +tive of u+ +θ and the partial derivatives of �U+ +θ : according to the chain rule, for any smooth +enough function F : Rn → C, one has +∀ x ∈ R, +d +dx[F(θ x)] = (Dθ F)(θ x), +with +Dθ = θ · ∇ = +n +� +i=1 +θi +∂ +∂yi +. +(3.3) +This leads us to introduce the n–dimensional PDE set on a half-space (see Remark 3.2) +−Dθ +�µp Dθ �U+ +θ +� − ρp ω2 �U+ +θ = 0, +for +yn > 0, +(3.4a) +where we recall that the coefficients µp, ρp : Rn → R are continuous and 1–periodic with +respect to each variable. In addition, the boundary condition in (3.1) can be lifted onto the +inhomogeneous Dirichlet boundary condition +�U+ +θ = �ϕ, +on +yn = 0, +(3.4b) +7 + +y1 +y2 +• +θ R+ +θ +0 +− d +dx +� +µθ +du+ +θ +dx +� +− ρθ ω2 u+ +θ = 0 +−Dθ +�µp Dθ �U+ +θ +� − ρp ω2 �U+ +θ = 0 +u+ +θ (0) = 1 +�U+ +θ = �ϕ +Figure 3: Illustration of the lifting approach for n = 2 +where the data �ϕ : Rn−1 → C could be chosen continuous and must satisfy �ϕ(0) = 1, for the +sake of consistency with the fact that u+ +θ (0) = 1. Furthermore, to exploit the periodicity of +the coefficients µp and ρp with respect to the transverse variables yj, j < n, we can impose +the following: +�ϕ is 1–periodic, +(3.5) +so that it is natural to impose that +�U+ +θ (ϕ) is 1–periodic with respect to the transverse variables yj, j < n. +(3.6) +In Section 3.3, we show how to reduce the above to a half-guide problem with periodic +coefficients. In order to do so, we shall need some preliminary materials, which is the object +of the next section. +Remark 3.2. (a). One could have defined the augmented problem (3.4) on other half-spaces +{y ∈ Rn, yi > 0}. The choice of the half-space is purely arbitrary. +(b). At first glance, one could imagine restricting the whole study to a constant boundary +data �ϕ = 1. Though, in practice, this can be the case, the method used to solve the half-guide +problem requires to investigate the structure of �U+ +θ ( �ϕ) for any �ϕ in an appropriate function +space (see Section 4 for more details). +3.2 +Preliminary material +The main objective of this section is to establish rigorously some Green’s formulas that are +formally obvious, such as the one of Proposition 3.10. This requires first to introduce the +adapted functional framework and, since Green’s formulas involve boundary integrals, to +establish relevant trace theorems. Section 3.2.1 is devoted to these trace theorems, while we +present the corresponding Green’s formulas in Section 3.2.2. Finally, Section 3.2.3 highlights +a simple but useful link between the derivative Dθ and a single partial derivative with respect +to one real variable, through a so-called oblique change of variables. +8 + +3.2.1 +Anisotropic Sobolev spaces and trace theorems +For any open set O ⊂ Rn, let us first define the directional Sobolev space +H1 +θ(O) := +�U ∈ L2(O) / Dθ U ∈ L2(O) +�, +(3.7) +which is a Hilbert space, provided with the scalar product +(U, V )H1 +θ(O) := +� +O +� +Dθ U Dθ V + U V +� +. +Let us denote ∥ · ∥H1 +θ(O) the induced norm. We begin with the following density property, +whose proof can be found in [29, Appendix 1]. +Lemma 3.3. The space C ∞ +0 (O) is dense in H1 +θ(O). +We denote the half-space Rn ++ := {y ∈ Rn, yn > 0} and the half-cylinder Ω♯ := (0, 1)n−1 × R+ +in the following. Let us introduce also the sets, for a ∈ {0, 1} and for any integer i ∈ �1, n�, +Σi,a = {y ∈ Rn ++, yi = a} +and +Σ♯ +i,a = {y ∈ Σi,a, yj ∈ (0, 1), j ∈ �1, n − 1�, j ̸= i}. +This definition is illustrated in Figure 4. Note that Σ♯ +n,a is bounded whereas Σ♯ +i,a for i ̸= n is +unbounded in the direction yn. Moreover, +∂Ω♯ = Σ♯ +n,0 ∪ +� n−1 +� +i=1 +�Σ♯ +i,0 ∪ Σ♯ +i,1 +�� +. +A trace operator can be defined from H1 +θ(Rn ++) on Σi,a. The main idea for doing so consists +in using a one-dimensional trace theorem on the θ–oriented line that starts from a point +(z1, . . . , zi−1, a, zi+1, . . . , zn) ∈ Σi,a, to obtain an inequality which will be integrated with +respect to zj, j ̸= i. The 1D trace theorem which will be used is the following. +Proposition 3.4. Let L ∈ R∗ ++ ∪ {+∞}. Then the mapping γL : u �→ u(0) is continuous from +H1(0, L) to C. Moreover, the operator norm of γL is given by +∥γL∥2 = eL + e−L +eL − e−L =: [tanh L]−1 for L > 0, +and +∥γ∞∥2 = 1. +(3.8) +Proof. +The continuity property is a classical result which can be proved by density. +By definition, ∥γL∥ := sup{|u(0)|, ∥u∥H1(0,L) = 1}. This corresponds to a constrained op- +timization problem. Using the standard theory, this leads to introduce a Lagrange multiplier +λ and to find a pair (λ, uL) ∈ C \ {0} × H1(0, L) such that ∥uL∥H1(0,L) = 1 and +∀ v ∈ H1(0, L) +λ uL(0) v(0) = +� L +0 +�duL +dx +dv +dx + uL v +� +dx, +(3.9) +in which case, we have ∥γL∥2 = λ. The explicit solution of this problem leads to the result. +■ +Note that, in particular, ∥γL∥2 ∼ +L→0 L−1. +We are now able to define traces on Σi,a in the following sense. +9 + +(a) n = 2 +Ω♯ +y2 +y1 +Σ1,0 += +Σ♯ +1,0 +Σ1,1 += +Σ♯ +1,1 +Σ2,0 +Σ♯ +2,0 +(b) n = 3 +Σ3,0 +Σ1,0 +Σ2,0 +y1 +y2 +y3 +Ω♯ +Σ♯ +2,0 +Σ♯ +2,1 +Σ♯ +1,0 +Σ♯ +1,1 +Σ♯ +3,0 +y1 +y2 +y3 +Figure 4: Domains Ω♯, Σi,a and Σ♯ +i,a for n = 2 (a) and n = 3 (b). +Proposition 3.5. Fix a ∈ {0, 1} and i ∈ �1, n�. The mapping γi,a : C ∞ +0 (Rn ++) → C ∞ +0 (Σi,a) +defined by γi,aU = U|Σi,a extends by continuity to a linear mapping still denoted γi,a, from +H1 +θ(Rn ++) to L2(Σi,a), and which satisfies the estimate +∀ U ∈ H1 +θ(Rn ++), +∥γi,aU∥2 +L2(Σi,a) ≤ 1 +θi +∥U∥2 +H1 +θ(Rn ++). +(3.10) +Proof. +One can simply prove the continuity estimate (3.10) for any function U ∈ C ∞ +0 (Rn ++) +and conclude using the density result of Proposition 3.3. +(i) Case i ∈ �1, n − 1�: Without loss of generality, we set i = 1. Define +Γ1,a := {z = (z2, . . . , zn), +(a, z) ∈ Σ1,a} ≡ Rn−1 ++ +, +where +(a, z) = (a, z2, . . . , zn). +(3.11) +For U ∈ C ∞ +0 (Rn ++) and given any z = (z2, . . . , zn) ∈ Γ1,a, consider the function +∀ x > 0, +uz,θ(x) = U(x θ + (a, z)). +(3.12) +As uz,θ belongs to H1(R∗ ++), Lemma 3.4 for L = +∞ combined with an integration with +respect to z ∈ Γ1,a leads to +� +Γ1,a +|uz,θ(0)|2 dz ≤ +� +Γ1,a +∥uz,θ∥2 +H1(R∗ ++)dz. +(3.13) +On the other hand, let us introduce the transformation +T : y �→ +�(y1 − a)/θ1, y2 − (y1 − a) θ2/θ1, · · · , yn − (y1 − a) θn/θ1 +�, +(3.14) +which defines a C 1–diffeomorphism with a Jacobian determinant det JT = 1/θ1 ̸= 0. Since +the inverse image {T−1(x, z), z ∈ Γ1,a, x > 0} is nothing but the polyhedron +Q1,a := {y ∈ Rn ++, y1 > a, yn > (y1 − a) θn/θ1} ⊂ Rn ++, +10 + +it follows from the chain rule and from the change of variables y �→ T y that +duz,θ +dx (x) = Dθ U(x θ + (a, z)) +and +� +Γ1,a +∥uz,θ∥2 +H1(R∗ ++) dz = 1 +θ1 +∥U∥2 +H1 +θ(Q1,a). +(3.15) +Finally, since uz,θ(0) = U(a, z2, · · · , zn), Equations (3.13) and (3.15) imply +∥U∥2 +L2(Σ1,a) ≤ 1 +θ1 +∥U∥2 +H1 +θ(Q1,a) ≤ 1 +θ1 +∥U∥2 +H1 +θ(Rn ++), +(3.16) +which is exactly the desired estimate. +(ii) Case i = n: starting from the function uz,θ(x) := U(x θ+(z, a)) defined for x > 0 and for +any z = (z1, . . . , zn−1) with (z, a) ∈ Σn,a, the proof uses the exact same arguments as above, +except the inverse image under T becomes the whole half-space Qn,a := {y ∈ Rn ++, yn > a}. +■ +The previous result does not hold in general for functions which are only H1 +θ in sub-domains +of the half-space Rn ++. In particular when it comes to the half-cylinder Ω♯, one is led to apply +the one-dimensional trace theorem on segments that become smaller in the neighbourhood of +the “corners”, i.e. the intersections of two faces. To overcome this difficulty, let us consider +the sets (see Figure 5) +∀ 0 < b < 1/2, +Σ♯,b +i,a = {y ∈ Σ♯ +i,a, +dist(y, ∂Σ♯ +i,a) := +inf +z ∈ ∂Σ♯ +i,a +|y − z| > b}. +(3.17) +Using these domains, the traces on Σ♯ +i,a can be defined as locally integrable functions in the +sense of the following proposition. +Σ♯,b +2,1 +Σ♯,b +1,0 +Σ♯,b +3,0 +y1 +y2 +y3 +b +Ω♯ +Tn +y1 +y2 +y3 +a +Ω♯ +a,− +Ω♯ +θ +y1 +y2 +y3 +Figure 5: From left to right: Σ♯,b +i,a (3.17), Tn (3.38), Ω♯ +a,− (3.37), and Ω♯ +θ (3.41) represented +for n = 3. +Proposition 3.6. Let a ∈ {0, 1} and i ∈ �1, n�. The mapping γ♯ +i,a : C ∞ +0 (Ω♯) → C ∞ +0 (Σ♯ +i,a) +defined by γ♯ +i,aU = U|Σ♯ +i,a extends by continuity to a linear mapping still denoted γ♯ +i,a, from +H1 +θ(Ω♯) to L2 +loc(Σ♯ +i,a), and which satisfies the estimate +∀ 0 < b < 1/2, +∃ Cb > 0, +∀ U ∈ H1 +θ(Ω♯), +∥γ♯ +i,aU∥2 +L2(Σ♯,b +i,a) ≤ Cb +θi +∥U∥2 +H1 +θ(Ω♯). +(3.18) +11 + +Proof. +Using the density result stated in Proposition 3.3, one only has to show (3.18) for +U ∈ C ∞ +0 (Ω♯). Let us assume that i = 1 and a = 0, the arguments in the following extending +without any difficulty to i ∈ �1, n� and a ∈ {0, 1}. Define +Γ♯ +1,0 := {z = (z2, . . . , zn), +(0, z) ∈ Σ♯ +1,0} ≡ (0, 1)n−1 × R+. +(3.19) +We introduce the length function defined by +∀ z ∈ Γ♯ +1,0, +λ1,0(z) := +��{θ R+(0, z)}∩Ω♯�� = sup{x > 0, x θ1 ≤ 1, x θi+zi ≤ 1 ∀ i ∈ �2, n−1�}. +We deduce easily that +λ1,0(z) = min +� 1 +θ1 +; +min +2≤j≤n−1 +�1 − zj +θj +�� +. +(3.20) +For U ∈ C ∞ +0 (Ω♯) and z ∈ Γ♯ +1,0, we define +∀ 0 < x < λ1,0(z), +uz,θ(x) = U(x θ + (0, z)). +(3.21) +Since uz,θ ∈ H1�0, λ1,0(z) +�, Lemma 3.4 and an integration with respect to z give +� +Γ♯ +1,0 +w1,0(z) |uz,θ(0)|2 dz ≤ +� +Γ♯ +1,0 +∥uz,θ∥2 +H1(0,γi,a(z)) dz, +with w1,0(z) = tanh[λ1,0(z)]. (3.22) +On the other hand, consider the C 1–diffeomorphism T given by (3.14). The set Q♯ +1,0 := +{T−1(x, z), +0 < x < λ1,0(z), z ∈ Γ♯ +1,0} is clearly included in Ω♯. Thus, by analogy with +(3.16) in the proof of Proposition 3.5, we have from (3.21), the chain rule, and the change of +variables y �→ T y that +� +Γ♯ +1,0 +w1,0(z) |U(0, z)|2 dz ≤ 1 +θ1 +∥U∥2 +H1 +θ(Ω♯). +(3.23) +More generally, we can show that γ♯ +i,a can be defined from H1 +θ(Ω♯) to the weighted space +L2(Σ♯ +i,a, wi,a dz), where the weight wi,a is given in (3.22) for i = 1 and a = 0. Now, the +expression (3.20) of λ1,0 implies that w1,0 degenerates at the neighbourhood of the corners +zj = 1. However, the weight w1,0 is bounded from below on Σ♯,b +1,0 with +inf +(0,z)∈Σ♯,b +1,0 +w1,0(z) = tanh +� +min +� 1 +θ1 +; b +min +2≤j≤n−1 +1 +θj +�� +> 0. +(3.24) +If we set Cb := [inf(0,z)∈Σ♯,b +1,0 w1,0(z)]−1 > 0, then (3.18) follows directly from (3.23) by inte- +grating with respect to {z, (0, z) ∈ Σ♯,b +1,0}, instead of Γ♯ +1,0. +■ +Remark 3.7. The best constant in the previous proposition necessarily blows up when b tends +to 0. The above proof shows that traces could be defined on the whole faces in appropriate +weighted L2-spaces. More details about traces in anisotropic spaces can be found in [18]. +12 + +3.2.2 +Green’s formulas +Let us now introduce the set H1 +θ,loc(Rn ++) of functions which are H1 +θ in any half-cylinder S ×R+ +where S is a bounded open set in Rn−1. More rigorously, we define for any ϕ ∈ C ∞ +0 (Rn−1) +the n–dimensional function ˇϕ ∈ C ∞(Rn) such that +ˇϕ(y1, . . . , yn−1, yn) = ϕ(y1, . . . , yn−1). +(3.25) +Note that for any U ∈ L2 +loc(Rn ++), the support of ˇϕ U is bounded in the directions yj, j ̸= n. +Starting from this remark, we define +H1 +θ,loc(Rn ++) := +� +U ∈ L2 +loc(Rn ++), +ˇϕ U ∈ H1 +θ(R+ +n ) ∀ϕ ∈ C ∞ +0 (Rn−1) +� +. +(3.26) +Let us introduce a 1D cut-off function χ ∈ C ∞ +0 (R) such that χ = 1 on (0, 1), from which we +define ˇχ♯ ∈ C ∞ +0 (Rn) as +ˇχ♯(y1, . . . , yn−1, yn) = χ(y1) . . . χ(yn−1). +(3.27) +We deduce in particular that +∀ U ∈ H1 +θ,loc(Rn ++), +U|Ω♯ = (ˇχ♯ U)|Ω♯ ∈ H1 +θ(Ω♯). +(3.28) +Moreover, by Proposition 3.5, it is obvious that we can define without any ambiguity the +trace map γ♯ +i,a to H1 +θ,loc(Rn ++) as follows +∀ U ∈ H1 +θ,loc(Rn ++), +γ♯ +i,aU := γi,a(ˇχ♯U)|Σ♯ +i,a ∈ L2(Σ♯ +i,a). +(3.29) +For simplicity, when considering traces on Σ♯ +i,a, we shall write U instead of γ♯ +i,aU. We can +now state the following Green’s formula. +Proposition 3.8. For any U, V ∈ H1 +θ,loc(Rn ++), we have the Green’s formula +� +Ω♯ +� +Dθ U V + U Dθ V +� +dy = 1 +θn +� +Σ♯ +n,0 +U V ds+ +n−1 +� +i=1 +1 +θi +� � +Σ♯ +i,1 +U V ds− +� +Σ♯ +i,0 +U V ds +� +. (3.30) +Proof. +Let U, V ∈ H1 +θ,loc(Rn ++). By definition, for any χ ∈ C ∞ +0 (R) such that χ = 1 on (0, 1), +the functions ˇχ♯ U and ˇχ♯ V belong to H1 +θ(Rn ++), where ˇχ♯ is defined in (3.27). Since Proposition +3.3 ensures that C ∞ +0 (Rn ++) is dense in H1 +θ(Rn ++), there exist two sequences (Uk)k∈N, (Vk)k∈N of +functions in C ∞ +0 (Rn ++), such that +Uk → ˇχ♯ U +and +Vk → ˇχ♯ V +in +H1 +θ(Rn ++), +k → +∞. +It follows from Green’s formula for smooth functions that Uk and Vk satisfy (3.30) for any +k ∈ N. Passing to the limit and using the trace continuity result stated in Propsition 3.5 +imply that (3.30) is satisfied by ˇχ♯ U and ˇχ♯ V , i.e. by U and V , since ˇχ♯ = 1 in Ω♯. +■ +We next focus on functions which are periodic with respect to their (n − 1) first variables. +More precisely, for any U ∈ L2(Ω♯) and any ϕ ∈ L2(Σ♯ +n,0), we introduce the respective +periodic extensions �U ∈ L2 +loc(Rn ++) and �ϕ ∈ L2 +loc(Σn,0) as defined for any i ∈ �1, n − 1� by +� +� +� +a. e. y ∈ Rn ++, +�U(y + ⃗ei) = �U(y) +and +�U|Ω♯ = U. +a. e. s ∈ Σn,0, +�ϕ(s + ⃗ei) = �ϕ(s) +and +�ϕ|Σ♯ +n,0 = ϕ. +(3.31) +13 + +An appropriate functional framework is provided by the space +H1 +θ,per(Ω♯) = +� +U ∈ L2(Ω♯), �U ∈ H1 +θ,loc(Rn ++) +� +⊂ H1 +θ(Ω♯), +(3.32) +where the inclusion follows from (3.28) and (3.31). If C ∞ +per(Ω♯) denotes the set of smooth +functions in C ∞(Ω♯) which are 1–periodic with respect to their first n − 1 variables, that is, +C ∞ +per(Ω♯) = +� +V ∈ C ∞(Ω♯), +�V ∈ C ∞(Rn ++) +� +, +(3.33) +then one can show the following result by adapting classical properties of H1 functions. +Lemma 3.9. The space C ∞ +per(Ω♯) is dense in H1 +θ,per(Ω♯). +Note that the traces of functions in H1 +θ,per(Ω♯) on Σ♯ +i,a are well-defined in L2 by (3.29). +Moreover, using the continuity estimate (3.10) we have +γ♯ +i,a ∈ L(H1 +θ,per(Ω♯), L2(Σ♯ +i,a)). +(3.34) +One has the characterization +H1 +θ,per(Ω♯) = +� +U ∈ H1 +θ(Ω♯), +γ♯ +i,0U = γ♯ +i,1U ∀ i ∈ �1, n − 1� +� +, +(3.35) +where the traces of functions in H1 +θ(Ω♯) are defined in Proposition 3.6 and the equality of +traces has to be understood up to the identification of functions on Σ♯ +i,0 and Σ♯ +i,1. It is clear +from (3.35) that H1 +θ,per(Ω♯) is a closed subspace of H1 +θ(Ω♯), thus it is an Hilbert space when +equipped with the norm of H1 +θ(Ω♯). From Proposition 3.8 and (3.35), we deduce the Green’s +formula on H1 +θ,per(Ω♯). +Proposition 3.10. For any U, V ∈ H1 +θ,per(Ω♯), we have the Green’s formula +� +Ω♯ +� +Dθ U V + U Dθ V +� +dy = 1 +θn +� +Σ♯ +n,0 +U V ds. +(3.36) +From the Green’s formula (3.36), we can easily deduce the following result. +Corollary 3.11. Let a > 0, and define the sets with common boundary Σ♯ +n,a (see Figure 5): +Ω♯ +a,+ := Ω♯ ∩ {yn > a} +and +Ω♯ +a,− := Ω♯ ∩ {yn < a}. +(3.37) +Consider a function U ∈ L2(Ω♯) such that U± := U|Ω♯ +a,± ∈ H1 +θ,per(Ω♯ +a,±), where H1 +θ,per(Ω♯ +a,±) +is defined as in (3.35). Then +U ∈ H1 +θ,per(Ω♯) +⇐⇒ +γ♯ +n,aU+ = γ♯ +n,aU−. +We finish this section with a more technical Green’s formula, used in the proof of Proposition +3.17, involving functions U that only belong to H1 +θ(Ω♯), provided that the test function V +vanishes in the neighborhood of the skeleton Tn defined by +T2 = Σ♯ +2,0 +and +Tn = Σ♯ +n,0 ∪ +� n−1 +� +j=1 +�∂Σ♯ +j,0 ∪ ∂Σ♯ +j,1 +�� +for n ≥ 3. +(3.38) +This domain is represented in Figure 5 for n = 3. +14 + +Proposition 3.12. For U ∈ H1 +θ(Ω♯) and V ∈ C ∞ +0 (Ω♯ \ Tn), the Green’s formula (3.30) still +holds. +Proof. +Consider U ∈ H1 +θ(Ω♯) and V ∈ C ∞ +0 (Ω♯ \ Tn). Since by Proposition 3.3, C ∞ +0 (Ω♯) is +dense in H1 +θ(Ω♯), there exists a sequence (Uk)k∈N of functions in C ∞ +0 (Ω♯) which tends to U. +It follows from Green’s formula in Ω♯ for smooth functions that Uk and V satisfy (3.30) for +any k ∈ N. For 0 < b < 1/2, let Ω♯,b be the domain +Ω♯,b = {y ∈ Ω♯, +dist(y, Tn) := inf +z ∈ Tn |y − z| > b}. +(3.39) +Since V ∈ C ∞ +0 (Ω♯ \ Tn), there exists a real number 0 < b < 1/2 such that V |Ω♯,b ∈ C ∞ +0 (Ω♯,b). +Consequently, for any i ∈ �1, n − 1�, the surface integral on Σ♯ +i,a is reduced to the set Σ♯,b +i,a +defined by (3.17). When k tends to +∞, we can then use the trace continuity result stated +in Proposition 3.6 on Σ♯,b +i,a, to deduce that (3.30) is satisfied by U and V . +■ +3.2.3 +An oblique change of variables +Before stating Proposition 3.14 which is the main result of this section, let us introduce the +change of variables in Rn ++: +(s, x) ∈ Rn ++ �→ y = (s, 0) + x θ ∈ Rn ++, +(3.40) +and denote by Ω♯ +θ the image of Ω♯ by the above transformation: +Ω♯ +θ := {(s, 0) + x θ, +s ∈ (0, 1)n−1, x > 0}. +(3.41) +This is illustrated in Figure 5 for n = 3 and in Figure 6 for n = 2 and |θ| = 1. The following +simple lemma will be used in the sequel. +Lemma 3.13. For any V ∈ L1(Ω♯), we have +� +Ω♯ +θ +�V (y) dy = +� +Ω♯ +�V (y) dy, +(3.42) +where �V ∈ L1 +loc(Rn ++) denotes the periodic extension of V , defined by (3.31). +Proof. +We will use the notation k = (k1, . . . , kd) ∈ Zd for a vector of integers. For any +set O ⊂ Rn, let 1O be the indicator function of O, that is, the function which equals 1 in +O and 0 elsewhere. By density, it suffices to prove (3.42) for V ∈ C ∞ +0 (Ω♯). By additivity of +integration, +� +Ω♯ +θ +�V (y) dy = +� +Rn ++ +1Ω♯ +θ(y) �V (y) dy = +� +k∈Zn−1 +� +Ω♯+(k,0) +1Ω♯ +θ(y) �V (y) dy, +where the sum over k ∈ Zn−1 is finite because of 1Ω♯ +θ and because V is compactly supported. +We then use the change of variables z �→ z + (k, 0) which leads to +� +Ω♯ +θ +�V (y) dy = +� +k∈Zn−1 +� +Ω♯ 1Ω♯ +θ(z + (k, 0)) �V (z) dz +because �V is periodic += +� +Ω♯ +� +� +k∈Zn−1 +1Ω♯ +θ−(k,0)(z) +� +�V (z) dz +by linearity. +(3.43) +15 + +Furthermore, by noticing that the collection of sets {Ω♯ +θ −(k, 0), k ∈ Zn−1} forms a partition +of Rn ++, it follows that +∀ z ∈ Ω♯, +� +k∈Zn−1 +1Ω♯ +θ−(k,0)(z) = 1Rn ++(z) = 1. +(3.44) +Combining (3.43) and (3.44) implies that (3.42) is satisfied for V ∈ C ∞ +0 (Ω♯). +■ +The inversion of the change of variables (3.40) leads us to introduce: +∀ y ∈ Rn, +sθ(y) := ˆy − (yn/θn) ˆθ ∈ Rn−1, +(3.45) +so that, +y = (s, 0) + x θ +⇐⇒ +s = sθ(y) +and +x = yn/θn. +(3.46) +The next proposition emphasizes the fact that through the change of variables (3.40), the +differential operator Dθ simply becomes the partial derivative with respect to yn (which is +obvious for smooth functions). +Proposition 3.14. Let Ψ ∈ L2(Ω♯). Then the periodic function Ψθ defined as +a. e. y ∈ Rn ++, +�Ψθ(y) := �Ψ(sθ(y), yn/θn), +(3.47) +(where �Ψ is the periodic extension of Ψ) belongs to L2(Ω♯) and +∥Ψθ∥L2(Ω♯) = +� +θn ∥Ψ∥L2(Ω♯). +(3.48) +Moreover, if ∂ynΨ ∈ L2(Ω♯), then Ψθ belongs to H1 +θ,per(Ω♯) with directional derivative +a. e. y ∈ Rn ++, +Dθ �Ψθ(y) = ∂ �Ψ +∂yn +(sθ(y), yn/θn). +(3.49) +Proof. +The map (s, x) �→ (s, 0) + x θ from Σ♯ +n,0 × R+ to Ω♯ +θ defines a C 1–diffeomorphism +with a non-vanishing Jacobian θn ̸= 0. Therefore, by using the definition (3.41) of Ω♯ +θ, a +change of variables as well as the property sθ((s, 0) + x θ) = s, we obtain that +� +Ω♯ +θ +|�Ψθ(y)|2 dy = θn +� +Σ♯ +n,0 +� +∞ +0 +|�Ψθ((s, 0) + x θ)|2 dx ds = θn +� +Σ♯ +n,0 +� +∞ +0 +|�Ψ(s, x)|2 dx ds. +We deduce from Lemma 3.13 that Ψθ ∈ L2(Ω♯), and that (3.48) holds. +Now in order to derive the expression of Dθ �Ψθ in the sense of distributions, consider a test +function Φ ∈ C ∞ +0 (Rn ++). The change of variables (s, x) �→ (s, 0) + x θ combined with Fubini’s +theorem for integrable functions leads to +� +Rn ++ +�Ψθ(y) DθΦ(y) dy = θn +� +Rn−1 +� +∞ +0 +�Ψ(s, x) DθΦ((s, 0) + x θ) dxds. +(3.50) +Furthermore the 1D function φs,θ defined by φs,θ(x) := Φ((s, 0) + x θ) belongs to C ∞ +0 (R+) +and we have [dφs,θ/dx](x) = DθΦ((s, 0) + x θ) from the chain rule. Since ∂ynΨ is in L2, we +16 + +can integrate by parts the inner integral in (3.50) to obtain +� +Rn ++ +�Ψθ(y) DθΦ(y) dy = −θn +� +Rn−1 +� +∞ +0 +∂Ψ +∂yn +(s, x) φs,θ(x) dxds += − +� +Rn ++ +∂Ψ +∂yn +(sθ(y), yn/θn) Φ(y) dy, +(3.51) +where the last equality comes from the change of variables y �→ (sθ(y), yn/θn). This gives +the expression of Dθ �Ψθ in (3.49). +■ +Remark 3.15. It will be often useful to use (3.49) in the form +a. e. (s, x) ∈ Rn ++, +Dθ �Ψθ((s, 0) + x θ) = ∂ �Ψ +∂yn +(s, x). +(3.52) +The previous proposition allows in particular to deduce the surjectivity of the trace operator +from H1 +θ,per(Ω♯) to L2(Σ♯ +n,0). +Corollary 3.16. Let ϕ ∈ L2(Σ♯ +n,0), and ψ ∈ H1(R+) such that ψ(0) = 1. Then the periodic +function defined by +a. e. y ∈ Rn ++, +Rϕ (y) := �ϕ(sθ(y)) ψ(yn/θn) +(3.53) +belongs to H1 +θ,per(Ω♯), and its trace is Rϕ|Σ♯ +n,0 = ϕ. Moreover, R defines a continuous map +from L2(Σ♯ +n,0) to H1 +θ,per(Ω♯). +3.3 +Link with a periodic half-guide problem +For any boundary data ϕ ∈ L2(Σ♯ +n,0), we can now introduce U+ +θ as the solution in H1 +θ(Ω♯) of +the half-guide problem +������������� +−Dθ +�µp Dθ U+ +θ +� − ρp ω2 U+ +θ = 0, +in +Ω♯, +U+ +θ |Σ♯ +n,0 = ϕ, +U+ +θ |Σ♯ +i,0 = U+ +θ |Σ♯ +i,1 +∀ i ∈ �1, n − 1�, +µp Dθ U+ +θ |Σ♯ +i,0 = µp Dθ U+ +θ |Σ♯ +i,1 +∀ i ∈ �1, n − 1�. +(3.54) +Note that the third equation above implies that U+ +θ ∈ H1 +θ,per(Ω♯), the first one implies that +µp Dθ U+ +θ ∈ H1 +θ(Ω♯), and finally the fourth one implies that µp Dθ U+ +θ ∈ H1 +θ,per(Ω♯). +The +space of the boundary data can seem surprising compared to the Helmholtz equation with +an elliptic principal part, but recall from Corollary 3.16 that the trace mapping on Σ♯ +n,0 is +surjective from H1 +θ,per(Ω♯) to L2(Σ♯ +n,0). +With the functional framework introduced in the previous section, we can now show that +Problem (3.54) is well-posed. +17 + +Proposition 3.17. For any ϕ ∈ L2(Σ♯ +n,0), Problem (3.54) is equivalent to the variational +formulation +������� +Find U+ +θ ∈ H1 +θ,per(Ω♯) such that U+ +θ |Σ♯ +n,0 = ϕ and +∀ V ∈ H1 +θ,per(Ω♯) such that V |Σ♯ +n,0 = 0, +� +Ω♯ +� +µp Dθ U+ +θ Dθ V − ρp ω2 U+ +θ V +� += 0, +(3.55) +for which Lax-Milgram’s theorem applies. +Proof. +The variational formulation is obtained by multiplying the first equation of (3.54) by +V ∈ H1 +θ,per(Ω♯), and by using Green’s formula (3.36). The application of the Lax-Milgram’s +theorem in {V ∈ H1 +θ,per(Ω♯), γn,0V = 0}, thanks to Corollary 3.16, is direct. +For the equivalence, as usual, one picks test functions V ∈ C ∞ +0 (Ω♯) to deduce that the +solution U+ +θ ∈ H1 +θ,per(Ω♯) of (3.55) satisfies the first equation of (3.54). This implies that +µp Dθ U+ +θ ∈ H1 +θ(Ω♯). The real difficulty is to show that U+ +θ satisfies the fourth equation in +(3.54) or equivalently that µp Dθ U+ +θ ∈ H1 +θ,per(Ω♯). According to Proposition 3.6, we have +∀ 1 ≤ i ≤ n − 1, +µp Dθ U+ +θ |Σ♯ +i,a ∈ L2 +loc(Σ♯ +i,a). +Therefore, Proposition 3.12 allows us to use Green’s formula (3.30) for U = µp Dθ U+ +θ and +for V ∈ C ∞ +0 (Ω♯ \ Tn) ∩ H1 +θ,per(Ω♯), where Tn is the skeleton defined in (3.38). By combining +this with the fact that U+ +θ solves (3.55) and the first equation of (3.54), one obtains that for +any integer i ∈ �1, n − 1�, +∀ V ∈ C ∞ +0 (Ω♯ \ Tn) ∩ H1 +θ,per(Ω♯), +� � +Σ♯ +i,1 +µp Dθ U+ +θ V ds − +� +Σ♯ +i,0 +µp Dθ U+ +θ V ds +� += 0. +Furthermore, C ∞ +0 (Σ♯ +i,0) is included in {V |Σ♯ +i,0, V ∈ C ∞ +0 (Ω♯ \ Tn) ∩ H1 +θ,per(Ω♯)}. +In fact, +any ψ ∈ C ∞ +0 (Σ♯ +i,0) admits the extension Ψ : y ∈ Ω♯ �→ ψ(y1, . . . , yi−1, yi+1, . . . , yn), which +belongs to C ∞ +0 (Ω♯ \ Tn) ∩ H1 +θ,per(Ω♯). Finally, since C ∞ +0 (Σ♯ +i,0) is dense in L2(Σ♯ +i,0), it is easy +to show that the fourth equation of (3.54) holds and that µp Dθ U+ +θ |Σ♯ +i,1 ∈ L2(Σ♯ +i,1) for any +i ∈ �1, n − 1�. +■ +We now make the link between U+ +θ (ϕ) and the solution of the half-line problem (3.1) that +fully justifies the introduction of the half-guide problem (3.54). +To do so, first, let us introduce the quasiperiodic coefficients defined for any s ∈ Rn−1 by +∀ x ∈ R, +µs,θ(x) := µp +�(s, 0) + x θ +� +and +ρs,θ(x) := ρp +�(s, 0) + x θ +�, +(3.56) +as well as the one-dimensional problems +������� +− d +dx +� +µs,θ +du+ +s,θ +dx +� +− ρs,θ ω2 u+ +s,θ = 0, +in +R+, +u+ +s,θ(0) = 1. +(3.57) +Note that (3.1) corresponds to (3.57) taken with s = 0. +18 + +y1 +y2 +•s +•y +• +x +θ +Ω♯ +Ω♯ +θ +0 +Σ♯ +n,0 +Σ♯ +n,1 +C♯ +0 +Σ♯ +n,2 +C♯ +1 +Σ♯ +n,3 +C♯ +2 +Figure 6: The half-cylinders Ω♯ and Ω♯ +θ (left), and the domains C♯ +ℓ and Σ♯ +n,k (right) for n = 2 +Under the assumptions (1.2) and (1.4), Problem (3.57) admits a unique solution u+ +s,θ in +H1(R+) for any s ∈ Rn−1. Moreover, u+ +s,θ decays exponentially at infinity, uniformly with +respect to s, that is, there exist constants α, c > 0 depending only on µ±, ρ± such that +∀ s ∈ Rn−1, +��e−α Im ω x u+ +s,θ +�� +H1(R+) ≤ c. +(3.58) +Furthermore, thanks to the continuity of µp and ρp, we can show that u+ +s,θ is continuous with +respect to s, as stated in the next proposition. +Proposition 3.18. The mapping s ∈ Rn−1 �→ u+ +s,θ, which associates with a real vector s the +solution in H1(R+) of the problem (3.57), defines a uniformly continuous function which is +periodic of period 1 in each direction. +Proof. +To show that s �→ u+ +s,θ is 1–periodic in each direction, one simply has to note that +since µs,θ and ρs,θ are 1–periodic with respect to each si, both u+ +s,θ and u+ +s+⃗ei,θ satisfy the +same half-line problem (3.57). Thus, by well-posedness of (3.57), u+ +s,θ = u+ +s+⃗ei,θ. +Now let us prove the regularity of s �→ u+ +s,θ. For any s1, s2 ∈ Rn−1, by writing the variational +formulations satisfied by u+ +s1,θ and u+ +s2,θ, and by substracting one from the other, we obtain +∀ v ∈ H1 +0(R+), +� +R+ +� +µs1,θ +d +dx(u+ +s1,θ − u+ +s2,θ) dv +dx − ρs1,θ ω2 (u+ +s1,θ − u+ +s2,θ) v +� += +� +R+ +� +(µs2,θ − µs1,θ) +du+ +s2,θ +dx +dv +dx − (ρs1,θ − ρs2,θ) ω2 u+ +s2,θ +� +. +Now choose v = u+ +s1,θ −u+ +s2,θ ∈ H1 +0(R+) in the above equality. The well-posedness of (3.57), a +Cauchy-Schwarz inequality applied to the right-hand side and (3.58) imply that there exists +a real number c > 0 independent of s and θ such that +��u+ +s1,θ − u+ +s2,θ +�� +H1(R+) ≤ c +� +∥µs2,θ − µs1,θ∥∞ + ∥ρs2,θ − ρs1,θ∥∞ +� +. +(3.59) +19 + +The functions µp and ρp are continuous and 1–periodic in each direction: from Heine-Cantor +theorem, they are uniformly continuous. Let us define the modulus of uniform continuity +∀ µ ∈ C 0(Rn), ∀ ε > 0, +δ(µ, ε) = sup +y,z {|µ(y) − µ(z)|, |y − z| < ε} +A function µ is uniformly continuous if δ(µ, ε) tends to 0 as ε tends to 0. It follows from +(3.59) that +��u+ +s1,θ − u+ +s2,θ +�� +H1(R+) ≤ c +� +δ(µp, |s1 − s2|) + δ(ρp, |s1 − s2|) +� +. +Therefore, s �→ u+ +s,θ is continuous from Rn−1 in H1(R+). +■ +Proposition 3.19. Let sθ be the mapping defined by (3.45), and �U+ +θ (resp. �ϕ) be the periodic +extension of U+ +θ (resp. ϕ) the solution of (3.54). Then, we have +a. e. y ∈ Rn ++, +�U+ +θ ( �ϕ)(y) = �ϕ +�sθ(y) +� u+ +sθ(y),θ(yn/θn), +(3.60) +or equivalently +a. e. (s, x) ∈ Rn−1 × R+, +�U+ +θ ( �ϕ)((s, 0) + θ x) = �ϕ(s) u+ +s,θ(x). +(3.61) +Moreover if �ϕ is continuous in the neighbourhood of 0 and satisfies �ϕ(0) = 1, then +a. e. x ∈ R, +u+ +θ (x) = �U+ +θ ( �ϕ)(x θ) +(3.62) +Proof. +We begin by proving (3.60). Let us denote for a. e. y ∈ Rn ++, U1(y) the right-hand +side of (3.60). Note that Ψ : (s, x) �→ �ϕ(s) u+ +s,θ(x) is 1–periodic with respect to s (thanks to +Proposition 3.18), and belongs to L2(Ω♯) since +∥Ψ∥2 +L2(Ω♯) = +� +Σ♯ +n,0 +|ϕ(s)|2 ∥u+ +s,θ∥2 +L2(R+) ds ≤ θn c2 ∥ϕ∥2 +L2(Σ♯ +n,0), +with c = sup +s ∥u+ +s,θ∥L2(R+). +Moreover, since for all s, u+ +s,θ ∈ H1(R+), ∂ynΨ is also in L2(Ω♯) (using similar inequalities to +the above). By Proposition 3.14, U1 belongs to H1 +θ,per(Ω♯) with +a. e. y ∈ Rn ++, +Dθ � +U1(y) = �ϕ +�sθ(y) +� du+ +sθ(y),θ +dx +(yn/θn). +Finally, since u+ +s,θ(0) = 1, it is clear that U1|Σ♯ +n,0 = ϕ. By repeating the same argument, we +can show that µpDθ U1 belongs to H1 +θ,per(Ω♯) with +a. e. y ∈ Rn ++, +Dθ [µp Dθ � +U1](y) = �ϕ +�sθ(y) +� d +dx +� +µsθ(y),θ +du+ +sθ(y),θ +dx +� +(yn/θn). +Since u+ +s,θ satisfies (3.57), it is clear that U1 satisfies (3.54). By well-posedness of (3.54), we +have U1 = U+ +θ . +The equivalence between (3.60) and (3.61) is directly obtained using the change of variables +(s, x) �→ ((s, 0) + θ x). Moreover, we have from Proposition 3.18 that s �→ u+ +s,θ is continuous. +If in addition to that, �ϕ is continuous in a neighbourhood of 0, then (3.61) becomes true for +any s in that neighbourhood. In particular, (3.61) can be written for s = 0, thus leading to +(3.62). +■ +20 + +In particular, we deduce from the above proprosition that +a. e. y ∈ Rn ++, +Dθ �U+ +θ ( �ϕ)(y) = �ϕ +�sθ(y) +� du+ +sθ(y),θ +dx +(yn/θn). +(3.63) +Remark 3.20. The half-guide solution U+ +θ +depends on ϕ whereas u+ +s,θ does not. +In this +sense, the relation (3.60) can seem surprising at first sight. Numerical results presented in +Section 5.5 will illustrate this property. +4 +Resolution of the half-guide problem +The advantage of the lifting process lies in the periodic nature of (3.54), which allows us to +exploit tools that are well-suited for periodic waveguides. In this paper, we use a DtN-based +method [10, 19], developed for the elliptic1 Helmholtz equation −∇ · (µp ∇U) − ρp ω2 U = 0 +in unbounded periodic guides. This method does not rely on decay properties, and therefore +remains robust when the absorption tends to 0. As we essentially transpose this method to +our directional Helmholtz equation, we will see below that the framework remains exactly +the same, although the analysis has to be adapted. Let us mention the recursive doubling +method [32, 8], suited for bounded periodic waveguides, and a method [33] based on the +Floquet-Bloch transform, although its extension to our non-elliptic equation seems unclear. +In what follows, C♯ +ℓ is the cell defined for every ℓ ∈ N by +C♯ +0 = (0, 1)n +and +C♯ +ℓ = C♯ +0 + ℓ⃗en, +so that +Ω♯ = +� +ℓ∈N +C♯ +ℓ. +(4.1) +For ℓ > 0, we call Σ♯ +n,ℓ the interface between the cells C♯ +ℓ and C♯ +ℓ+1, that is, Σ♯ +n,ℓ = Σ♯ +n,0 + ℓ⃗en. +By periodicity, each cell C♯ +ℓ can be identified to C♯ +0. Similarly, each interface Σ♯ +n,ℓ can be +identified to Σ♯ +n,0. The cells and interfaces are represented in Figure 6. +4.1 +Structure of the solution +The solution U+ +θ (ϕ) of (3.54) has a particular structure that we explain in this section. +Denote by P ∈ L +�L2(Σ♯ +n,0) +� the operator +∀ ϕ ∈ L2(Σ♯ +n,0), +Pϕ := U+ +θ (ϕ)|Σ♯ +n,1, +(4.2) +where L2(Σ♯ +n,1) and L2(Σ♯ +n,0) have been identified to each other in an obvious manner. This +identification will be used systematically in what follows, even if not mentioned. Note that +the operator P is well-defined, due to the continuity of the trace operator on Σ♯ +i,a (3.34). +Proposition 4.1. For any ϕ in L2(Σ♯ +n,0), we have +∀ ℓ ∈ N, a. e. y ∈ Ω♯, +U+ +θ (ϕ)(y + ℓ⃗en) = U+ +θ (Pℓϕ)(y). +(4.3) +Moreover, the spectral radius of P is strictly less than one. +1By elliptic Helmholtz equation, we refer to the Helmholtz equation with an elliptic principal part. +21 + +Proof. +We only present the outline of the proof, which is quite similar to the one in [10, +19]. Given ϕ ∈ L2(Σ♯ +n,0), consider the function U1 defined in Ω♯ by U1(y) = U+ +θ (ϕ)(y + ⃗en) +for almost any y ∈ Ω♯. Since the coefficients µp and ρp are periodic, one deduces that U1 +satisfies the volume equation as well as the periodicity condition in (3.54). Furthermore, +U1|Σ♯ +n,0 = U+ +θ (ϕ)|Σ♯ +n,1 = Pϕ. +Thus, by well-posedness of (3.54), we have (4.3) for ℓ = 1. The result (4.3) for ℓ ≥ 2 is proved +by induction. +It remains to show that the spectral radius is strictly less than 1. To this end, by analogy +with (3.58), one can show the existence of constants α, c > 0 such that +∀ ϕ ∈ L2(Σ♯ +n,0), +��eα Im ω yn/θn U+ +θ +�� +H1 +θ(Ω♯) ≤ c ∥ϕ∥L2(Σ♯ +n,0). +(4.4) +Since Pℓϕ = U+ +θ (ϕ)(·, ℓ), the estimate above implies that ∥Pℓ∥ ≤ c e−α Im ω ℓ/θn. Hence, using +Gelfand’s formula [26, §10.3], the spectral radius can be estimated as follows: +ρ(P) = +lim +ℓ→+∞ ∥Pℓ∥1/ℓ ≤ e−β Im ω/θn < 1. +■ +The operator P is called the propagation operator, as it describes how the solution of (3.54) +evolves from one interface to another. Provided that P is known, the solution U+ +θ (ϕ) may +be constructed using local cell problems. Let us first introduce the appropriate functional +framework in a periodicity cell +H1 +θ,per(C♯ +0) := +� +U ∈ H1 +θ(C♯ +0), �U ∈ H1 +θ,loc(B0) +� +, +(4.5) +where B0 := Rn ++ ∩ {0 < yn < 1}. Similarly to Section 3.2.1, one can show that any function +of H1 +θ,per(C♯ +0) has a L2 trace on the boundary of C♯ +0. We can prove in particular that +H1 +θ,per(C♯ +0) = +� +U ∈ H1 +θ(C♯ +0) / U|yi=0 = U|yi=1, ∀ i ∈ �1, n − 1� +� +. +We can now introduce the local cell problems: for all ϕ ∈ L2(Σ♯ +n,0), for j ∈ {0, 1}, let +Ej(ϕ) ∈ H1 +θ,per(C♯ +0) satisfy +����� +−Dθ +�µp Dθ Ej� − ρp ω2 Ej = 0, +in +C♯ +0, +µp Dθ Ej|yi=0 = µp Dθ Ej|yi=1 +∀ i ∈ �1, n − 1�, +(4.6) +defined for j = 0, 1, with the boundary conditions +����� +E0|Σ♯ +n,0 = ϕ +and +E0|Σ♯ +n,1 = 0, +E1|Σ♯ +n,0 = 0 +and +E1|Σ♯ +n,1 = ϕ. +(4.7) +A variational formulation can be derived as in Proposition 3.17, and the well-posedness follows +once again from a lifting argument (see Proposition 3.14) combined with Lax-Milgram’s +theorem in H1 +θ,per(C♯ +0). +22 + +Proposition 4.1 implies that U+ +θ (ϕ)(· + ℓ⃗en)|Σ♯ +n,0 = Pℓϕ. Hence, if the propagation operator +P is known, by linearity, the solution of the half-guide problem can be entirely constructed +cell by cell as follows: +∀ ℓ ∈ N, +U+ +θ (ϕ)(· + ℓ⃗en)|C♯ +0 = E0(Pℓϕ) + E1(Pℓ+1ϕ). +(4.8) +4.2 +Characterization of the propagation operator: the Riccati equation +In the sequel, ⟨·, ·⟩ denotes the canonical L2 scalar product on Σ♯ +n,0 (or equivalently on Σ♯ +n,1). +In order to characterize the propagation operator P, it is useful to introduce the local DtN +operators T jk ∈ L(L2(Σ♯ +n,0)), defined for j, k = 0, 1 by +∀ ϕ ∈ L2(Σ♯ +n,0), +T jkϕ = (−1)k+1 θn +� +µp Dθ Ej(ϕ) +� +|Σ♯ +n,k. +(4.9) +where Ej(ϕ) satisfies (4.6)-(4.7). By Green’s formula (3.30), note that for all j, k = 0, 1 and +for (ϕ, ψ) ∈ L2(Σ♯ +n,0)2, these operators satisfy +� +T jkϕ, ψ +� += +� +C♯ +0 +� +µp Dθ Ej(ϕ) Dθ Ek(ψ) − ρp ω2 Ej(ϕ) Ek(ψ) +� +. +(4.10) +Before deriving other useful properties of the local DtN operators, we need to introduce some +additional notations. For any closed operator A ∈ L(L2(Σ♯ +n,0)), we denote A∗ the adjoint of +A, and A its « complex conjugate », that is, +∀ ϕ ∈ L2(Σ♯ +n,0), +Aϕ = Aϕ. +It is not difficult to see that A∗ = A∗, and A = A. +Proposition 4.2. The local DtN operators T jk satisfy +� +T 00�∗ = T 00, +� +T 11�∗ = T 11, +� +T 01�∗ = T 10, +� +T 10�∗ = T 01. +(4.11) +Furthermore, the operators T 00, T 11, and T 00 + T 11 are invertible. +Proof. +The property (4.11) follows from Green’s formula applied to Ej(ϕ) and Ek(ψ), see +for instance [10, Proposition 2.2.4] in the case of the Helmholtz equation. +The operators T 00, T 11, and T 00 + T 11 are bounded. We are going to show that they are +also coercive. Their invertibility will then follow from Lax-Milgram’s theorem. +From the expression (4.10), one has the existence of a constant c ≡ c(µ−, ρ−, |ω|) > 0 +such that +−|ω| Im +� 1 +ω +� +T kkϕ, ϕ +�� +≥ c Im ω ∥Ek(ϕ)∥2 +H1 +θ(C♯ +0) ≥ ˜c Im ω ∥ϕ∥2 +L2(Σ♯ +n,0), +since from (3.34), the trace application from H1 +θ,per(C♯ +0) to L2(Σ♯ +n,0) is continuous. It follows +that the operators T 00 and T 11 are coercive, and therefore invertible. The inequalities above +summed for k = 0, 1 imply the coercivity and hence the invertibility of T 00 +T 11 as well. +■ +23 + +As seen earlier, the solution of the half-guide problem (3.54) is given by (4.8). Now let us +use the characterization of H1 +per,θ(Ω♯), namely, Corollary 3.11 with a = 1, so that Ω♯ +a,− = C♯ +0 +and Ω♯ +a,+ = Ω♯ \ C♯ +0. Since µp Dθ U+ +θ (ϕ) belongs to H1 +θ,per(Ω♯), the directional derivative of +U+ +θ (ϕ) is continuous across the interface Σ♯ +n,1, i.e. +� +µp Dθ U+ +θ (ϕ) +� +|Σ♯ +n,1 = +� +µp Dθ U+ +θ (ϕ)((· + ⃗en) +� +|Σ♯ +n,0, +(4.12) +or equivalently, +� +µp Dθ E0(ϕ) +� +|Σ♯ +n,1 + +� +µp Dθ E1(Pϕ) +� +|Σ♯ +n,1 += +� +µp Dθ E0(Pϕ) +� +|Σ♯ +n,0 + +� +µp Dθ E1(P2ϕ) +� +|Σ♯ +n,0. +(4.13) +By using the definition of the local DtN operators T jk, (4.13) leads to the following charac- +terization. +Proposition 4.3. The propagation operator P defined by (4.2) is the unique solution of the +constrained Riccati equation +������ +Find P ∈ L(L2(Σ♯ +n,0)) such that ρ(P) < 1 and +T 10P2 + (T 00 + T 11) P + T 01 = 0. +(4.14) +Proof. +The proof is identical to the one for the elliptic Helmholtz equation [19, Theorem +4.1]. We know from Proposition 4.1 that P has a spectral radius which is strictly less than +1. Moreover (4.13) ensures that P satisfies the Riccati equation. +In order to prove the uniqueness, let us consider an operator P1 which satisfies (4.14). The +function defined cell by cell by +∀ ϕ ∈ L2(Σ♯ +n,0), +∀ ℓ ∈ N∗, +U1(ϕ)(· + ℓ⃗en)|C♯ +0 = E0(Pℓ +1ϕ) + E1(Pℓ+1 +1 +ϕ), +solves (3.54) in each cell Cℓ and is continuous across each interface Σ♯ +n,ℓ, by definition (4.6), +(4.7) of E0 and E1. By Corollary 3.11, U1 is locally H1 +θ in Ω♯. +Moreover, since P1 satisfies (4.14), the directional derivative µpDθ U1 is continuous across +each interface. Thus, using Corollary 3.11, we deduce that U1 satisfies (3.54) in Ω♯. +Furthermore, given that ρ(P1) < 1, Gelfand’s formula and the well-posedness of the cell +problems ensure that there exist positive constants c, ρ∗, with ρ∗ < 1 such that, for ℓ ∈ N +large enough, +∥U1(ϕ)∥H1 +θ(C♯ +ℓ) ≤ c ρℓ +∗ ∥ϕ∥L2(Σ♯ +n,0). +Hence U1(ϕ) belongs to H1 +θ,per(Ω♯) and satisfies the half-guide problem (3.54). +By well- +posedness of (3.54), U1(ϕ) and U+ +θ (ϕ) coincide, and thus have the same trace on Σ♯ +n,1, that +is P1ϕ = Pϕ for any ϕ ∈ L2(Σ♯ +n,0). +■ +24 + +As a consequence, the propagation operator can be obtained by solving the Riccati equation +in (4.14), and by choosing the unique solution whose spectral radius is strictly less than 1. +One important thing to retain from the above is that both the propagation operator and the +solution of the half-guide problem only require the computation of E0, E1, and the operators +T 00, T 10, T 01, and T 11, which involve problems defined on a periodicity cell. However, the +resolution of the constrained Riccati equation (4.14) is not obvious at all. The properties of +this equation are investigated in further details in Section 4.4. +4.3 +The DtN operator and the DtN coefficient +The goal of this part is to see how the half-guide problem and the local cell problems can be +used to compute the DtN coefficient λ+. We recall that +λ+ = −µθ(0) du+ +θ +dx (0). +Therefore, it is natural to introduce the DtN operator Λ ∈ L(L2(Σ♯ +n,0)) defined by +∀ ϕ ∈ L2(Σ♯ +n,0), +Λϕ := −θn +� +µp Dθ U+ +θ (ϕ) +� +|Σ♯ +n,0. +(4.15) +This operator also has the following properties, whose proof is exactly identical to the one of +Proposition 4.2. +Proposition 4.4. One has Λ∗ = Λ. Moreover, Λ and Λ + T 11 are invertible operators. +Taking the directional derivative of (4.8) (for ℓ = 0) on Σ♯ +n,0 and using the definition (4.9) of +the local DtN operators T 00 and T 10 leads to +Λ = T 00 + T 10P. +(4.16) +Besides, by writing the formula (3.63) after multiplication by µp, and by evaluating it for +y = (s, 0), so that sθ(y) = s, we obtain +Λϕ(s) = θn λθ(s) ϕ(s), +with +λθ(s) = − +� +µs,θ +du+ +s,θ +dx +� +(0), +(4.17) +namely, Λ is a multiplication operator. We deduce from (4.17) the DtN coefficient λ+. +Proposition 4.5. The function λθ : Rn−1 → C defined by (4.17) is continuous. Moreover, +if ϕ ∈ Cper(Rn−1) is a given function which satisfies ϕ(0) = 1, then we have +λ+ = λθ(0) = 1 +θn +(Λϕ)(0). +(4.18) +Proof. +Using Green’s formula, we have that for all s ∈ Rn−1 +λθ(s) = as(u+ +s,θ, u+ +s,θ), +with +as(u, v) = +� +R+ +� +µs,θ +du +dx +dv +dx − ρs,θ ω2 u v +� +. +The continuity of u �→ as(u, u) results directly from the properties of the coefficients µp and +ρp. Moreover, Proposition 3.18 ensures that the function s �→ u+ +s,θ is continuous. Therefore, +as the composition of these two functions, λθ is also continuous. +If in addition ϕ is continuous, then Λϕ is also continuous. Hence, (Λϕ)(0) = θn λθ(0)ϕ(0) +which yields the desired result. +■ +25 + +4.4 +Spectral properties of the Riccati equation +We now present some properties regarding Equation (4.14). These properties will be exploited +for the numerical resolution of the Riccati equation, by constructing the operator P from its +eigenpairs (this will be done in Section 5.3 after space discretization). For this reason, it is +worhwhile to reformulate a spectral version (Proposition 4.7) of the Riccati equation that +would characterize these eigenpairs, while taking into account the spectral radius constraint. +This is precisely the purpose of this section. +Recall that T (P) = 0, where T is the bounded operator defined by +∀ X ∈ L +�L2(Σ♯ +n,0) +�, +T (X) = T 10X2 + (T 00 + T 11)X + T 01. +(4.19) +In the sequel, we will write T (λ) for T (λI). We begin with the following factorization lemma. +Lemma 4.6. Let P be the propagation operator defined by (4.2). For any number λ ∈ C, +T (λ) = (λP∗ − I) (Λ + T 11) (P − λ), +(4.20) +where T 11 is defined by (4.9) and Λ is defined by (4.15). +Proof. +Let λ ∈ C. Since the propagation operator satisfies T (P) = 0, one obtains that +T (λ) = T (λ) − T (P) += +� +T 10(λ + P) + T 00 + T 11� +(λ − P) += (λT 10 + Λ + T 11) (λ − P), +from (4.16). +(4.21) +We use once again the fact that T (P) = 0 which, by the expression (4.16), is equivalent to +T 01 = −(Λ + T 11) P. By transposing this equation, and by taking the complex conjugate, +one obtains that [T 01]∗ = −P∗ (Λ + T 11)∗. Since +�T 11�∗ = T 11 and +�T 01�∗ = T 10 as ensured +by Proposition 4.2, and since Λ∗ = Λ from Proposition 4.4, it follows that +T 10 = −P∗ (Λ + T 11). +Inserting this expression of T 10 in (4.21) therefore leads to +T (λ) = +� +−λP∗ (Λ + T 11) + Λ + T 11� +(λ − P) = (I − λP∗) (Λ + T 11) (λ − P). +which is the desired result. +■ +The previous factorization lemma allows one to characterize the spectrum of the propagation +operator as follows. +Proposition 4.7. For any complex number λ, one has +λ ∈ σ(P) +⇐⇒ +0 ∈ σ +�T (λ) +� and |λ| < 1. +(4.22) +26 + +Proof. +Proving (4.22) amounts to showing that for any λ ∈ C such that |λ| < 1, P − λ +is invertible if and only if T (λ) is invertible. To this end, using Lemma 4.6, it is sufficient +to prove that (λP∗ − I) (Λ + T 11) is an invertible operator. Proposition 4.4 ensures the +invertibility of Λ + T 11 already. It thus remains to show that λP∗ − I is invertible, which is +true when |λ| < 1. +Indeed, if λ = 0, then λP∗−I = −I is obviously invertible. Otherwise, it is not difficult to +see that P and P∗ have the same spectrum. Hence, given that |1/λ| > 1 > ρ(P∗), it follows +that 1/λ does not belong to σ(P∗). In other words, P∗ −(1/λ) I is an invertible operator. +■ +Remark 4.8. Note that the property (4.22) can be proved easily (and without Lemma 4.6) +for the point spectrum: +λ ∈ σp(P) +⇐⇒ +0 ∈ σp +�T (λ) +� and |λ| < 1. +(4.23) +This property was already proved in [19] for the Helmholtz equation. In this context, this was +sufficient since the operator P was compact, which is no longer the case here. +Finally, it is worth noting that the values λ ̸= 0 for which 0 ∈ σ +�T (λ) +� can be paired in the +following way. +Proposition 4.9. For any complex number λ ̸= 0, one has the following equivalence: +0 ∈ σ +�T (λ) +� +⇐⇒ +0 ∈ σ +�T (1/λ) +�. +(4.24) +Proof. +Let λ ∈ C∗. From the properties of the local DtN operators (see Proposition 4.2), +we deduce that +[T (λ)]∗ = λ2 T 01 + λ(T 00 + T 11) + T 10 = λ2 T (1/λ). +(4.25) +The operators T (λ) and [T (λ)]∗ have the same spectrum, hence the result. +■ +Remark 4.10. As Proposition 4.9 shows, the values λ ̸= 0 for which +0 ∈ σ +�T (λ) +� +come by pairs (λ, λ−1). From a numerical point of view, it suffices to choose λ such that +|λ| < 1 and discard λ−1. +4.5 +Spectral properties of the propagation operator +This section, contrary to Section 4.4 is not related to the construction of our numerical +method; it is of theoretical interest. On one hand, the result of this section, that is Proposition +4.11, is useful for interpreting some of the numerical results in Section 5.5.3. On the other +hand, it emphasizes the differences between the spectral properties of P, and the ones of +the corresponding operator for classical waveguide problems. +For the elliptic Helmholtz +equation, P is compact (see [19, Theorem 3.1]) and its spectrum hence consists only in +isolated eigenvalues which accumulate to 0. However, the picture is completely different in +this case, because the spectrum has no isolated points. +27 + +One useful way to study the properties of the propagation operator (especially its spectrum) +is through an analytic formula: according to (3.60), P can be expressed for all ϕ in L2(Σ♯ +n,0) +and for s ∈ Rn−1 as +Pϕ(s) = pθ(s) �ϕ +�s − δ +�, +with +pθ(s) = u+ +s−δ,θ(1/θn) +and +δ = ˆθ /θn ∈ Rn−1. +(4.26) +Note that since θ is an irrational vector, δ is also an irrational vector. +The properties of the mapping s �→ u+ +s,θ stated in Proposition 3.18 imply that the fonction +pθ is continuous and 1-periodic in each direction. +Operators that can be written under the form (4.26) are known as weighted shift operators, +and have been studied for instance in [2]. In particular, the spectral properties of P are given +by the following result. +Proposition 4.11. Let pθ : Σ♯ +n,0 → C be the function defined in (4.26). Then, pθ(s) ̸= 0 for +all s in Σ♯ +n,0, and the spectral radius of P is given by +ρ(P) = exp +�� +Σ♯ +n,0 +log |pθ(s)| ds +� +. +(4.27) +Moreover, the spectrum of P is a circle of radius ρ(P). +This result can be found in [2, Theorem 2.1] for n = 2. We give below the proof for n > 2, +which requires the following lemma (see Theorem 6.1 and Example 6.1 of [21]), known as a +particular case of Birkhoff’s ergodic theorem for continuous functions. +Lemma 4.12. Let ψ : Σ♯ +n,0 → C be continuous and 1–periodic in each direction. Let α ∈ Rn−1 +be an irrational vector. Then, we have the following uniform convergence: +lim +ℓ→+∞ +���1 +ℓ +ℓ−1 +� +m=0 +ψ(· − mα) − +� +Σ♯ +n,0 +ψ +��� +∞ = 0. +Proof of Proposition 4.11. +Let us first show by contradiction that pθ or equivalently +the function s �→ u+ +s,θ(1/θn) is nowhere vanishing. To do so, we use an argument of unique +continuation. In fact, assume that there exists s ∈ Σ♯ +n,0 such that u+ +s,θ(1/θn) = 0. Then u+ +s,θ +satisfies the problem +− d +dx +� +µs,θ +du+ +s,θ +dx +� +− ρs,θ ω2 u+ +s,θ = 0, in (1/θn, +∞), +and +u+ +s,θ(1/θn) = 0. +From the well-posedness of this problem, it follows that u+ +s,θ = 0 in (1/θn, +∞). Therefore, +by unique continuation, one deduces that u+ +s,θ = 0 in R+, which contradicts the boundary +condition u+ +s,θ(0) = 1. +We now establish the expression of the spectral radius ρ(P). One has ρ(P) = +lim +ℓ→+∞ ∥Pℓ∥1/ℓ +from Gelfand’s formula, and by induction, Pℓ can be expressed under the form +Pℓϕ(s) = p(ℓ) +θ (s) ϕ(s − ℓδ), +with +p(ℓ) +θ (s) = +ℓ−1 +� +m=0 +pθ(s − mδ). +28 + +Since the translation operator ϕ �→ ϕ(· − ℓδ) is isometric and bijective, the norm of Pℓ is +equal to the norm of the multiplication operator ϕ �→ p(ℓ) +θ ϕ, that is ∥p(ℓ) +θ ∥∞. Hence, given +that pθ(s) ̸= 0 for all s, one has +ρ(P) = +lim +ℓ→+∞ +��� +ℓ−1 +� +m=0 +pθ(· − mδ) +��� +1/ℓ +∞ = +lim +ℓ→+∞ exp +���1 +ℓ +ℓ−1 +� +m=0 +log +�|pθ(· − mδ)| +���� +∞ +Since θ is an irrational vector, δ = ˆθ/θn is also an irrational vector. Therefore, Lemma 4.12 +can be applied with α = δ, and ψ : s �→ log |pθ(s)|, which is well-defined and continuous. +Hence the spectral radius is given by +ρ(P) = Mlog(pθ) := exp +�� +Σ♯ +n,0 +log |pθ(s)| ds +� +. +Let us now characterize the spectrum. To begin, note that the inverse of P is well-defined, +since pθ vanishes nowhere: for all ϕ ∈ L2(Σ♯ +n,0), P−1ϕ(s) := pθ(s)−1 �ϕ +�s + δ +�. Therefore, all +the computations above can be applied to P−1, thus yielding +ρ(P−1) = Mlog(p−1 +θ ) = +1 +Mlog(pθ) = +1 +ρ(P) +Since the spectrum of P is always included in the annulus ρ(P−1)−1 ≤ |z| ≤ ρ(P), it follows +that σ(P) is included in the circle |z| = ρ(P) = Mlog(pθ). +Conversely, for k ∈ Zn−1, let Sk be the multiplication operator by s ∈ Rn−1 �→ exp(2iπ k · s). +From the expression (4.26) of the propagation operator, we obtain that +Sk P S−1 +k += e2iπ k · δ P. +The operators P and e2iπk · δ P are similar, and thus have the same spectrum. Now consider +an element λ0 of σ(P). Then, |λ0| = Mlog(pθ), and λk := e2iπk · δ λ0 also belongs to σ(P) for +all k ∈ Zn−1. Since δ is irrational, we have from Kronecker’s theorem (Theorem 2.2) that +the set {λk, k ∈ Zn−1} is dense in the circle |z| = |λ0| = Mlog(pθ). Consequently, this whole +circle is included in the spectrum, since the latter is a closed set. +■ +5 +Resolution algorithm and discretization issues for n = 2 +In order to compute the solution of Equation (1.1), the previous sections provide an algorithm +which sums up as follows: +1. Compute the solution u+ +θ of (1.8) and the DtN coefficient λ+ defined by (1.7) by using +the following procedure: +(a). for any boundary data ϕ ∈ L2(Σ♯ +n,0), compute the solutions E0(ϕ), E1(ϕ) of the +local cell problems (4.6); +(b). compute the local DtN operators (T 00, T 01, T 10, T 11), defined by (4.9)–(4.10); +(c). compute the propagation operator P as the unique solution of the constrained +Riccati equation (4.14); +29 + +(d). using an arbitrarily chosen boundary data ϕ ∈ Cper(Rn−1) which satisfies ϕ(0) = 1, +• from (4.8), construct the solution U+ +θ of the half-guide problem cell by cell; +• deduce the half-line solution u+ +θ via the formula (3.62); +(e). compute the DtN operator Λ defined by (4.16), and deduce λ+ from (4.18). +2. Compute the solution u− +θ of (1.8) and the DtN coefficient λ− defined by (1.7) by using +exactly the same procedure as in Step 1 (but independently from Step 1). +3. Finally, solve the interior problem (1.9) in (−a, a), and extend the solution everywhere +by using (1.10), as well as Step 1 and Step 2. +For convenience sake, the quasiperiodicity order is set to n = 2. The most original aspects of +the algorithm are the steps (1.a)–(1.d), and the rest of this section focuses on the discretiza- +tion of these four steps. We present in Sections 5.1 and 5.2 two different methods that are +linked to a choice of discretization of the step (1.a), which influences the implementation of +the steps (1.b) and (1.d). The treatment of the step (1.c) is independent of this choice, and +will be presented in Section 5.3. +per +per +Figure 7: Two-dimensional mesh for the 2D method (left), and family of one-dimensional +meshes for the quasi-1D method (right) +5.1 +A fully two-dimensional method +The first method is inspired from the resolution of the elliptic Helmholtz equation (see [10] +for instance), and consists in solving directly the local cell problems on an unstructured mesh +of the periodicity cell C♯ +0 = (0, 1)2 (see Figure 7). +We start from a triangular mesh Th(C♯ +0) of C♯ +0 = (0, 1)2 with a mesh step h > 0. We assume +that this mesh is periodic, in the sense that one can identify the mesh nodes on the boundary +yi = 0 with those on yi = 1, for 1 ≤ i ≤ 2. In particular for i = 1, this condition allows us to +handle the periodic boundary conditions. +Now let Vh(C♯ +0) be the usual H1–conforming approximation by Lagrange finite elements of +order d > 0. We also introduce +Vh,per(C♯ +0) := +�V ∈ Vh(C♯ +0) / V |y1=0 = V |y1=1 +� +as an internal approximation of H1 +θ,per(C♯ +0). Finally, to approximate L2(Σ♯ +2,0) and L2(Σ♯ +2,1), +we consider the following subspaces: +∀ a ∈ {0, 1}, +Vh,per(Σ♯ +2,a) = +�Vh|Σ♯ +2,a / Vh ∈ Vh,per(C♯ +0) +�. +30 + +Since the mesh nodes on Σ♯ +2,0 and Σ♯ +2,1 can be identified to each other by periodicity of +Th(C♯ +0), we can also make the identification Vh,per(Σ♯ +2,0) ≡ Vh,per(Σ♯ +2,1) ≡ Vh,per(0, 1), as in the +continuous case. Set N := dim Vh,per(0, 1), and consider a basis (ϕp)1≤p≤N. +For any data ϕh ∈ Vh,per(0, 1), we denote by E0 +h(ϕh), E1 +h(ϕh) ∈ Vh,per(C♯ +0) the solutions of +the discrete counterpart of the local cell problems (4.6)–(4.7) defined in a weak sense. In +practice, one has to compute Ej +h(ϕp), where (ϕp)1≤p≤N is a basis of Vh,per(0, 1). +Similarly to the weak expression (4.10) of the continuous local DtN operators, the discrete +local DtN operators T jk +h +∈ L(Vh,per(0, 1)), j, k = 0, 1, are defined for any ϕh, ψh ∈ Vh,per(0, 1) +as follows: +� +T jk +h ϕh, ψh +� += +� +C♯ +0 +� +µp Dθ Ej +h(ϕh) Dθ Ek +h(ψh) − ρp ω2 Ej +h(ϕh) Ek +h(ψh) +� +. +In practice, these operators are represented as N × N matrices Tjk whose components are +given by Tjk +pq = +�T jk +h ϕq, ϕp +�, for p, q ∈ �1, N�. +Let ϕh ∈ Vh,per(0, 1) ⊂ Cper(R) such that ϕh(0) = 1. The computation of the propagation +operator Ph ∈ L(Vh,per(0, 1)) is presented in Subsection 5.3. Once this operator is determined, +the solution of the half-guide problem (3.54) can be approximated with the function defined +cell by cell by +∀ ℓ ∈ N, +U+ +θ,h(ϕh)(· + ℓ⃗en)|C♯ +0 = E0 +h(Pℓ +h ϕh) + E1 +h(Pℓ+1 +h +ϕh). +Finally, a suitable approximation of the solution of the half-line problem 3.1 is provided by +∀ x ∈ R, +u+ +θ,h(x) = U+ +θ,h(ϕ)(θ x). +5.2 +A quasi one-dimensional method +Though easy to implement, the two-dimensional approach described in the previous section +does not exploit the fibered properties of the directional derivative Dθ . However, the periodic +half-guide problem can be seen as a concatenation in a certain sense of one-dimensional half- +line problems. This fibered structure is the core of the method presented in this section. +5.2.1 +Presentation +For any s ∈ R, we consider the one-dimensional cell problems +���������� +− d +dx +� +µs,θ +dej +s,θ +dx +� +− ρs,θ ω2 ej +s,θ = 0, +in +(0, 1/θ2) := Iθ, +e0 +s,θ(0) = 1 +and +e0 +s,θ(1/θ2) = 0, +e1 +s,θ(0) = 0 +and +e1 +s,θ(1/θ2) = 1. +(5.1) +Then, by analogy with Proposition 3.19, one easily shows that the local cell problems are +concatenations of one-dimensional cell problems, in the following sense. +31 + +Proposition 5.1. For any boundary data ϕ in L2(0, 1), the solutions E0(ϕ) and E1(ϕ) of +the local cell problems (4.6) are given by +a. e. y ∈ C♯ +0, +Ej(ϕ)(y) = �ϕ +�sθ(y) + j θ1/θ2 +� ej +sθ(y),θ +�y2 +θ2 +� +, +(5.2) +where ej +s,θ denotes the solution of the cell problems (5.1). +Proposition 5.1 also highlights the structure of the local DtN operators. To see this, let us +introduce the local DtN functions tjk +θ defined for j, k = 0, 1, by +∀ s ∈ R, +tjk +θ (s) = (−1)k+1θ2 +� +µs,θ +dej +s,θ +dx +�� j +θ2 +� +. +(5.3) +Note that by periodicity of µp and ρp, the maps s �→ ej +s,θ and tjk +θ are 1–periodic. +By applying the directional derivative operator Dθ to (5.2), and by using the relationship +between Dθ Ej(ϕ) and dej +s,θ/dx given by (3.52), it follows that the local DtN operators +defined by (4.9) are weighted translation operators, similarly to the propagation operator. +Proposition 5.2. The operators T jk can be written for ϕ ∈ L2(0, 1) and s ∈ (0, 1) as +T 00ϕ(s) = t00 +θ (s) �ϕ(s) +and +T 10ϕ(s) = t10 +θ (s) �ϕ(s + θ1/θ2), +T 11ϕ(s) = t11 +θ (s − θ1/θ2) �ϕ(s) +and +T 01ϕ(s) = t01 +θ (s − θ1/θ2) �ϕ(s − θ1/θ2), +(5.4) +where we recall that �ϕ denotes the periodic extension of ϕ on R, defined by (3.31). +Finally, the solution u+ +θ of the half-line problem (3.1) can be computed directly from the +functions ej +s,θ and from the propagation operator. In fact, given a function ϕ ∈ Cper(Σ♯ +n,0) +such that ϕ(0) = 1, taking formally the trace along θ R in (4.8) leads to +∀ ℓ ∈ N, +u+ +θ (· + ℓ/θ2)|Iθ = ( � +Pℓϕ)(ℓ θ1/θ2) e0 +ℓθ1/θ2,θ + ( � +Pℓ+1ϕ)((ℓ + 1) θ1/θ2) e1 +ℓθ1/θ2,θ. (5.5) +The proof of this result is similar to those of (4.8) and Proposition 4.1. +Expressions (5.2), (5.4), and (5.5) form the basis of the quasi one-dimensional or quasi-1D +strategy, which consists in approximating the solutions ej +s,θ as well as the functions tjk +θ and +finally the local DtN operators T jk. Then once the propagation operator is computed by +solving the constrained Riccati equation (4.14), the solution u+ +θ may be constructed directly +cell by cell using (5.5). +5.2.2 +Discretization +The quasi-1D approach requires two distinct approximate spaces associated to the transverse +and the θ–oriented directions (see Figure 7). +32 + +Transverse direction. +We begin with a one-dimensional mesh Th(0, 1) of Σ♯ +2,0 ≡ (0, 1) +with a mesh step h > 0. Let Vh(0, 1) be the approximation space of H1(0, 1) by Lagrange +finite elements of order d > 0. We denote by (ϕp)0≤p≤N the usual nodal basis, which satisfies +in particular ϕp(sq) = δp,q, where (sp)0≤p≤N are points (including the mesh vertices) such +that 0 = s0 < · · · < sN = 1. Then an internal approximation of L2(0, 1) is +Vh,per(0, 1) := Span{ϕ0 + ϕN, ϕ1, . . . , ϕN−1}, +which is chosen so that Vh,per(0, 1) ⊂ Cper(0, 1). In particular, from the definition of the basis +functions ϕi, one has the following decomposition +∀ϕh ∈ Vh,per(0, 1), +ϕh = +N +� +p=0 +ϕh(sp) ϕp, +with +ϕh(s0) = ϕh(sN). +(5.6) +θ–oriented direction. +Let Thθ(Iθ) denote a mesh of the line segment Iθ with a mesh step +hθ > 0. Set Vhθ(Iθ) as the approximation space of H1(Iθ) by Lagrange finite elements of +order dθ > 0 and define Vhθ,0(Iθ) := Vhθ(Iθ) ∩ H1 +0(Iθ). +The approximation of e0 +s,θ and e1 +s,θ can be seen as a two-step process. First, for any s ∈ R, +consider the solution ej +s,θ,hθ of the discrete variational formulation associated to (5.1). +In practice, the solution ej +s,θ,hθ can only be computed for a finite number of s ∈ (0, 1). This +is where the discretization in the transverse direction comes into play: given x ∈ Iθ, the +function s �→ ej +s,θ,hθ(x) may be interpolated in Vh,per(0, 1). +The interpolation process requires to compute the discrete solution ej +s,θ,hθ only for s = sp, +p ∈ �0, N − 1�. Then, using the decomposition formula (5.6), ej +s,θ shall be approximated by +∀ (s, x) ∈ (0, 1) × Iθ, +ej +s,θ,h(x) = +N +� +p=0 +ej +sp,θ,hθ(x) ϕp(s), +with +h = (h, hθ). +(5.7) +where ej +0,θ,hθ = ej +1,θ,hθ (because ej +s,θ is 1–periodic with respect to s). +From the solutions ej +s,θ,h, we introduce the discrete local DtN functions +∀ s ∈ (0, 1), +tjk +θ,h(s) = θn +� 1/θn +0 +� +µs,θ +dej +s,θ,h +dx +dek +s,θ,h +dx +− ρs,θ ω2 ej +s,θ,h ek +s,θ,h +� +, +which are inspired from the weak expression (5.3) of the local DtN functions tjk +θ . Then, by +analogy with (5.4), we define the discrete DtN operators T jk +h +∈ L(Vh,per(0, 1)) for any ϕh, +ψh ∈ Vh,per(0, 1) as follows: +� +T jk +h ϕh, ψh +� += +� 1 +0 +tjk +θ,h(s − k θ1/θ2) ϕh(s + (j − k) θ1/θ2) ψh(s) ds. +(5.8) +These discrete DtN operators, when computed for ϕh, ψh being the basis functions of +Vh,per(0, 1), are represented as N × N matrices, where N = dim Vh,per(0, 1). The integrals +33 + +which appear in (5.8) are evaluated in practice using a specifically designed quadrature rule +whose description is omitted here. +Finally, let ϕh ∈ Vh,per(0, 1) ⊂ Cper(R) such that ϕh(0) = 1. Then using (5.5), the solution of +the half-line problem (3.1) can be approximated with the function defined cell by cell by +∀ ℓ ∈ N, +u+ +θ,h(· + ℓ/θ2)|Iθ = (Pℓ +hϕh)(ℓ θ1/θ2) e0 +ℓθ1/θ2,θ,h + (Pℓ+1 +h +ϕh)((ℓ + 1) θ1/θ2) e1 +ℓθ1/θ2,θ,h. +where Ph ∈ L(Vh,per(0, 1)) corresponds to a suitable discrete RN×N approximation of P. The +computation of such an operator is the subject of the next subsection. +5.3 +Approximation of the propagation operator +In order to find a suitable approximation Ph ∈ L(Vh,per(0, 1)) of the propagation operator P, +it is natural to introduce the discrete constrained Riccati equation +������ +Find Ph ∈ L(Vh,per(0, 1)) such that ρ(Ph) < 1 and Th(Ph) = 0, where +Th(Ph) := T 10 +h P2 +h + (T 00 +h ++ T 11 +h ) Ph + T 01 +h , +(5.9) +and where (T 00 +h , T 01 +h , T 10 +h , T 11 +h ) are obtained via one of the methods described in Sections +5.1 and 5.2. Using the same arguments as for the elliptic Helmholtz equation [10], it can be +proved that this discrete equation admits a unique solution. +In order to solve (5.9), two methods have been proposed in [19]: a spectral decomposition +method, and a modified Newton method. Here, we only describe the spectral approach. +The spectral decomposition method consists in characterizing Ph by means of its eigenpairs +(λi, ψi) of Ph. Doing so however raises an important question: is Ph completely defined by +its eigenpairs? This is equivalent to wondering if Ph is diagonalizable or not. The diagonaliz- +ability of Ph is an open question, but for the sake of simplicity, we will assume in the sequel +that this is the case, namely +The family of eigenfunctions (ψi)1≤i≤N forms a basis of Vh,per(0, 1). +In practice, this is the situation that we always have encountered. Moreover, in the case where +this assumption fails to be true, one can still adapt the method, and recover Ph through a +Jordan decomposition. (See [10, Section 2.3.2.3] for more details.) +The spectral approach relies on the results presented in Section 4.4, which remain true for the +discrete equation. In particular, by analogy with Proposition 4.7, (λh, ψh) ∈ C × Vh,per(0, 1) +is an eigenpair of Ph if and only if it satisfies +Th(λh) ψh = 0, +with +ψh ̸= 0 +and +|λh| < 1. +Solving the Riccati equation hence comes down to solving a quadratic eigenvalue problem: +������ +Find (λh, ψh) ∈ C × Vh,per(0, 1) such that ψh ̸= 0, |λh| < 1 and +λ2 +h T 10 +h ψh + λh (T 00 +h ++ T 11 +h )ψh + T 01 +h ψh = 0. +(5.10) +34 + +If one sets N = dim Vh,per(0, 1), then (5.10) can be reduced to a 2N × 2N linear eigenvalue +problem, thus yielding 2N eigenvalues. In order to pick the N eigenvalues of the propagation +operator, we need a criterion. To do so, note that with the 2D or the quasi-1D method, +the properties of the local DtN operators (Proposition 4.2) remain preserved for the discrete +operators T jk +h . Hence Proposition 4.9 admits the following discrete version: +Ker Th(λh) ̸= {0} +⇐⇒ +Ker Th(1/λh) ̸= {0}. +Therefore, as already expected with Remark 4.10, the solutions of (5.10) can be grouped into +pairs (λh, 1/λh), where 0 < |λh| < 1. Consequently, in order to compute Ph, one can solve +(5.10) (using for instance linearization techniques), and choose the N eigenpairs (λh, ψh) +which satisfy |λh| < 1. +5.4 +The DtN coefficient +Finally, consider a function ϕh ∈ Vh,per(0, 1) ⊂ Cper(R) which satisfies ϕh(0) = 1. Then by +analogy with (4.16), and in the spirit of Proposition 4.5, we define the discrete DtN operator +and the discrete DtN coefficient as follows: +Λh = T 10 +h Ph + T 00 +h +and +λ+ +h = (Λhϕh)(0) +θ2 +, +where T 10 +h +and T 00 +h +are computed using one of the methods presented in Sections 5.1 and 5.2, +and where Ph is the solution of the discrete Riccati equation (5.9). +5.5 +Numerical results +We present some numerical results to validate the method, to illustrate its efficiency, and to +compare the multi-dimensional and the quasi one-dimensional methods in the case where the +order of quasiperiodicity is set to n = 2. Simulations will be carried out with the periodic +coefficients µp and ρp, defined for y = (y1, y2) ∈ R2 by +µp(y) = 1.5 + cos(2πy1) cos(2πy2) +and +ρp(y) = 1.5 + 0.5 sin(2πy1) + 0.5 sin(2πy2). +We set θ = (cos π/3, sin π/3). As the ratio θ2/θ1 = +√ +3 is irrational, θ is an irrational vector. +For a = 1, the source term f is the cut-off function +∀ x ∈ R, +f(x) = exp +� +100 +�1 − 1/(1 − x2) +�� +χ(−1,1), +and the local perturbations µi and ρi are defined as piecewise constants, so that the coefficients +µ and ρ of the model problem (1.1) are represented in Figure 8. +35 + +−6 +−4 +−2 +0 +2 +4 +6 +1 +2 +µ +−6 +−4 +−2 +0 +2 +4 +6 +1 +2 +ρ +−6 +−4 +−2 +0 +2 +4 +6 +0.5 +f +Figure 8: The locally perturbed quasiperiodic coefficients µ and ρ, and the source term f. +5.5.1 +The half-line and the half-guide solutions +The model problem (1.1) is solved by computing the solutions of the half-line problems (1.8), +as well as the DtN coefficients λ±. In this part, only results regarding the numerical resolution +of the problem (3.1) are going to be presented, as the problem set on (−∞, −a) provides the +same overall results. +Error analysis +In order to validate the method, we introduce for L > 0 the unique function +u+ +θ,L in H1(0, L) that satisfies Problem (3.1) on the truncated domain (0, L), with u+ +θ,L(L) = 0. +Similarly, define ΩL := (0, 1)n−1 × (0, L), and for any ϕ ∈ L2(Σ♯ +n,0), let U+ +θ,L(ϕ) ∈ H1 +θ(ΩL) +denote the unique function that satisfies (3.54) on ΩL, with U+ +θ,L(ϕ)|y2=L = 0. +In presence of absorption, the solutions u+ +θ and U+ +θ (ϕ) decay exponentially at infinity (see +(3.58) and (4.4)), and by studying the problems satisfied by u+ +θ,L − u+ +θ and U+ +θ,L(ϕ) − U+ +θ (ϕ), +it can be proved as in [11] that there exist constants α, c > 0 such that for any L > 0, +∥u+ +θ,L − u+ +θ ∥H1(0,L) ≤ c e−α Im ωL ∥u+ +θ ∥H1(0,L) +∥U+ +θ,L(ϕ) − U+ +θ (ϕ)∥H1 +θ(ΩL) ≤ c e−α Im ωL ∥U+ +θ (ϕ)∥H1 +θ(ΩL). +(5.11) +with α = +� +ρ−/µ+. In particular, if L is chosen large enough, then u+ +θ,L and U+ +θ,L(ϕ) can be +viewed as suitable approximations of u+ +θ and U+ +θ (ϕ), and thus can serve as reference solutions. +In the upcoming results, to make the truncation errors in (5.11) negligible with respect to +the errors induced by the numerical method, we choose L so that +exp +� − +� +ρ−/µ+ Im ω L +� ≤ 10−10. +(5.12) +The corresponding solutions u+ +θ,L and U+ +θ,L(ϕ), which will be denoted by u+ +ref and U+ +ref(ϕ) +respectively, are computed via P1 Lagrange finite elements, with a mesh step h = 5 × 10−4. +36 + +In the following, the boundary data is fixed to ϕ = 1, and is omitted in the notation of U+ +θ +and U+ +ref. Also, we only plot relative errors corresponding to the 1D solution, as the errors +for the 2D solution behave similarly. In Figure 9, the relative error +ε(u+ +θ ) := +∥u+ +θ,h − u+ +ref ∥H1(0,4/θ2) +∥u+ +ref ∥H1(0,4/θ2) +(5.13) +is represented with respect to the mesh step h, and for both the 2D and the quasi-1D method +(with hθ = h for the quasi-1D method). The solutions are computed using Lagrange finite +elements of degree 1. +One sees that the errors tend to 0 as h at least, as expected for P1 Lagrange finite elements. +With the quasi-1D method however, ε(u+ +θ ) behaves as h2. This is a special superconvergence +phenomenon, which is probably due to the fact that the problems solved in practice with +the quasi-1D method are one-dimensional. Note also that in general, the quasi-1D method +appears to be more accurate than the 2D method. +−2.03 +−1.46 +32 +64 +128 +256 +512 +10−3 +10−2 +10−1 +100 +Relative errors +−1.98 +−1.64 +32 +64 +128 +256 +512 +Discretization parameter 1/h +2D method +Quasi-1D method +(a) ω = 8 + 0.25 i +(b) ω = 20 + 0.25 i +Figure 9: Relative error in H1 norm of the half-line solution for different values of ω. +For a fixed mesh step, the relative error increases with the real frequency Re ω. This is a well- +known particularity of the Helmholtz equation: since Re ω represents the spatial frequency +of the time-harmonic waves, the discretization parameter h has to be adapted in order to +take their oscillations into account. +Representation of the half-guide solution +The half-guide solution is represented in +Figure 10 for different values of ω, when ϕ = 1. +37 + +0 +0.5 +1 +0 +1 +2 +3 +4 +−1 +0 +1 +0 +0.5 +1 +0 +1 +2 +3 +4 +−1 +0 +1 +0 +0.5 +1 +0 +1 +2 +3 +4 +−1 +0 +1 +(a) ω = 8 + 0.25 i +(b) ω = 20 + 0.25 i +(c) ω = 20 + 0.05 i +Figure 10: Real part of the half-guide solution computed using the quasi-1D approach, with +P1 Lagrange finite elements and h = 2 × 10−3, and for different values of ω. +Dependence with respect to the boundary data +The goal of this part is to see how +the half-line and the half-guide solutions depend on the boundary data ϕ. To do so, we +choose three different datas: +ϕ1(s) = 1, +ϕ2(s) = cos(2πs), +and +ϕ3(s) = 1 − 1[1/3,2/3](s). +(5.14) +We set ω = 8 + 0.25 i, and we display results obtained with the quasi-1D method, knowing +that the 2D method yields the same conclusions. The computations are carried out using P1 +Lagrange finite elements, with mesh steps h = hθ = 2 × 10−3. +Size of periodicity cell +0 +1 +2 +3 +4 +−1 +0 +1 +ϕ1 +ϕ2 +ϕ3 +Figure 11: Real part of the half-line solution computed using the quasi-1D approach, with +P1 Lagrange finite elements and h = 2 × 10−3, and for different values of ϕ. +38 + +0 +0.5 +1 +0 +1 +2 +3 +4 +−1 +0 +1 +0 +0.5 +1 +0 +1 +2 +3 +4 +−1 +0 +1 +0 +0.5 +1 +0 +1 +2 +3 +4 +−1 +0 +1 +(a) ϕ1 +(b) ϕ2 +(c) ϕ3 +Figure 12: Real part of the half-guide solution computed using the quasi-1D approach, with +P1 Lagrange finite elements and h = 2 × 10−3, and for different values of ϕ. +As expected, and as Figures 11 and 12a–12c show, the aspect of half-guide solution changes +extensively with respect to the boundary data, whereas the half-line solution looks invariant. +5.5.2 +The whole line problem +The solutions u± +θ of the half-line problems (1.8) allow one to compute the DtN coefficients +λ±, to solve (1.9), and then to compute the solution u of Problem (1.1) using (1.10). Recall +that the coefficients µ, ρ, and the source term f are shown in Figure 8. The solution of (1.1) +is represented in Figure 13 for different values of ω. +(a) ω = 8 + 0.25 i +−6 +−4 +−2 +0 +2 +4 +6 +−1 +0 +1 +39 + +(b) ω = 20 + 0.25 i +−6 +−4 +−2 +0 +2 +4 +6 +−1 +0 +1 +(c) ω = 20 + 0.05 i +−6 +−4 +−2 +0 +2 +4 +6 +−1 +0 +1 +Figure 13: Real part of the solution of (1.1) computed using the quasi-1D approach, with P1 +Lagrange finite elements and h = 2 × 10−3, and for different values of ω. +5.5.3 +About the dependence with respect to the absorption +We come back to the numerical resolution of Problem (3.1), and we study the convergence +of the 2D and quasi-1D methods depending on the absorption, especially when it tends to +0. As in Section 5.5.1, the solutions are computed with Lagrange finite elements of degree 1. +The relative error ε(u+ +θ ) defined (5.13) is represented in Figure 14 for both the 2D and the +quasi-1D method, and for different values of Im ω. +−2.03 +−1.46 +32 +64 +128 +256 +512 +10−3 +10−2 +10−1 +100 +Relative errors +−1.7 +−1.07 +32 +64 +128 +256 +512 +−0.91 +−0.3 +32 +64 +128 +256 +512 +Discretization parameter 1/h +2D method +Quasi-1D method +(a) ω = 8 + 0.25 i +(b) ω = 8 + 0.01 i +(c) ω = 8 + 0.001 i +Figure 14: Relative error in H1 norm of the half-line solution for different values of ω. +40 + +As Figure 14 shows, the error deteriorates with Im ω. It would mean that the numerical +method becomes less efficient as the absorption decreases. This issue is closely related to the +well-posedness of the local cell problems with Dirichlet boundary conditions when Im ω = 0. +In fact, for the elliptic Helmholtz equation, it is known (see [10, Section 3.2.1.1] for instance) +that the local cell problems are well-posed except for a countable set of frequencies which +correspond to the eigenvalues of the associated differential operator. In our case however, as +the differential operator has a non-elliptic principal part, it also has a continuous spectrum, +and one can show that when µp and ρp are non-constant, the local cell problems are well- +posed only for frequencies in a bounded set (that can even be empty). An alternative to avoid +this problem is to use a Robin-to-Robin operator instead of the DtN operator, which would +involve solving well-posed local cell problems with Robin boundary conditions, as it is done +in [12] for periodic media. This will be done in a forthcoming paper for quasiperiodic media. +5.5.4 +About the spectral approximation of the propagation operator +As explained in Subsection 5.3, the discrete propagation operator Ph is computed by means +of its eigenpairs. In this section, the eigenvalues of Ph are compared with the spectrum of +the exact propagation operator which, according to Proposition 4.11, is a circle of radius +Mlog(pθ) = exp +� � 1 +0 +log |pθ(s)| ds +� +, +with +pθ(s) = u+ +s−θ1/θ2,θ(1/ sin θ2). +To compute this radius, u+ +s,θ is approximated by the unique function u+ +s,θ,L that satisfies (3.57) +on a truncated domain (0, L), with u+ +s,θ,L(L) = 0. One can show similar estimates to (5.11), +and if L is chosen large enough (for instance, if L satisfies (5.12)), then u+ +s,θ,L can be used +as a reference solution. In practice, u+ +s,θ,L is computed for several s, and finally the integral +that defines Mlog(pθ) is evaluated using a rectangular quadrature rule. +The spectra of Ph and P are shown in Figure 16 for ω = 8 + 0.25 i, and for different values of +the discretization parameter h (with hθ = h for the quasi-1D method). Figure 15 represents +the number Nh of eigenvalues of Ph that are close by 5% to σ(P), namely +Nh = # +� +λh ∈ σ(Ph) +� ���� +|λh| − Mlog(pθ) +Mlog(pθ) +���� ≤ 5% +� +. +(5.15) +In Figure 15, one sees that Nh increases with 1/h, which means that more and more eigen- +values of Ph are close to σ(P) when h decreases. In other words, a finer discretization leads +to a better approximation of the spectrum. The number Nh of such eigenvalues also seems +to increase linearly with 1/h (up to a subsequence for the quasi-1D method). Finally, note +that Nh is higher with the quasi-1D method than with the 2D method. +As Figure 16 shows, the eigenvalues of Ph are all included in the disk of radius ρ(P), but one +observes some spectral pollution. This is a classical phenomenon when one approximates the +spectrum of an operator which is neither compact nor self-adjoint. What is striking however, +is that the pollution behaviours are very different depending on the method used. +On one hand, the eigenvalues obtained with the 2D approach tend to accumulate to 0. A +likely explanation for this phenomenon is that solving the local cell problems on 2D meshes +does not take their directional structure into account. Since the location of the eigenvalues +41 + +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +220 +240 +50 +100 +150 +200 +Discretization parameter 1/h +Nh +2D +Quasi-1D +Figure 15: Number of eigenvalues of Ph that are close by 5% to σ(P) with respect to h. +−1 +0 +1 +−1 +0 +1 +2D method +1/h = 32 +−1 +0 +1 +1/h = 64 +−1 +0 +1 +1/h = 129 +−1 +0 +1 +1/h = 258 +−1 +0 +1 +−1 +0 +1 +Quasi-1D method +−1 +0 +1 +−1 +0 +1 +−1 +0 +1 +Figure 16: Eigenvalues of the discrete propagation operator (circle-shaped markers) compared +to the spectrum of the exact propagation operator (circle in dashed line) for ω = 8 + 0.25 i, +and for different values of the discretization parameter. +42 + +of Ph is similar to the one obtained in the elliptic case, for which P is compact (see [19, +Theorem 3.1]), we believe the 2D method somehow regularizes the half-guide problem (3.54) +by introducing an elliptic (discrete) approximation of the corresponding differential operator. +On the other hand, with the quasi-1D approach, the spectrum of Ph “oscillates” as the +discretization parameter h tends to 0. This phenomenon has to do with the particular nature +of P which is a weighted translation operator. We strongly suspect that one can extract a +subsequence (Ph′) whose spectrum converges towards σ(P) in a sense to be defined precisely, +as it is suggested by the peaks in Figure 15. The investigation of this assumption as well as +the construction of such a subsequence are subject to ongoing works. +With both approaches, it has been observed numerically that the eigenfunctions associated +to the spurious eigenvalues were highly oscillating functions that were badly approximated +by the discretization, whereas the components of the half-guide solution with respect to these +eigenfunctions are very small. This might explain why the spectral pollution does not have +a visible influence on the approximation of the half-guide and the half-line solutions, as the +errors in Figure 9 seem to suggest. +6 +Perspectives and ongoing works +A numerical method has been proposed to solve Helmholtz equation in 1D unbounded +quasiperiodic media. Using the presence of absorption, we justified that this equation could +be lifted onto a higher-dimensional problem which, in turn, can be solved using a Dirichlet- +to-Neumann approach. +For the discretization, we presented a multi-dimensional method, +as well as a so-called quasi one-dimensional method. As shown by numerical simulations, +both methods provide a suitable approximation of the solution as long as there is absorption. +However, the quasi-1D method proved to be more efficient than the 2D method, as it takes +the anisotropy of the problems involved into account. +The method presented opens up numerous perspectives, and raises multiple questions that are +subject to ongoing works. For instance, it would be interesting to approximate efficiently the +spectrum of the propagation operator, even though the spectral pollution seems to have no +major impact on the efficiency of the overall method. Another key extension concerns the case +where the absorption tends to 0. 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Almost periodic functions and +differential equations. Cambridge University Press, 1982. +[23] +Yves Meyer. “Quasicrystals, Diophantine approximation and algebraic numbers”. In: +Beyond quasicrystals. Springer, 1995, pp. 3–16. +[24] +Roger Penrose. “Pentaplexity A Class of Non-Periodic Tilings of the Plane”. In: The +Mathematical Intelligencer 2.1 (Mar. 1979), pp. 32–37. issn: 0343-6993. +[25] +Maria Radosz. “New limiting absorption and limit amplitude principles for periodic +operators”. In: Zeitschrift für angewandte Mathematik und Physik 66.2 (2015), pp. 253– +275. +[26] +Walter Rudin. Functional Analysis. International series in pure and applied mathemat- +ics. McGraw-Hill, 1991. isbn: 9780070542365. +[27] +Marjorie Senechal. Quasicrystals and geometry. CUP Archive, 1996. +[28] +Dan Shechtman et al. “Metallic Phase with Long-Range Orientational Order and No +Translational Symmetry”. In: Phys. Rev. Lett. 53 (20 Nov. 1984), pp. 1951–1953. doi: +10.1103/PhysRevLett.53.1951. +[29] +Roger Temam. “Sur la stabilité et la convergence de la méthode des pas fractionnaires”. +In: Annali di Matematica pura ed applicata 79.1 (1968), pp. 191–379. +[30] +Niklas Wellander, Sébastien Guenneau, and Elena Cherkaev. “Homogenization of quasiperi- +odic structures and two-scale cut-and-projection convergence”. In: (Nov. 2019). +[31] +Calvin H. Wilcox. “Wave operators and asymptotic solutions of wave propagation prob- +lems of classical physics”. In: Archive for Rational Mechanics and Analysis 22.1 (1966), +pp. 37–76. +[32] +Lijun Yuan and Ya Yan Lu. “A recursive-doubling Dirichlet-to-Neumann-map method +for periodic waveguides”. In: Journal of Lightwave Technology 25.11 (2007), pp. 3649– +3656. +[33] +Ruming Zhang. “Numerical methods for scattering problems in periodic waveguides”. +In: Numerische Mathematik 148.4 (2021), pp. 959–996. +45 + diff --git a/m9AzT4oBgHgl3EQfN_u0/content/tmp_files/load_file.txt b/m9AzT4oBgHgl3EQfN_u0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d98159fb8f1b4cdbdf1d5e55e77c56724d51bbe --- /dev/null +++ b/m9AzT4oBgHgl3EQfN_u0/content/tmp_files/load_file.txt @@ -0,0 +1,1592 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf,len=1591 +page_content='Wave propagation in one-dimensional quasiperiodic media Pierre Amenoagbadji, Sonia Fliss, Patrick Joly Abstract This work is devoted to the resolution of the Helmholtz equation −(µ u′)′ −ρ ω2u = f in a one-dimensional unbounded medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We assume the coefficients of this equation to be local perturbations of quasiperiodic functions, namely the traces along a particular line of higher-dimensional periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Using the definition of quasiperiodicity, the problem is lifted onto a higher-dimensional problem with periodic coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The periodicity of the augmented problem allows us to extend the ideas of the DtN-based method developed in [10, 19] for the elliptic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' However, the associated mathematical and numerical analysis of the method are more delicate because the augmented PDE is degenerate, in the sense that the principal part of its operator is no longer elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We also study the numerical resolution of this PDE, which relies on the resolution of Dirichlet cell problems as well as a constrained Riccati equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 1 Introduction and motivation We consider the Helmholtz equation − d dx � µ du dx � − ρ ω2 u = f in R, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) where the coefficients µ and ρ have positive upper and lower bounds: ∃ µ±, ρ±, ∀ x ∈ R, 0 < µ− ≤ µ(x) ≤ µ+ and 0 < ρ− ≤ ρ(x) ≤ ρ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2) The source term f belongs to L2(R) and is assumed to have a compact support: ∃ a > 0, supp f ⊂ (−a, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3) Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) is encountered when one is looking for time-harmonic solutions u(x) eiωt of the linear wave equation in heterogeneous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For real frequencies ω, the well-posedness of this problem is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In fact, on one hand, one expects that the physical solution u, if it exists, may not belong to H1(R) due to possible wave propagation phenomena and a lack of decay at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' On the other hand, uniqueness of a solution in H1 loc(R) does not hold in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In this case, one needs a so-called a radiation condition that imposes the behaviour of the solution at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Such a condition can be obtained in practice using the limiting absorption principle, which consists in (i) adding some absorption – that is some imaginary part to ω: Im ω, and (ii) studying the limit of the solution u ≡ u(ω) as the absorption tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The limiting absorption principle is a classical approach to study time-harmonic wave propagation problems in unbounded domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' see for instance [1, 9, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' More recently, it has been successfully applied for locally perturbed periodic media [10, 17, 20, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='01159v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='AP] 3 Jan 2023 In this paper, we will only address the case with absorption, that is the frequency ω satisfies Im ω > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4) Under these assumptions, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) admits a unique solution in H1(R) by Lax-Milgram’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover, it can be shown (using for instance an argument similar to the one in [7]) that this solution satisfies a sharp exponential decay property ∃ c, α > 0, ∀ x ∈ R, |u(x)| ≤ c e−α Im ω|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5) Exploiting (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5), a naive numerical method for treating the unboundedness would consist in truncating the computational domain (with homogeneous Dirichlet boundary conditions for instance) at a certain distance related to Im ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' However the cost and the accuracy of the method would deteriorate when Im ω tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Our objective in this paper is to develop a numerical method which is robust when Im ω tends to 0, in the particular case of locally perturbed quasiperiodic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' More precisely, we solve the problem in the bounded domain (−a, a) (which is independent of Im ω) by constructing transparent boundary conditions of Dirichlet-to-Neumann type: ± µ du dx + λ± u = 0 on x = ±a, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6) where λ± are called Dirichlet-to-Neumann (DtN) coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' These coefficients are defined by λ± = ∓ � µ du± dx � (±a), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7) where u± is the unique solution in H1(±a, ±∞) of ������� − d dx � µ du± dx � − ρ ω2 u± = 0, for ±x > a, u±(±a) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8) Knowing λ±, one is then reduced to compute u|(−a,a) by solving the problem ��������� − d dx � µ dui dx � − ρ ω2 ui = f, for x ∈ (−a, a), � ± µ dui dx + λ± ui� (±a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9) The well-posedness of this problem is a direct consequence of the sign property Im λ± < 0, which, through a Green’s formula, results itself from the presence of dissipation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then the solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) is given by ∀ x ∈ R, u(x) = � � � � � � � � � ui(−a) u−(x), x < −a, ui(x), x ∈ (−a, a), ui(a) u+(x), x > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10) In general, the problem is that computing λ±, that is to say solving (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8), is as difficult as the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' However, this is no longer true when the exterior medium (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' outside (−a, a)) has a certain structure: 2 if the exterior medium is homogeneous (ρ and µ are constant), these coefficients can be computed explicitly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' if the exterior medium is periodic (ρ and µ are periodic), several methods for the computation of these DtN coefficients are developed in [10, 19, 20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' if the exterior medium is a weakly random perturbation of a periodic medium, the coefficients can be approximated via an asymptotic analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' see [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Our main objective in this paper is to compute the DtN coefficients for a quasiperiodic exterior medium, in order to develop a numerical method according to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The outline of the rest of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In Section 2, we introduce the fundamental notion of quasiperiodic functions (in 1D) and define what is a locally perturbed quasiperiodic medium in the context of the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Sections 3 and 4 are the most important sections of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In Section 3, we link the solution of the 1D half-line problem with quasiperiodic coefficients to the solution of a degenerate directional Helmholtz equation posed in dimension n, with n > 1 defined as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This is the so-called lifting approach whose principle is presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' More precisely, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3, we characterize the solution of the 1D quasiperiodic problem as the trace along a (broken) line of a nD problem posed in a domain with the geometry of a half-waveguide: (0, 1)n−1 × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In between, we need to dedicate the (rather long) Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 to fix the notations used in the rest of the paper and present some useful preliminary material about an adapted functional framework for the rigorous setting of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This concerns anisotropic Sobolev spaces with an emphasis on trace theorems and related Green’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In Section 4, we provide a method for solving the half-waveguide problem of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1, we describe the structure of the solution with the help of a propagation operator P and local cell problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2, we show that the operator P is characterized as a particular solution of a Riccati equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3, we first build a directional DtN operator Λ for the half-waveguide problem, from which we deduce the DtN coefficients λ± we are looking for (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Finally, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4, we analyze the Riccati equation from a spectral point of view and in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 we describe the spectrum of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In Section 5 devoted to numerical results, we restrict ourselves to n = 2 for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The first two subsections are devoted to the discretization of the cell problems evoked above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We have considered two approaches: one, natural but naive, consists in using 2D Lagrange finite elements (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) while the other, called the quasi-1D method, is better fitted to the anisotropy of the problem (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3, we explain how we can construct a discrete propagation operator from a discrete Riccati equation that we choose to solve via a spectral approach, while Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4 simply mimics Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3 at the discrete level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 is devoted to numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In the first three subsections, we provide various validations of our method for the half-line problem (Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3) and the whole line problem (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' At last, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4, we address the question of the approximation of the spectrum of the propagation operator P by the one of its discrete approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Particular notation used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In what follows, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' the equality modulo 1 is denoted by ∀ y ∈ R, z = y [1] ⇐⇒ z ∈ [0, 1) and y − z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 3 and for all p, q ∈ N, p < q, we set �p, q� := {j ∈ N, p ≤ j ≤ q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We introduce Cper(Rn) as the space of continuous functions F : Rn → R that are 1– periodic with respect to each variable, and C ∞ 0 (O) as the space of smooth functions that are compactly supported in O ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For i ∈ �1, n�, we denote by ⃗ei the i-th unit vector from the canonical basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For any element y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , yn) in Rn, we define ˆy as the vector (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , yn−1) ∈ Rn−1, so that y = (ˆy, yn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , yn) and z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , zn), the Euclidean inner product of y and z is denoted y · z := y1 z1 + · · · yn zn, and the associated norm is |y| := √y · y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 2 Quasiperiodicity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 Quasiperiodic functions of one real variable In this section, we present a brief overview of the main properties of quasiperiodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We refer to [3, 5, 22] for more complete presentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Quasiperiodicity is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' A continuous function f : R → R is said to be quasiperiodic of order n > 1 if there exist a constant real vector θ = (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , θn), with θi > 0 for all i ∈ �1, n�, and a continuous function F : Rn → R, 1–periodic with respect to each variable, such that ∀ x ∈ R, f(x) = F(x θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) The vector θ is called a cut direction, and F is a periodic extension of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' A geometrical interpretation of this definition is to see the one-dimensional function f as the trace of a n-dimensional function F along the line passing through (0, 0) and parallel to the vector θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This is illustrated in Figure 1 for n = 2 and θ = (1, √ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' θ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8 0 Size of periodicity cell −4 −2 0 2 4 −2 0 2 Figure 1: Function F : (y1, y2) �→ cos 2πy1 + cos 2πy2 in its periodicity cell (left), and whose trace along θ = (1, √ 2) leads to a quasiperiodic function (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Periodic functions are obviously quasiperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Other examples of quasiperiodic functions are finite sums or products of periodic functions: if f1 and f2 are periodic, then f1 + f2 and f1f2 can be expressed under the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Note that f1 + f2 and f1f2 are not periodic if f1 and f2 are continuous functions with non-commensurable least periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For instance, with 4 f1(x) = cos 2πx and f2(x) = cos 2π √ 2x, one easily checks that the sum f1 + f2, represented in Figure 1, is not periodic since it equals 2 only when x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1, it is easy to see that neither the periodic extension nor the cut direction are uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Given (F, θ), it is always possible to lower the value of n, and change the function F accordingly, so that the coefficients θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , θn are linearly independent over the integers (see [22, Chapter 2]), that is ∀ k ∈ Zn, k · θ = 0 ⇐⇒ k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2) For n = 2 and θ = (θ1, θ2), the above condition amounts to saying that the ratio θ1/θ2 is irrational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Due to this observation, vectors that satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2) will be abusively referred to as irrational vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' A consequence of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2) is given by Kronecker’s approximation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 ([16, Theorem 444]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' If θ is an irrational vector, then the set θ R + Nn is dense in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' If θ is an irrational vector, and if F ∈ Cper(Rn) satisfies F(θ R) = 0, then Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 ensures that F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In other words, under the linear independence assumption, F is uniquely determined by its restriction on the line θ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For n = 2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 implies that the broken line �(x θ1[1], x θ2[1]), x ∈ R � is dense in the unit cell (0, 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' To illustrate this, Figure 2 represents the set �(x θ1[1], x θ2[1]), x ∈ (0, M) � in the unit cell for different values of M, when (1) θ1/θ2 ∈ Q (see the first row), and when (2) θ1/θ2 ∈ R \\ Q (see the second row for θ = ( √ 2, 1) and the third one for θ = (π, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For M large enough, in the first case, this set is reduced to a finite union of segments, whereas in the second case, it seems to fill the unit cell without ever passing through the same positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' It is also interesting to see that for θ = ( √ 2, 1), the unit cell is somehow filled uniformly, contrary to the case where θ = (π, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Finally, it is worth mentioning that Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 extends to higher-dimensional continuous functions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover, the notion of quasiperiodicty can be defined at a discrete level, to describe the properties of tilings that are cuts and projections of higher-dimensional periodic tilings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' These quasiperiodic tilings have been extensively studied [13, 23, 24, 27], and are used for modelling quasicrystals [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 Locally perturbed quasiperiodic media A locally perturbed quasiperiodic medium is a medium corresponding to functions µ and ρ that satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2) and that are quasiperiodic outside a bounded interval, which can be supposed to be (−a, a) (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3)) without any loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' More precisely, µ(x) = ����� µi(x) x ∈ (−a, a) µp(x θ) x ∈ R \\ (−a, a) and ρ(x) = ����� ρi(x) x ∈ (−a, a) ρp(x θ) x ∈ R \\ (−a, a), where the functions µp, ρp belong to Cper(Rn) with n > 1, and θ ∈ Rn is an irrational vector (see Condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 5 0 1 0 1 θ = (3, 1) M = 1/3 0 1 M = 2/3 0 1 M = 1 0 1 M ≥ 1 0 1 0 1 θ = ( √ 2, 1) M = 1 0 1 M = 20 0 1 M = 40 0 1 M = 80 0 1 0 1 θ = (π, 1) M = 1 0 1 M = 20 0 1 M = 40 0 1 M = 80 Figure 2: Representation of the set �(x θ1[1], x θ2[1]), x ∈ (0, M) � in (0, 1)2 for different values of M, when θ1/θ2 ∈ Q (first row), and when θ1/θ2 ∈ R \\ Q (second row for θ = ( √ 2, 1) and third row for θ = (π, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Since θ is an irrational vector, Kronecker’s approximation theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 ensures that the functions µp and ρp are entirely determined by their restrictions on the line R θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Therefore, µp and ρp satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2) with respectively the same bounds as µ and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The present study can be extended without difficulty to the case where µ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ρ) coincides with two different quasiperiodic functions in (−∞, −a) and in (a, +∞): for ± x > ±a, µ(x) = µ± p (x θ± ) and ρ(x) = ρ± p (x θ± ), where µ± p , ρ± p belong to Cper(Rn±) with n± > 1, and where θ± ∈ Rn± are irrational vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 6 3 The half-line quasiperiodic problem We now focus on the half-line quasiperiodic problems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' As these problems are very similar to each other, it is sufficient to study the half-line problem set on (a, +∞) and suppose without loss of generality that a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let µθ := µp(θ ·) and ρθ := ρp(θ ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Therefore, the problem we consider in this section is the following: ������� − d dx � µθ du+ θ dx � − ρθ ω2 u+ θ = 0, in R+, u+ θ (0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The function u+ θ corresponds exactly to the solution u+ of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8) that was introduced in Section 1 for very general media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The reason why this solution is relabeled u+ θ is due to the fact that, because we consider here quasiperiodic media, the coefficients µ and ρ that appear in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8) have been replaced by µθ and ρθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 Lifting in a higher-dimensional periodic problem We wish to exhibit some structure of the solution u+ θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' As the coefficients µθ and ρθ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) are by definition traces of n–dimensional functions along the half-line θ R+, it is natural to seek u+ θ as the trace along the same line of a function y ∈ Rn �→ �U+ θ (y), that is to say: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' x ∈ R, u+ θ (x) = �U+ θ (x θ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2) where �U+ θ shall be characterized as the solution of a n–dimensional PDE (in some sense, an “augmented” problem in which y is the augmented space variable) with periodic coefficients, as illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This so-called lifting approach has been used in the homogenization setting for the analysis of some correctors in presence of periodic halfspaces [14, 15] or periodic structures separated by an interface [4], as well as for the homogenization of quasicrystals and Penrose tilings [6, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' However, to our knowledge, very little seems to have been done in other contexts (such as wave propagation), and in particular for numerical analysis and simulation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' To build a higher-dimensional PDE, one has to exploit the correspondence between the deriva- tive of u+ θ and the partial derivatives of �U+ θ : according to the chain rule, for any smooth enough function F : Rn → C, one has ∀ x ∈ R, d dx[F(θ x)] = (Dθ F)(θ x), with Dθ = θ · ∇ = n � i=1 θi ∂ ∂yi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3) This leads us to introduce the n–dimensional PDE set on a half-space (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2) −Dθ �µp Dθ �U+ θ � − ρp ω2 �U+ θ = 0, for yn > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4a) where we recall that the coefficients µp, ρp : Rn → R are continuous and 1–periodic with respect to each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In addition, the boundary condition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) can be lifted onto the inhomogeneous Dirichlet boundary condition �U+ θ = �ϕ, on yn = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4b) 7 y1 y2 θ R+ θ 0 − d dx � µθ du+ θ dx � − ρθ ω2 u+ θ = 0 −Dθ �µp Dθ �U+ θ � − ρp ω2 �U+ θ = 0 u+ θ (0) = 1 �U+ θ = �ϕ Figure 3: Illustration of the lifting approach for n = 2 where the data �ϕ : Rn−1 → C could be chosen continuous and must satisfy �ϕ(0) = 1, for the sake of consistency with the fact that u+ θ (0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Furthermore, to exploit the periodicity of the coefficients µp and ρp with respect to the transverse variables yj, j < n, we can impose the following: �ϕ is 1–periodic, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5) so that it is natural to impose that �U+ θ (ϕ) is 1–periodic with respect to the transverse variables yj, j < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6) In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3, we show how to reduce the above to a half-guide problem with periodic coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In order to do so, we shall need some preliminary materials, which is the object of the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' One could have defined the augmented problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4) on other half-spaces {y ∈ Rn, yi > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The choice of the half-space is purely arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' At first glance, one could imagine restricting the whole study to a constant boundary data �ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Though, in practice, this can be the case, the method used to solve the half-guide problem requires to investigate the structure of �U+ θ ( �ϕ) for any �ϕ in an appropriate function space (see Section 4 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 Preliminary material The main objective of this section is to establish rigorously some Green’s formulas that are formally obvious, such as the one of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This requires first to introduce the adapted functional framework and, since Green’s formulas involve boundary integrals, to establish relevant trace theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 is devoted to these trace theorems, while we present the corresponding Green’s formulas in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Finally, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3 highlights a simple but useful link between the derivative Dθ and a single partial derivative with respect to one real variable, through a so-called oblique change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 Anisotropic Sobolev spaces and trace theorems For any open set O ⊂ Rn, let us first define the directional Sobolev space H1 θ(O) := �U ∈ L2(O) / Dθ U ∈ L2(O) �, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7) which is a Hilbert space, provided with the scalar product (U, V )H1 θ(O) := � O � Dθ U Dθ V + U V � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let us denote ∥ · ∥H1 θ(O) the induced norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We begin with the following density property, whose proof can be found in [29, Appendix 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The space C ∞ 0 (O) is dense in H1 θ(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We denote the half-space Rn + := {y ∈ Rn, yn > 0} and the half-cylinder Ω♯ := (0, 1)n−1 × R+ in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let us introduce also the sets, for a ∈ {0, 1} and for any integer i ∈ �1, n�, Σi,a = {y ∈ Rn +, yi = a} and Σ♯ i,a = {y ∈ Σi,a, yj ∈ (0, 1), j ∈ �1, n − 1�, j ̸= i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This definition is illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Note that Σ♯ n,a is bounded whereas Σ♯ i,a for i ̸= n is unbounded in the direction yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover, ∂Ω♯ = Σ♯ n,0 ∪ � n−1 � i=1 �Σ♯ i,0 ∪ Σ♯ i,1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' A trace operator can be defined from H1 θ(Rn +) on Σi,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The main idea for doing so consists in using a one-dimensional trace theorem on the θ–oriented line that starts from a point (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , zi−1, a, zi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , zn) ∈ Σi,a, to obtain an inequality which will be integrated with respect to zj, j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The 1D trace theorem which will be used is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let L ∈ R∗ + ∪ {+∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then the mapping γL : u �→ u(0) is continuous from H1(0, L) to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover, the operator norm of γL is given by ∥γL∥2 = eL + e−L eL − e−L =: [tanh L]−1 for L > 0, and ∥γ∞∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The continuity property is a classical result which can be proved by density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' By definition, ∥γL∥ := sup{|u(0)|, ∥u∥H1(0,L) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This corresponds to a constrained op- timization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Using the standard theory, this leads to introduce a Lagrange multiplier λ and to find a pair (λ, uL) ∈ C \\ {0} × H1(0, L) such that ∥uL∥H1(0,L) = 1 and ∀ v ∈ H1(0, L) λ uL(0) v(0) = � L 0 �duL dx dv dx + uL v � dx, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9) in which case, we have ∥γL∥2 = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The explicit solution of this problem leads to the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ Note that, in particular, ∥γL∥2 ∼ L→0 L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We are now able to define traces on Σi,a in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 9 (a) n = 2 Ω♯ y2 y1 Σ1,0 = Σ♯ 1,0 Σ1,1 = Σ♯ 1,1 Σ2,0 Σ♯ 2,0 (b) n = 3 Σ3,0 Σ1,0 Σ2,0 y1 y2 y3 Ω♯ Σ♯ 2,0 Σ♯ 2,1 Σ♯ 1,0 Σ♯ 1,1 Σ♯ 3,0 y1 y2 y3 Figure 4: Domains Ω♯, Σi,a and Σ♯ i,a for n = 2 (a) and n = 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Fix a ∈ {0, 1} and i ∈ �1, n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The mapping γi,a : C ∞ 0 (Rn +) → C ∞ 0 (Σi,a) defined by γi,aU = U|Σi,a extends by continuity to a linear mapping still denoted γi,a, from H1 θ(Rn +) to L2(Σi,a), and which satisfies the estimate ∀ U ∈ H1 θ(Rn +), ∥γi,aU∥2 L2(Σi,a) ≤ 1 θi ∥U∥2 H1 θ(Rn +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' One can simply prove the continuity estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10) for any function U ∈ C ∞ 0 (Rn +) and conclude using the density result of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (i) Case i ∈ �1, n − 1�: Without loss of generality, we set i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Define Γ1,a := {z = (z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , zn), (a, z) ∈ Σ1,a} ≡ Rn−1 + , where (a, z) = (a, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11) For U ∈ C ∞ 0 (Rn +) and given any z = (z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , zn) ∈ Γ1,a, consider the function ∀ x > 0, uz,θ(x) = U(x θ + (a, z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='12) As uz,θ belongs to H1(R∗ +), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4 for L = +∞ combined with an integration with respect to z ∈ Γ1,a leads to � Γ1,a |uz,θ(0)|2 dz ≤ � Γ1,a ∥uz,θ∥2 H1(R∗ +)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='13) On the other hand, let us introduce the transformation T : y �→ �(y1 − a)/θ1, y2 − (y1 − a) θ2/θ1, · · · , yn − (y1 − a) θn/θ1 �, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14) which defines a C 1–diffeomorphism with a Jacobian determinant det JT = 1/θ1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Since the inverse image {T−1(x, z), z ∈ Γ1,a, x > 0} is nothing but the polyhedron Q1,a := {y ∈ Rn +, y1 > a, yn > (y1 − a) θn/θ1} ⊂ Rn +, 10 it follows from the chain rule and from the change of variables y �→ T y that duz,θ dx (x) = Dθ U(x θ + (a, z)) and � Γ1,a ∥uz,θ∥2 H1(R∗ +) dz = 1 θ1 ∥U∥2 H1 θ(Q1,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='15) Finally, since uz,θ(0) = U(a, z2, · · · , zn), Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='15) imply ∥U∥2 L2(Σ1,a) ≤ 1 θ1 ∥U∥2 H1 θ(Q1,a) ≤ 1 θ1 ∥U∥2 H1 θ(Rn +), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='16) which is exactly the desired estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (ii) Case i = n: starting from the function uz,θ(x) := U(x θ+(z, a)) defined for x > 0 and for any z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , zn−1) with (z, a) ∈ Σn,a, the proof uses the exact same arguments as above, except the inverse image under T becomes the whole half-space Qn,a := {y ∈ Rn +, yn > a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ The previous result does not hold in general for functions which are only H1 θ in sub-domains of the half-space Rn +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In particular when it comes to the half-cylinder Ω♯, one is led to apply the one-dimensional trace theorem on segments that become smaller in the neighbourhood of the “corners”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' the intersections of two faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' To overcome this difficulty, let us consider the sets (see Figure 5) ∀ 0 < b < 1/2, Σ♯,b i,a = {y ∈ Σ♯ i,a, dist(y, ∂Σ♯ i,a) := inf z ∈ ∂Σ♯ i,a |y − z| > b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='17) Using these domains, the traces on Σ♯ i,a can be defined as locally integrable functions in the sense of the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Σ♯,b 2,1 Σ♯,b 1,0 Σ♯,b 3,0 y1 y2 y3 b Ω♯ Tn y1 y2 y3 a Ω♯ a,− Ω♯ θ y1 y2 y3 Figure 5: From left to right: Σ♯,b i,a (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='17), Tn (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='38), Ω♯ a,− (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='37), and Ω♯ θ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='41) represented for n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let a ∈ {0, 1} and i ∈ �1, n�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The mapping γ♯ i,a : C ∞ 0 (Ω♯) → C ∞ 0 (Σ♯ i,a) defined by γ♯ i,aU = U|Σ♯ i,a extends by continuity to a linear mapping still denoted γ♯ i,a, from H1 θ(Ω♯) to L2 loc(Σ♯ i,a), and which satisfies the estimate ∀ 0 < b < 1/2, ∃ Cb > 0, ∀ U ∈ H1 θ(Ω♯), ∥γ♯ i,aU∥2 L2(Σ♯,b i,a) ≤ Cb θi ∥U∥2 H1 θ(Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='18) 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Using the density result stated in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3, one only has to show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='18) for U ∈ C ∞ 0 (Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let us assume that i = 1 and a = 0, the arguments in the following extending without any difficulty to i ∈ �1, n� and a ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Define Γ♯ 1,0 := {z = (z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , zn), (0, z) ∈ Σ♯ 1,0} ≡ (0, 1)n−1 × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='19) We introduce the length function defined by ∀ z ∈ Γ♯ 1,0, λ1,0(z) := ��{θ R+(0, z)}∩Ω♯�� = sup{x > 0, x θ1 ≤ 1, x θi+zi ≤ 1 ∀ i ∈ �2, n−1�}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We deduce easily that λ1,0(z) = min � 1 θ1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' min 2≤j≤n−1 �1 − zj θj �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='20) For U ∈ C ∞ 0 (Ω♯) and z ∈ Γ♯ 1,0, we define ∀ 0 < x < λ1,0(z), uz,θ(x) = U(x θ + (0, z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='21) Since uz,θ ∈ H1�0, λ1,0(z) �, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4 and an integration with respect to z give � Γ♯ 1,0 w1,0(z) |uz,θ(0)|2 dz ≤ � Γ♯ 1,0 ∥uz,θ∥2 H1(0,γi,a(z)) dz, with w1,0(z) = tanh[λ1,0(z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='22) On the other hand, consider the C 1–diffeomorphism T given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The set Q♯ 1,0 := {T−1(x, z), 0 < x < λ1,0(z), z ∈ Γ♯ 1,0} is clearly included in Ω♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Thus, by analogy with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='16) in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5, we have from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='21), the chain rule, and the change of variables y �→ T y that � Γ♯ 1,0 w1,0(z) |U(0, z)|2 dz ≤ 1 θ1 ∥U∥2 H1 θ(Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='23) More generally, we can show that γ♯ i,a can be defined from H1 θ(Ω♯) to the weighted space L2(Σ♯ i,a, wi,a dz), where the weight wi,a is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='22) for i = 1 and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Now, the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='20) of λ1,0 implies that w1,0 degenerates at the neighbourhood of the corners zj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' However, the weight w1,0 is bounded from below on Σ♯,b 1,0 with inf (0,z)∈Σ♯,b 1,0 w1,0(z) = tanh � min � 1 θ1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' b min 2≤j≤n−1 1 θj �� > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='24) If we set Cb := [inf(0,z)∈Σ♯,b 1,0 w1,0(z)]−1 > 0, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='18) follows directly from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='23) by inte- grating with respect to {z, (0, z) ∈ Σ♯,b 1,0}, instead of Γ♯ 1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The best constant in the previous proposition necessarily blows up when b tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The above proof shows that traces could be defined on the whole faces in appropriate weighted L2-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' More details about traces in anisotropic spaces can be found in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 Green’s formulas Let us now introduce the set H1 θ,loc(Rn +) of functions which are H1 θ in any half-cylinder S ×R+ where S is a bounded open set in Rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' More rigorously, we define for any ϕ ∈ C ∞ 0 (Rn−1) the n–dimensional function ˇϕ ∈ C ∞(Rn) such that ˇϕ(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , yn−1, yn) = ϕ(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , yn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='25) Note that for any U ∈ L2 loc(Rn +), the support of ˇϕ U is bounded in the directions yj, j ̸= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Starting from this remark, we define H1 θ,loc(Rn +) := � U ∈ L2 loc(Rn +), ˇϕ U ∈ H1 θ(R+ n ) ∀ϕ ∈ C ∞ 0 (Rn−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='26) Let us introduce a 1D cut-off function χ ∈ C ∞ 0 (R) such that χ = 1 on (0, 1), from which we define ˇχ♯ ∈ C ∞ 0 (Rn) as ˇχ♯(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , yn−1, yn) = χ(y1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' χ(yn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='27) We deduce in particular that ∀ U ∈ H1 θ,loc(Rn +), U|Ω♯ = (ˇχ♯ U)|Ω♯ ∈ H1 θ(Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='28) Moreover, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5, it is obvious that we can define without any ambiguity the trace map γ♯ i,a to H1 θ,loc(Rn +) as follows ∀ U ∈ H1 θ,loc(Rn +), γ♯ i,aU := γi,a(ˇχ♯U)|Σ♯ i,a ∈ L2(Σ♯ i,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='29) For simplicity, when considering traces on Σ♯ i,a, we shall write U instead of γ♯ i,aU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We can now state the following Green’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For any U, V ∈ H1 θ,loc(Rn +), we have the Green’s formula � Ω♯ � Dθ U V + U Dθ V � dy = 1 θn � Σ♯ n,0 U V ds+ n−1 � i=1 1 θi � � Σ♯ i,1 U V ds− � Σ♯ i,0 U V ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='30) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let U, V ∈ H1 θ,loc(Rn +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' By definition, for any χ ∈ C ∞ 0 (R) such that χ = 1 on (0, 1), the functions ˇχ♯ U and ˇχ♯ V belong to H1 θ(Rn +), where ˇχ♯ is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Since Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3 ensures that C ∞ 0 (Rn +) is dense in H1 θ(Rn +), there exist two sequences (Uk)k∈N, (Vk)k∈N of functions in C ∞ 0 (Rn +), such that Uk → ˇχ♯ U and Vk → ˇχ♯ V in H1 θ(Rn +), k → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' It follows from Green’s formula for smooth functions that Uk and Vk satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='30) for any k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Passing to the limit and using the trace continuity result stated in Propsition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 imply that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='30) is satisfied by ˇχ♯ U and ˇχ♯ V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' by U and V , since ˇχ♯ = 1 in Ω♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ We next focus on functions which are periodic with respect to their (n − 1) first variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' More precisely, for any U ∈ L2(Ω♯) and any ϕ ∈ L2(Σ♯ n,0), we introduce the respective periodic extensions �U ∈ L2 loc(Rn +) and �ϕ ∈ L2 loc(Σn,0) as defined for any i ∈ �1, n − 1� by � � � a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' y ∈ Rn +, �U(y + ⃗ei) = �U(y) and �U|Ω♯ = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' s ∈ Σn,0, �ϕ(s + ⃗ei) = �ϕ(s) and �ϕ|Σ♯ n,0 = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='31) 13 An appropriate functional framework is provided by the space H1 θ,per(Ω♯) = � U ∈ L2(Ω♯), �U ∈ H1 θ,loc(Rn +) � ⊂ H1 θ(Ω♯), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='32) where the inclusion follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='28) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' If C ∞ per(Ω♯) denotes the set of smooth functions in C ∞(Ω♯) which are 1–periodic with respect to their first n − 1 variables, that is, C ∞ per(Ω♯) = � V ∈ C ∞(Ω♯), �V ∈ C ∞(Rn +) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='33) then one can show the following result by adapting classical properties of H1 functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The space C ∞ per(Ω♯) is dense in H1 θ,per(Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Note that the traces of functions in H1 θ,per(Ω♯) on Σ♯ i,a are well-defined in L2 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover, using the continuity estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10) we have γ♯ i,a ∈ L(H1 θ,per(Ω♯), L2(Σ♯ i,a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='34) One has the characterization H1 θ,per(Ω♯) = � U ∈ H1 θ(Ω♯), γ♯ i,0U = γ♯ i,1U ∀ i ∈ �1, n − 1� � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='35) where the traces of functions in H1 θ(Ω♯) are defined in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6 and the equality of traces has to be understood up to the identification of functions on Σ♯ i,0 and Σ♯ i,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' It is clear from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='35) that H1 θ,per(Ω♯) is a closed subspace of H1 θ(Ω♯), thus it is an Hilbert space when equipped with the norm of H1 θ(Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='35), we deduce the Green’s formula on H1 θ,per(Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For any U, V ∈ H1 θ,per(Ω♯), we have the Green’s formula � Ω♯ � Dθ U V + U Dθ V � dy = 1 θn � Σ♯ n,0 U V ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='36) From the Green’s formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='36), we can easily deduce the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let a > 0, and define the sets with common boundary Σ♯ n,a (see Figure 5): Ω♯ a,+ := Ω♯ ∩ {yn > a} and Ω♯ a,− := Ω♯ ∩ {yn < a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='37) Consider a function U ∈ L2(Ω♯) such that U± := U|Ω♯ a,± ∈ H1 θ,per(Ω♯ a,±), where H1 θ,per(Ω♯ a,±) is defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then U ∈ H1 θ,per(Ω♯) ⇐⇒ γ♯ n,aU+ = γ♯ n,aU−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We finish this section with a more technical Green’s formula, used in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='17, involving functions U that only belong to H1 θ(Ω♯), provided that the test function V vanishes in the neighborhood of the skeleton Tn defined by T2 = Σ♯ 2,0 and Tn = Σ♯ n,0 ∪ � n−1 � j=1 �∂Σ♯ j,0 ∪ ∂Σ♯ j,1 �� for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='38) This domain is represented in Figure 5 for n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 14 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For U ∈ H1 θ(Ω♯) and V ∈ C ∞ 0 (Ω♯ \\ Tn), the Green’s formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='30) still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Consider U ∈ H1 θ(Ω♯) and V ∈ C ∞ 0 (Ω♯ \\ Tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Since by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3, C ∞ 0 (Ω♯) is dense in H1 θ(Ω♯), there exists a sequence (Uk)k∈N of functions in C ∞ 0 (Ω♯) which tends to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' It follows from Green’s formula in Ω♯ for smooth functions that Uk and V satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='30) for any k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For 0 < b < 1/2, let Ω♯,b be the domain Ω♯,b = {y ∈ Ω♯, dist(y, Tn) := inf z ∈ Tn |y − z| > b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='39) Since V ∈ C ∞ 0 (Ω♯ \\ Tn), there exists a real number 0 < b < 1/2 such that V |Ω♯,b ∈ C ∞ 0 (Ω♯,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Consequently, for any i ∈ �1, n − 1�, the surface integral on Σ♯ i,a is reduced to the set Σ♯,b i,a defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' When k tends to +∞, we can then use the trace continuity result stated in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6 on Σ♯,b i,a, to deduce that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='30) is satisfied by U and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3 An oblique change of variables Before stating Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14 which is the main result of this section, let us introduce the change of variables in Rn +: (s, x) ∈ Rn + �→ y = (s, 0) + x θ ∈ Rn +, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='40) and denote by Ω♯ θ the image of Ω♯ by the above transformation: Ω♯ θ := {(s, 0) + x θ, s ∈ (0, 1)n−1, x > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='41) This is illustrated in Figure 5 for n = 3 and in Figure 6 for n = 2 and |θ| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The following simple lemma will be used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For any V ∈ L1(Ω♯), we have � Ω♯ θ �V (y) dy = � Ω♯ �V (y) dy, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='42) where �V ∈ L1 loc(Rn +) denotes the periodic extension of V , defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We will use the notation k = (k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , kd) ∈ Zd for a vector of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For any set O ⊂ Rn, let 1O be the indicator function of O, that is, the function which equals 1 in O and 0 elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' By density, it suffices to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='42) for V ∈ C ∞ 0 (Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' By additivity of integration, � Ω♯ θ �V (y) dy = � Rn + 1Ω♯ θ(y) �V (y) dy = � k∈Zn−1 � Ω♯+(k,0) 1Ω♯ θ(y) �V (y) dy, where the sum over k ∈ Zn−1 is finite because of 1Ω♯ θ and because V is compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We then use the change of variables z �→ z + (k, 0) which leads to � Ω♯ θ �V (y) dy = � k∈Zn−1 � Ω♯ 1Ω♯ θ(z + (k, 0)) �V (z) dz because �V is periodic = � Ω♯ � � k∈Zn−1 1Ω♯ θ−(k,0)(z) � �V (z) dz by linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='43) 15 Furthermore, by noticing that the collection of sets {Ω♯ θ −(k, 0), k ∈ Zn−1} forms a partition of Rn +, it follows that ∀ z ∈ Ω♯, � k∈Zn−1 1Ω♯ θ−(k,0)(z) = 1Rn +(z) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='44) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='43) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='44) implies that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='42) is satisfied for V ∈ C ∞ 0 (Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ The inversion of the change of variables (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='40) leads us to introduce: ∀ y ∈ Rn, sθ(y) := ˆy − (yn/θn) ˆθ ∈ Rn−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='45) so that, y = (s, 0) + x θ ⇐⇒ s = sθ(y) and x = yn/θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='46) The next proposition emphasizes the fact that through the change of variables (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='40), the differential operator Dθ simply becomes the partial derivative with respect to yn (which is obvious for smooth functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let Ψ ∈ L2(Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then the periodic function Ψθ defined as a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' y ∈ Rn +, �Ψθ(y) := �Ψ(sθ(y), yn/θn), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='47) (where �Ψ is the periodic extension of Ψ) belongs to L2(Ω♯) and ∥Ψθ∥L2(Ω♯) = � θn ∥Ψ∥L2(Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='48) Moreover, if ∂ynΨ ∈ L2(Ω♯), then Ψθ belongs to H1 θ,per(Ω♯) with directional derivative a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' y ∈ Rn +, Dθ �Ψθ(y) = ∂ �Ψ ∂yn (sθ(y), yn/θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='49) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The map (s, x) �→ (s, 0) + x θ from Σ♯ n,0 × R+ to Ω♯ θ defines a C 1–diffeomorphism with a non-vanishing Jacobian θn ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Therefore, by using the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='41) of Ω♯ θ, a change of variables as well as the property sθ((s, 0) + x θ) = s, we obtain that � Ω♯ θ |�Ψθ(y)|2 dy = θn � Σ♯ n,0 � +∞ 0 |�Ψθ((s, 0) + x θ)|2 dx ds = θn � Σ♯ n,0 � +∞ 0 |�Ψ(s, x)|2 dx ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We deduce from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='13 that Ψθ ∈ L2(Ω♯), and that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='48) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Now in order to derive the expression of Dθ �Ψθ in the sense of distributions, consider a test function Φ ∈ C ∞ 0 (Rn +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The change of variables (s, x) �→ (s, 0) + x θ combined with Fubini’s theorem for integrable functions leads to � Rn + �Ψθ(y) DθΦ(y) dy = θn � Rn−1 � +∞ 0 �Ψ(s, x) DθΦ((s, 0) + x θ) dxds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='50) Furthermore the 1D function φs,θ defined by φs,θ(x) := Φ((s, 0) + x θ) belongs to C ∞ 0 (R+) and we have [dφs,θ/dx](x) = DθΦ((s, 0) + x θ) from the chain rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Since ∂ynΨ is in L2, we 16 can integrate by parts the inner integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='50) to obtain � Rn + �Ψθ(y) DθΦ(y) dy = −θn � Rn−1 � +∞ 0 ∂Ψ ∂yn (s, x) φs,θ(x) dxds = − � Rn + ∂Ψ ∂yn (sθ(y), yn/θn) Φ(y) dy, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='51) where the last equality comes from the change of variables y �→ (sθ(y), yn/θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This gives the expression of Dθ �Ψθ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' It will be often useful to use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='49) in the form a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (s, x) ∈ Rn +, Dθ �Ψθ((s, 0) + x θ) = ∂ �Ψ ∂yn (s, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='52) The previous proposition allows in particular to deduce the surjectivity of the trace operator from H1 θ,per(Ω♯) to L2(Σ♯ n,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let ϕ ∈ L2(Σ♯ n,0), and ψ ∈ H1(R+) such that ψ(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then the periodic function defined by a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' y ∈ Rn +, Rϕ (y) := �ϕ(sθ(y)) ψ(yn/θn) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='53) belongs to H1 θ,per(Ω♯), and its trace is Rϕ|Σ♯ n,0 = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover, R defines a continuous map from L2(Σ♯ n,0) to H1 θ,per(Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3 Link with a periodic half-guide problem For any boundary data ϕ ∈ L2(Σ♯ n,0), we can now introduce U+ θ as the solution in H1 θ(Ω♯) of the half-guide problem ������������� −Dθ �µp Dθ U+ θ � − ρp ω2 U+ θ = 0, in Ω♯, U+ θ |Σ♯ n,0 = ϕ, U+ θ |Σ♯ i,0 = U+ θ |Σ♯ i,1 ∀ i ∈ �1, n − 1�, µp Dθ U+ θ |Σ♯ i,0 = µp Dθ U+ θ |Σ♯ i,1 ∀ i ∈ �1, n − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54) Note that the third equation above implies that U+ θ ∈ H1 θ,per(Ω♯), the first one implies that µp Dθ U+ θ ∈ H1 θ(Ω♯), and finally the fourth one implies that µp Dθ U+ θ ∈ H1 θ,per(Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The space of the boundary data can seem surprising compared to the Helmholtz equation with an elliptic principal part, but recall from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='16 that the trace mapping on Σ♯ n,0 is surjective from H1 θ,per(Ω♯) to L2(Σ♯ n,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' With the functional framework introduced in the previous section, we can now show that Problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54) is well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 17 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For any ϕ ∈ L2(Σ♯ n,0), Problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54) is equivalent to the variational formulation ������� Find U+ θ ∈ H1 θ,per(Ω♯) such that U+ θ |Σ♯ n,0 = ϕ and ∀ V ∈ H1 θ,per(Ω♯) such that V |Σ♯ n,0 = 0, � Ω♯ � µp Dθ U+ θ Dθ V − ρp ω2 U+ θ V � = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='55) for which Lax-Milgram’s theorem applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The variational formulation is obtained by multiplying the first equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54) by V ∈ H1 θ,per(Ω♯), and by using Green’s formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The application of the Lax-Milgram’s theorem in {V ∈ H1 θ,per(Ω♯), γn,0V = 0}, thanks to Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='16, is direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For the equivalence, as usual, one picks test functions V ∈ C ∞ 0 (Ω♯) to deduce that the solution U+ θ ∈ H1 θ,per(Ω♯) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='55) satisfies the first equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This implies that µp Dθ U+ θ ∈ H1 θ(Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The real difficulty is to show that U+ θ satisfies the fourth equation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54) or equivalently that µp Dθ U+ θ ∈ H1 θ,per(Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' According to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6, we have ∀ 1 ≤ i ≤ n − 1, µp Dθ U+ θ |Σ♯ i,a ∈ L2 loc(Σ♯ i,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Therefore, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='12 allows us to use Green’s formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='30) for U = µp Dθ U+ θ and for V ∈ C ∞ 0 (Ω♯ \\ Tn) ∩ H1 θ,per(Ω♯), where Tn is the skeleton defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' By combining this with the fact that U+ θ solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='55) and the first equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54), one obtains that for any integer i ∈ �1, n − 1�, ∀ V ∈ C ∞ 0 (Ω♯ \\ Tn) ∩ H1 θ,per(Ω♯), � � Σ♯ i,1 µp Dθ U+ θ V ds − � Σ♯ i,0 µp Dθ U+ θ V ds � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Furthermore, C ∞ 0 (Σ♯ i,0) is included in {V |Σ♯ i,0, V ∈ C ∞ 0 (Ω♯ \\ Tn) ∩ H1 θ,per(Ω♯)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In fact, any ψ ∈ C ∞ 0 (Σ♯ i,0) admits the extension Ψ : y ∈ Ω♯ �→ ψ(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , yi−1, yi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , yn), which belongs to C ∞ 0 (Ω♯ \\ Tn) ∩ H1 θ,per(Ω♯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Finally, since C ∞ 0 (Σ♯ i,0) is dense in L2(Σ♯ i,0), it is easy to show that the fourth equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54) holds and that µp Dθ U+ θ |Σ♯ i,1 ∈ L2(Σ♯ i,1) for any i ∈ �1, n − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ We now make the link between U+ θ (ϕ) and the solution of the half-line problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) that fully justifies the introduction of the half-guide problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' To do so, first, let us introduce the quasiperiodic coefficients defined for any s ∈ Rn−1 by ∀ x ∈ R, µs,θ(x) := µp �(s, 0) + x θ � and ρs,θ(x) := ρp �(s, 0) + x θ �, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='56) as well as the one-dimensional problems ������� − d dx � µs,θ du+ s,θ dx � − ρs,θ ω2 u+ s,θ = 0, in R+, u+ s,θ(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='57) Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) corresponds to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='57) taken with s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 18 y1 y2 s y x θ Ω♯ Ω♯ θ 0 Σ♯ n,0 Σ♯ n,1 C♯ 0 Σ♯ n,2 C♯ 1 Σ♯ n,3 C♯ 2 Figure 6: The half-cylinders Ω♯ and Ω♯ θ (left), and the domains C♯ ℓ and Σ♯ n,k (right) for n = 2 Under the assumptions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4), Problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='57) admits a unique solution u+ s,θ in H1(R+) for any s ∈ Rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover, u+ s,θ decays exponentially at infinity, uniformly with respect to s, that is, there exist constants α, c > 0 depending only on µ±, ρ± such that ∀ s ∈ Rn−1, ��e−α Im ω x u+ s,θ �� H1(R+) ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='58) Furthermore, thanks to the continuity of µp and ρp, we can show that u+ s,θ is continuous with respect to s, as stated in the next proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The mapping s ∈ Rn−1 �→ u+ s,θ, which associates with a real vector s the solution in H1(R+) of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='57), defines a uniformly continuous function which is periodic of period 1 in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' To show that s �→ u+ s,θ is 1–periodic in each direction, one simply has to note that since µs,θ and ρs,θ are 1–periodic with respect to each si, both u+ s,θ and u+ s+⃗ei,θ satisfy the same half-line problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Thus, by well-posedness of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='57), u+ s,θ = u+ s+⃗ei,θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Now let us prove the regularity of s �→ u+ s,θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For any s1, s2 ∈ Rn−1, by writing the variational formulations satisfied by u+ s1,θ and u+ s2,θ, and by substracting one from the other, we obtain ∀ v ∈ H1 0(R+), � R+ � µs1,θ d dx(u+ s1,θ − u+ s2,θ) dv dx − ρs1,θ ω2 (u+ s1,θ − u+ s2,θ) v � = � R+ � (µs2,θ − µs1,θ) du+ s2,θ dx dv dx − (ρs1,θ − ρs2,θ) ω2 u+ s2,θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Now choose v = u+ s1,θ −u+ s2,θ ∈ H1 0(R+) in the above equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The well-posedness of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='57), a Cauchy-Schwarz inequality applied to the right-hand side and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='58) imply that there exists a real number c > 0 independent of s and θ such that ��u+ s1,θ − u+ s2,θ �� H1(R+) ≤ c � ∥µs2,θ − µs1,θ∥∞ + ∥ρs2,θ − ρs1,θ∥∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='59) 19 The functions µp and ρp are continuous and 1–periodic in each direction: from Heine-Cantor theorem, they are uniformly continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let us define the modulus of uniform continuity ∀ µ ∈ C 0(Rn), ∀ ε > 0, δ(µ, ε) = sup y,z {|µ(y) − µ(z)|, |y − z| < ε} A function µ is uniformly continuous if δ(µ, ε) tends to 0 as ε tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='59) that ��u+ s1,θ − u+ s2,θ �� H1(R+) ≤ c � δ(µp, |s1 − s2|) + δ(ρp, |s1 − s2|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Therefore, s �→ u+ s,θ is continuous from Rn−1 in H1(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let sθ be the mapping defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='45), and �U+ θ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' �ϕ) be the periodic extension of U+ θ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ϕ) the solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then, we have a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' y ∈ Rn +, �U+ θ ( �ϕ)(y) = �ϕ �sθ(y) � u+ sθ(y),θ(yn/θn), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='60) or equivalently a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (s, x) ∈ Rn−1 × R+, �U+ θ ( �ϕ)((s, 0) + θ x) = �ϕ(s) u+ s,θ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='61) Moreover if �ϕ is continuous in the neighbourhood of 0 and satisfies �ϕ(0) = 1, then a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' x ∈ R, u+ θ (x) = �U+ θ ( �ϕ)(x θ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='62) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We begin by proving (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let us denote for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' y ∈ Rn +, U1(y) the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Note that Ψ : (s, x) �→ �ϕ(s) u+ s,θ(x) is 1–periodic with respect to s (thanks to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='18), and belongs to L2(Ω♯) since ∥Ψ∥2 L2(Ω♯) = � Σ♯ n,0 |ϕ(s)|2 ∥u+ s,θ∥2 L2(R+) ds ≤ θn c2 ∥ϕ∥2 L2(Σ♯ n,0), with c = sup s ∥u+ s,θ∥L2(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover, since for all s, u+ s,θ ∈ H1(R+), ∂ynΨ is also in L2(Ω♯) (using similar inequalities to the above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14, U1 belongs to H1 θ,per(Ω♯) with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' y ∈ Rn +, Dθ � U1(y) = �ϕ �sθ(y) � du+ sθ(y),θ dx (yn/θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Finally, since u+ s,θ(0) = 1, it is clear that U1|Σ♯ n,0 = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' By repeating the same argument, we can show that µpDθ U1 belongs to H1 θ,per(Ω♯) with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' y ∈ Rn +, Dθ [µp Dθ � U1](y) = �ϕ �sθ(y) � d dx � µsθ(y),θ du+ sθ(y),θ dx � (yn/θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Since u+ s,θ satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='57), it is clear that U1 satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' By well-posedness of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54), we have U1 = U+ θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The equivalence between (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='60) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='61) is directly obtained using the change of variables (s, x) �→ ((s, 0) + θ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover, we have from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='18 that s �→ u+ s,θ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' If in addition to that, �ϕ is continuous in a neighbourhood of 0, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='61) becomes true for any s in that neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In particular, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='61) can be written for s = 0, thus leading to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ 20 In particular, we deduce from the above proprosition that a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' y ∈ Rn +, Dθ �U+ θ ( �ϕ)(y) = �ϕ �sθ(y) � du+ sθ(y),θ dx (yn/θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='63) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The half-guide solution U+ θ depends on ϕ whereas u+ s,θ does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In this sense, the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='60) can seem surprising at first sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Numerical results presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 will illustrate this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 4 Resolution of the half-guide problem The advantage of the lifting process lies in the periodic nature of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54), which allows us to exploit tools that are well-suited for periodic waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In this paper, we use a DtN-based method [10, 19], developed for the elliptic1 Helmholtz equation −∇ · (µp ∇U) − ρp ω2 U = 0 in unbounded periodic guides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This method does not rely on decay properties, and therefore remains robust when the absorption tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' As we essentially transpose this method to our directional Helmholtz equation, we will see below that the framework remains exactly the same, although the analysis has to be adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let us mention the recursive doubling method [32, 8], suited for bounded periodic waveguides, and a method [33] based on the Floquet-Bloch transform, although its extension to our non-elliptic equation seems unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In what follows, C♯ ℓ is the cell defined for every ℓ ∈ N by C♯ 0 = (0, 1)n and C♯ ℓ = C♯ 0 + ℓ⃗en, so that Ω♯ = � ℓ∈N C♯ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) For ℓ > 0, we call Σ♯ n,ℓ the interface between the cells C♯ ℓ and C♯ ℓ+1, that is, Σ♯ n,ℓ = Σ♯ n,0 + ℓ⃗en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' By periodicity, each cell C♯ ℓ can be identified to C♯ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Similarly, each interface Σ♯ n,ℓ can be identified to Σ♯ n,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The cells and interfaces are represented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 Structure of the solution The solution U+ θ (ϕ) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54) has a particular structure that we explain in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Denote by P ∈ L �L2(Σ♯ n,0) � the operator ∀ ϕ ∈ L2(Σ♯ n,0), Pϕ := U+ θ (ϕ)|Σ♯ n,1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2) where L2(Σ♯ n,1) and L2(Σ♯ n,0) have been identified to each other in an obvious manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This identification will be used systematically in what follows, even if not mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Note that the operator P is well-defined, due to the continuity of the trace operator on Σ♯ i,a (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For any ϕ in L2(Σ♯ n,0), we have ∀ ℓ ∈ N, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' y ∈ Ω♯, U+ θ (ϕ)(y + ℓ⃗en) = U+ θ (Pℓϕ)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3) Moreover, the spectral radius of P is strictly less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 1By elliptic Helmholtz equation, we refer to the Helmholtz equation with an elliptic principal part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We only present the outline of the proof, which is quite similar to the one in [10, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Given ϕ ∈ L2(Σ♯ n,0), consider the function U1 defined in Ω♯ by U1(y) = U+ θ (ϕ)(y + ⃗en) for almost any y ∈ Ω♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Since the coefficients µp and ρp are periodic, one deduces that U1 satisfies the volume equation as well as the periodicity condition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Furthermore, U1|Σ♯ n,0 = U+ θ (ϕ)|Σ♯ n,1 = Pϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Thus, by well-posedness of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3) for ℓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The result (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3) for ℓ ≥ 2 is proved by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' It remains to show that the spectral radius is strictly less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' To this end, by analogy with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='58), one can show the existence of constants α, c > 0 such that ∀ ϕ ∈ L2(Σ♯ n,0), ��eα Im ω yn/θn U+ θ �� H1 θ(Ω♯) ≤ c ∥ϕ∥L2(Σ♯ n,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4) Since Pℓϕ = U+ θ (ϕ)(·, ℓ), the estimate above implies that ∥Pℓ∥ ≤ c e−α Im ω ℓ/θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Hence, using Gelfand’s formula [26, §10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3], the spectral radius can be estimated as follows: ρ(P) = lim ℓ→+∞ ∥Pℓ∥1/ℓ ≤ e−β Im ω/θn < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ The operator P is called the propagation operator, as it describes how the solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54) evolves from one interface to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Provided that P is known, the solution U+ θ (ϕ) may be constructed using local cell problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let us first introduce the appropriate functional framework in a periodicity cell H1 θ,per(C♯ 0) := � U ∈ H1 θ(C♯ 0), �U ∈ H1 θ,loc(B0) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5) where B0 := Rn + ∩ {0 < yn < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Similarly to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1, one can show that any function of H1 θ,per(C♯ 0) has a L2 trace on the boundary of C♯ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We can prove in particular that H1 θ,per(C♯ 0) = � U ∈ H1 θ(C♯ 0) / U|yi=0 = U|yi=1, ∀ i ∈ �1, n − 1� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We can now introduce the local cell problems: for all ϕ ∈ L2(Σ♯ n,0), for j ∈ {0, 1}, let Ej(ϕ) ∈ H1 θ,per(C♯ 0) satisfy ����� −Dθ �µp Dθ Ej� − ρp ω2 Ej = 0, in C♯ 0, µp Dθ Ej|yi=0 = µp Dθ Ej|yi=1 ∀ i ∈ �1, n − 1�, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6) defined for j = 0, 1, with the boundary conditions ����� E0|Σ♯ n,0 = ϕ and E0|Σ♯ n,1 = 0, E1|Σ♯ n,0 = 0 and E1|Σ♯ n,1 = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7) A variational formulation can be derived as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='17, and the well-posedness follows once again from a lifting argument (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14) combined with Lax-Milgram’s theorem in H1 θ,per(C♯ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 22 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 implies that U+ θ (ϕ)(· + ℓ⃗en)|Σ♯ n,0 = Pℓϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Hence, if the propagation operator P is known, by linearity, the solution of the half-guide problem can be entirely constructed cell by cell as follows: ∀ ℓ ∈ N, U+ θ (ϕ)(· + ℓ⃗en)|C♯ 0 = E0(Pℓϕ) + E1(Pℓ+1ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 Characterization of the propagation operator: the Riccati equation In the sequel, ⟨·, ·⟩ denotes the canonical L2 scalar product on Σ♯ n,0 (or equivalently on Σ♯ n,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In order to characterize the propagation operator P, it is useful to introduce the local DtN operators T jk ∈ L(L2(Σ♯ n,0)), defined for j, k = 0, 1 by ∀ ϕ ∈ L2(Σ♯ n,0), T jkϕ = (−1)k+1 θn � µp Dθ Ej(ϕ) � |Σ♯ n,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9) where Ej(ϕ) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' By Green’s formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='30), note that for all j, k = 0, 1 and for (ϕ, ψ) ∈ L2(Σ♯ n,0)2, these operators satisfy � T jkϕ, ψ � = � C♯ 0 � µp Dθ Ej(ϕ) Dθ Ek(ψ) − ρp ω2 Ej(ϕ) Ek(ψ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10) Before deriving other useful properties of the local DtN operators, we need to introduce some additional notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For any closed operator A ∈ L(L2(Σ♯ n,0)), we denote A∗ the adjoint of A, and A its « complex conjugate », that is, ∀ ϕ ∈ L2(Σ♯ n,0), Aϕ = Aϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' It is not difficult to see that A∗ = A∗, and A = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The local DtN operators T jk satisfy � T 00�∗ = T 00, � T 11�∗ = T 11, � T 01�∗ = T 10, � T 10�∗ = T 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11) Furthermore, the operators T 00, T 11, and T 00 + T 11 are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11) follows from Green’s formula applied to Ej(ϕ) and Ek(ψ), see for instance [10, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4] in the case of the Helmholtz equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The operators T 00, T 11, and T 00 + T 11 are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We are going to show that they are also coercive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Their invertibility will then follow from Lax-Milgram’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' From the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10), one has the existence of a constant c ≡ c(µ−, ρ−, |ω|) > 0 such that −|ω| Im � 1 ω � T kkϕ, ϕ �� ≥ c Im ω ∥Ek(ϕ)∥2 H1 θ(C♯ 0) ≥ ˜c Im ω ∥ϕ∥2 L2(Σ♯ n,0), since from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='34), the trace application from H1 θ,per(C♯ 0) to L2(Σ♯ n,0) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' It follows that the operators T 00 and T 11 are coercive, and therefore invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The inequalities above summed for k = 0, 1 imply the coercivity and hence the invertibility of T 00 +T 11 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ 23 As seen earlier, the solution of the half-guide problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54) is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Now let us use the characterization of H1 per,θ(Ω♯), namely, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11 with a = 1, so that Ω♯ a,− = C♯ 0 and Ω♯ a,+ = Ω♯ \\ C♯ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Since µp Dθ U+ θ (ϕ) belongs to H1 θ,per(Ω♯), the directional derivative of U+ θ (ϕ) is continuous across the interface Σ♯ n,1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' � µp Dθ U+ θ (ϕ) � |Σ♯ n,1 = � µp Dθ U+ θ (ϕ)((· + ⃗en) � |Σ♯ n,0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='12) or equivalently, � µp Dθ E0(ϕ) � |Σ♯ n,1 + � µp Dθ E1(Pϕ) � |Σ♯ n,1 = � µp Dθ E0(Pϕ) � |Σ♯ n,0 + � µp Dθ E1(P2ϕ) � |Σ♯ n,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='13) By using the definition of the local DtN operators T jk, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='13) leads to the following charac- terization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The propagation operator P defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2) is the unique solution of the constrained Riccati equation ������ Find P ∈ L(L2(Σ♯ n,0)) such that ρ(P) < 1 and T 10P2 + (T 00 + T 11) P + T 01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The proof is identical to the one for the elliptic Helmholtz equation [19, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We know from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 that P has a spectral radius which is strictly less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='13) ensures that P satisfies the Riccati equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In order to prove the uniqueness, let us consider an operator P1 which satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The function defined cell by cell by ∀ ϕ ∈ L2(Σ♯ n,0), ∀ ℓ ∈ N∗, U1(ϕ)(· + ℓ⃗en)|C♯ 0 = E0(Pℓ 1ϕ) + E1(Pℓ+1 1 ϕ), solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54) in each cell Cℓ and is continuous across each interface Σ♯ n,ℓ, by definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7) of E0 and E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11, U1 is locally H1 θ in Ω♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover, since P1 satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14), the directional derivative µpDθ U1 is continuous across each interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Thus, using Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11, we deduce that U1 satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54) in Ω♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Furthermore, given that ρ(P1) < 1, Gelfand’s formula and the well-posedness of the cell problems ensure that there exist positive constants c, ρ∗, with ρ∗ < 1 such that, for ℓ ∈ N large enough, ∥U1(ϕ)∥H1 θ(C♯ ℓ) ≤ c ρℓ ∗ ∥ϕ∥L2(Σ♯ n,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Hence U1(ϕ) belongs to H1 θ,per(Ω♯) and satisfies the half-guide problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' By well- posedness of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54), U1(ϕ) and U+ θ (ϕ) coincide, and thus have the same trace on Σ♯ n,1, that is P1ϕ = Pϕ for any ϕ ∈ L2(Σ♯ n,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ 24 As a consequence, the propagation operator can be obtained by solving the Riccati equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14), and by choosing the unique solution whose spectral radius is strictly less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' One important thing to retain from the above is that both the propagation operator and the solution of the half-guide problem only require the computation of E0, E1, and the operators T 00, T 10, T 01, and T 11, which involve problems defined on a periodicity cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' However, the resolution of the constrained Riccati equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14) is not obvious at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The properties of this equation are investigated in further details in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3 The DtN operator and the DtN coefficient The goal of this part is to see how the half-guide problem and the local cell problems can be used to compute the DtN coefficient λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We recall that λ+ = −µθ(0) du+ θ dx (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Therefore, it is natural to introduce the DtN operator Λ ∈ L(L2(Σ♯ n,0)) defined by ∀ ϕ ∈ L2(Σ♯ n,0), Λϕ := −θn � µp Dθ U+ θ (ϕ) � |Σ♯ n,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='15) This operator also has the following properties, whose proof is exactly identical to the one of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' One has Λ∗ = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover, Λ and Λ + T 11 are invertible operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Taking the directional derivative of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8) (for ℓ = 0) on Σ♯ n,0 and using the definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9) of the local DtN operators T 00 and T 10 leads to Λ = T 00 + T 10P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='16) Besides, by writing the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='63) after multiplication by µp, and by evaluating it for y = (s, 0), so that sθ(y) = s, we obtain Λϕ(s) = θn λθ(s) ϕ(s), with λθ(s) = − � µs,θ du+ s,θ dx � (0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='17) namely, Λ is a multiplication operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='17) the DtN coefficient λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The function λθ : Rn−1 → C defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='17) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover, if ϕ ∈ Cper(Rn−1) is a given function which satisfies ϕ(0) = 1, then we have λ+ = λθ(0) = 1 θn (Λϕ)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='18) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Using Green’s formula, we have that for all s ∈ Rn−1 λθ(s) = as(u+ s,θ, u+ s,θ), with as(u, v) = � R+ � µs,θ du dx dv dx − ρs,θ ω2 u v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The continuity of u �→ as(u, u) results directly from the properties of the coefficients µp and ρp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='18 ensures that the function s �→ u+ s,θ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Therefore, as the composition of these two functions, λθ is also continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' If in addition ϕ is continuous, then Λϕ is also continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Hence, (Λϕ)(0) = θn λθ(0)ϕ(0) which yields the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4 Spectral properties of the Riccati equation We now present some properties regarding Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' These properties will be exploited for the numerical resolution of the Riccati equation, by constructing the operator P from its eigenpairs (this will be done in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3 after space discretization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For this reason, it is worhwhile to reformulate a spectral version (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7) of the Riccati equation that would characterize these eigenpairs, while taking into account the spectral radius constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This is precisely the purpose of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Recall that T (P) = 0, where T is the bounded operator defined by ∀ X ∈ L �L2(Σ♯ n,0) �, T (X) = T 10X2 + (T 00 + T 11)X + T 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='19) In the sequel, we will write T (λ) for T (λI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We begin with the following factorization lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let P be the propagation operator defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For any number λ ∈ C, T (λ) = (λP∗ − I) (Λ + T 11) (P − λ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='20) where T 11 is defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9) and Λ is defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let λ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Since the propagation operator satisfies T (P) = 0, one obtains that T (λ) = T (λ) − T (P) = � T 10(λ + P) + T 00 + T 11� (λ − P) = (λT 10 + Λ + T 11) (λ − P), from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='21) We use once again the fact that T (P) = 0 which, by the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='16), is equivalent to T 01 = −(Λ + T 11) P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' By transposing this equation, and by taking the complex conjugate, one obtains that [T 01]∗ = −P∗ (Λ + T 11)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Since �T 11�∗ = T 11 and �T 01�∗ = T 10 as ensured by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2, and since Λ∗ = Λ from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4, it follows that T 10 = −P∗ (Λ + T 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Inserting this expression of T 10 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='21) therefore leads to T (λ) = � −λP∗ (Λ + T 11) + Λ + T 11� (λ − P) = (I − λP∗) (Λ + T 11) (λ − P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' which is the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ The previous factorization lemma allows one to characterize the spectrum of the propagation operator as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For any complex number λ, one has λ ∈ σ(P) ⇐⇒ 0 ∈ σ �T (λ) � and |λ| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='22) 26 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proving (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='22) amounts to showing that for any λ ∈ C such that |λ| < 1, P − λ is invertible if and only if T (λ) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' To this end, using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6, it is sufficient to prove that (λP∗ − I) (Λ + T 11) is an invertible operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4 ensures the invertibility of Λ + T 11 already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' It thus remains to show that λP∗ − I is invertible, which is true when |λ| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Indeed, if λ = 0, then λP∗−I = −I is obviously invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Otherwise, it is not difficult to see that P and P∗ have the same spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Hence, given that |1/λ| > 1 > ρ(P∗), it follows that 1/λ does not belong to σ(P∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In other words, P∗ −(1/λ) I is an invertible operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Note that the property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='22) can be proved easily (and without Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6) for the point spectrum: λ ∈ σp(P) ⇐⇒ 0 ∈ σp �T (λ) � and |λ| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='23) This property was already proved in [19] for the Helmholtz equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In this context, this was sufficient since the operator P was compact, which is no longer the case here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Finally, it is worth noting that the values λ ̸= 0 for which 0 ∈ σ �T (λ) � can be paired in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For any complex number λ ̸= 0, one has the following equivalence: 0 ∈ σ �T (λ) � ⇐⇒ 0 ∈ σ �T (1/λ) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='24) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let λ ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' From the properties of the local DtN operators (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2), we deduce that [T (λ)]∗ = λ2 T 01 + λ(T 00 + T 11) + T 10 = λ2 T (1/λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='25) The operators T (λ) and [T (λ)]∗ have the same spectrum, hence the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' As Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9 shows, the values λ ̸= 0 for which 0 ∈ σ �T (λ) � come by pairs (λ, λ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' From a numerical point of view, it suffices to choose λ such that |λ| < 1 and discard λ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 Spectral properties of the propagation operator This section, contrary to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4 is not related to the construction of our numerical method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' it is of theoretical interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' On one hand, the result of this section, that is Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11, is useful for interpreting some of the numerical results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' On the other hand, it emphasizes the differences between the spectral properties of P, and the ones of the corresponding operator for classical waveguide problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For the elliptic Helmholtz equation, P is compact (see [19, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1]) and its spectrum hence consists only in isolated eigenvalues which accumulate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' However, the picture is completely different in this case, because the spectrum has no isolated points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 27 One useful way to study the properties of the propagation operator (especially its spectrum) is through an analytic formula: according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='60), P can be expressed for all ϕ in L2(Σ♯ n,0) and for s ∈ Rn−1 as Pϕ(s) = pθ(s) �ϕ �s − δ �, with pθ(s) = u+ s−δ,θ(1/θn) and δ = ˆθ /θn ∈ Rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='26) Note that since θ is an irrational vector, δ is also an irrational vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The properties of the mapping s �→ u+ s,θ stated in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='18 imply that the fonction pθ is continuous and 1-periodic in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Operators that can be written under the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='26) are known as weighted shift operators, and have been studied for instance in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In particular, the spectral properties of P are given by the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let pθ : Σ♯ n,0 → C be the function defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then, pθ(s) ̸= 0 for all s in Σ♯ n,0, and the spectral radius of P is given by ρ(P) = exp �� Σ♯ n,0 log |pθ(s)| ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='27) Moreover, the spectrum of P is a circle of radius ρ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This result can be found in [2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1] for n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We give below the proof for n > 2, which requires the following lemma (see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 and Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 of [21]), known as a particular case of Birkhoff’s ergodic theorem for continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let ψ : Σ♯ n,0 → C be continuous and 1–periodic in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let α ∈ Rn−1 be an irrational vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then, we have the following uniform convergence: lim ℓ→+∞ ���1 ℓ ℓ−1 � m=0 ψ(· − mα) − � Σ♯ n,0 ψ ��� ∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let us first show by contradiction that pθ or equivalently the function s �→ u+ s,θ(1/θn) is nowhere vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' To do so, we use an argument of unique continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In fact, assume that there exists s ∈ Σ♯ n,0 such that u+ s,θ(1/θn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then u+ s,θ satisfies the problem − d dx � µs,θ du+ s,θ dx � − ρs,θ ω2 u+ s,θ = 0, in (1/θn, +∞), and u+ s,θ(1/θn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' From the well-posedness of this problem, it follows that u+ s,θ = 0 in (1/θn, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Therefore, by unique continuation, one deduces that u+ s,θ = 0 in R+, which contradicts the boundary condition u+ s,θ(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We now establish the expression of the spectral radius ρ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' One has ρ(P) = lim ℓ→+∞ ∥Pℓ∥1/ℓ from Gelfand’s formula, and by induction, Pℓ can be expressed under the form Pℓϕ(s) = p(ℓ) θ (s) ϕ(s − ℓδ), with p(ℓ) θ (s) = ℓ−1 � m=0 pθ(s − mδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 28 Since the translation operator ϕ �→ ϕ(· − ℓδ) is isometric and bijective, the norm of Pℓ is equal to the norm of the multiplication operator ϕ �→ p(ℓ) θ ϕ, that is ∥p(ℓ) θ ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Hence, given that pθ(s) ̸= 0 for all s, one has ρ(P) = lim ℓ→+∞ ��� ℓ−1 � m=0 pθ(· − mδ) ��� 1/ℓ ∞ = lim ℓ→+∞ exp ���1 ℓ ℓ−1 � m=0 log �|pθ(· − mδ)| ���� ∞ Since θ is an irrational vector, δ = ˆθ/θn is also an irrational vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Therefore, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='12 can be applied with α = δ, and ψ : s �→ log |pθ(s)|, which is well-defined and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Hence the spectral radius is given by ρ(P) = Mlog(pθ) := exp �� Σ♯ n,0 log |pθ(s)| ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let us now characterize the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' To begin, note that the inverse of P is well-defined, since pθ vanishes nowhere: for all ϕ ∈ L2(Σ♯ n,0), P−1ϕ(s) := pθ(s)−1 �ϕ �s + δ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Therefore, all the computations above can be applied to P−1, thus yielding ρ(P−1) = Mlog(p−1 θ ) = 1 Mlog(pθ) = 1 ρ(P) Since the spectrum of P is always included in the annulus ρ(P−1)−1 ≤ |z| ≤ ρ(P), it follows that σ(P) is included in the circle |z| = ρ(P) = Mlog(pθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Conversely, for k ∈ Zn−1, let Sk be the multiplication operator by s ∈ Rn−1 �→ exp(2iπ k · s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' From the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='26) of the propagation operator, we obtain that Sk P S−1 k = e2iπ k · δ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The operators P and e2iπk · δ P are similar, and thus have the same spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Now consider an element λ0 of σ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then, |λ0| = Mlog(pθ), and λk := e2iπk · δ λ0 also belongs to σ(P) for all k ∈ Zn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Since δ is irrational, we have from Kronecker’s theorem (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2) that the set {λk, k ∈ Zn−1} is dense in the circle |z| = |λ0| = Mlog(pθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Consequently, this whole circle is included in the spectrum, since the latter is a closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' ■ 5 Resolution algorithm and discretization issues for n = 2 In order to compute the solution of Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1), the previous sections provide an algorithm which sums up as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Compute the solution u+ θ of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8) and the DtN coefficient λ+ defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7) by using the following procedure: (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' for any boundary data ϕ ∈ L2(Σ♯ n,0), compute the solutions E0(ϕ), E1(ϕ) of the local cell problems (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' compute the local DtN operators (T 00, T 01, T 10, T 11), defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' compute the propagation operator P as the unique solution of the constrained Riccati equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 29 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' using an arbitrarily chosen boundary data ϕ ∈ Cper(Rn−1) which satisfies ϕ(0) = 1, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8), construct the solution U+ θ of the half-guide problem cell by cell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' deduce the half-line solution u+ θ via the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='62);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' compute the DtN operator Λ defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='16), and deduce λ+ from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Compute the solution u− θ of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8) and the DtN coefficient λ− defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7) by using exactly the same procedure as in Step 1 (but independently from Step 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Finally, solve the interior problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9) in (−a, a), and extend the solution everywhere by using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10), as well as Step 1 and Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For convenience sake, the quasiperiodicity order is set to n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The most original aspects of the algorithm are the steps (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='a)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='d), and the rest of this section focuses on the discretiza- tion of these four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We present in Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 two different methods that are linked to a choice of discretization of the step (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='a), which influences the implementation of the steps (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='b) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The treatment of the step (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='c) is independent of this choice, and will be presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' per per Figure 7: Two-dimensional mesh for the 2D method (left), and family of one-dimensional meshes for the quasi-1D method (right) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 A fully two-dimensional method The first method is inspired from the resolution of the elliptic Helmholtz equation (see [10] for instance), and consists in solving directly the local cell problems on an unstructured mesh of the periodicity cell C♯ 0 = (0, 1)2 (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We start from a triangular mesh Th(C♯ 0) of C♯ 0 = (0, 1)2 with a mesh step h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We assume that this mesh is periodic, in the sense that one can identify the mesh nodes on the boundary yi = 0 with those on yi = 1, for 1 ≤ i ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In particular for i = 1, this condition allows us to handle the periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Now let Vh(C♯ 0) be the usual H1–conforming approximation by Lagrange finite elements of order d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We also introduce Vh,per(C♯ 0) := �V ∈ Vh(C♯ 0) / V |y1=0 = V |y1=1 � as an internal approximation of H1 θ,per(C♯ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Finally, to approximate L2(Σ♯ 2,0) and L2(Σ♯ 2,1), we consider the following subspaces: ∀ a ∈ {0, 1}, Vh,per(Σ♯ 2,a) = �Vh|Σ♯ 2,a / Vh ∈ Vh,per(C♯ 0) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 30 Since the mesh nodes on Σ♯ 2,0 and Σ♯ 2,1 can be identified to each other by periodicity of Th(C♯ 0), we can also make the identification Vh,per(Σ♯ 2,0) ≡ Vh,per(Σ♯ 2,1) ≡ Vh,per(0, 1), as in the continuous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Set N := dim Vh,per(0, 1), and consider a basis (ϕp)1≤p≤N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For any data ϕh ∈ Vh,per(0, 1), we denote by E0 h(ϕh), E1 h(ϕh) ∈ Vh,per(C♯ 0) the solutions of the discrete counterpart of the local cell problems (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7) defined in a weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In practice, one has to compute Ej h(ϕp), where (ϕp)1≤p≤N is a basis of Vh,per(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Similarly to the weak expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10) of the continuous local DtN operators, the discrete local DtN operators T jk h ∈ L(Vh,per(0, 1)), j, k = 0, 1, are defined for any ϕh, ψh ∈ Vh,per(0, 1) as follows: � T jk h ϕh, ψh � = � C♯ 0 � µp Dθ Ej h(ϕh) Dθ Ek h(ψh) − ρp ω2 Ej h(ϕh) Ek h(ψh) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In practice, these operators are represented as N × N matrices Tjk whose components are given by Tjk pq = �T jk h ϕq, ϕp �, for p, q ∈ �1, N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let ϕh ∈ Vh,per(0, 1) ⊂ Cper(R) such that ϕh(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The computation of the propagation operator Ph ∈ L(Vh,per(0, 1)) is presented in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Once this operator is determined, the solution of the half-guide problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54) can be approximated with the function defined cell by cell by ∀ ℓ ∈ N, U+ θ,h(ϕh)(· + ℓ⃗en)|C♯ 0 = E0 h(Pℓ h ϕh) + E1 h(Pℓ+1 h ϕh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Finally, a suitable approximation of the solution of the half-line problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 is provided by ∀ x ∈ R, u+ θ,h(x) = U+ θ,h(ϕ)(θ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 A quasi one-dimensional method Though easy to implement, the two-dimensional approach described in the previous section does not exploit the fibered properties of the directional derivative Dθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' However, the periodic half-guide problem can be seen as a concatenation in a certain sense of one-dimensional half- line problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This fibered structure is the core of the method presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 Presentation For any s ∈ R, we consider the one-dimensional cell problems ���������� − d dx � µs,θ dej s,θ dx � − ρs,θ ω2 ej s,θ = 0, in (0, 1/θ2) := Iθ, e0 s,θ(0) = 1 and e0 s,θ(1/θ2) = 0, e1 s,θ(0) = 0 and e1 s,θ(1/θ2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) Then, by analogy with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='19, one easily shows that the local cell problems are concatenations of one-dimensional cell problems, in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 31 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For any boundary data ϕ in L2(0, 1), the solutions E0(ϕ) and E1(ϕ) of the local cell problems (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6) are given by a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' y ∈ C♯ 0, Ej(ϕ)(y) = �ϕ �sθ(y) + j θ1/θ2 � ej sθ(y),θ �y2 θ2 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2) where ej s,θ denotes the solution of the cell problems (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 also highlights the structure of the local DtN operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' To see this, let us introduce the local DtN functions tjk θ defined for j, k = 0, 1, by ∀ s ∈ R, tjk θ (s) = (−1)k+1θ2 � µs,θ dej s,θ dx �� j θ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3) Note that by periodicity of µp and ρp, the maps s �→ ej s,θ and tjk θ are 1–periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' By applying the directional derivative operator Dθ to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2), and by using the relationship between Dθ Ej(ϕ) and dej s,θ/dx given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='52), it follows that the local DtN operators defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9) are weighted translation operators, similarly to the propagation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The operators T jk can be written for ϕ ∈ L2(0, 1) and s ∈ (0, 1) as T 00ϕ(s) = t00 θ (s) �ϕ(s) and T 10ϕ(s) = t10 θ (s) �ϕ(s + θ1/θ2), T 11ϕ(s) = t11 θ (s − θ1/θ2) �ϕ(s) and T 01ϕ(s) = t01 θ (s − θ1/θ2) �ϕ(s − θ1/θ2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4) where we recall that �ϕ denotes the periodic extension of ϕ on R, defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Finally, the solution u+ θ of the half-line problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) can be computed directly from the functions ej s,θ and from the propagation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In fact, given a function ϕ ∈ Cper(Σ♯ n,0) such that ϕ(0) = 1, taking formally the trace along θ R in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8) leads to ∀ ℓ ∈ N, u+ θ (· + ℓ/θ2)|Iθ = ( � Pℓϕ)(ℓ θ1/θ2) e0 ℓθ1/θ2,θ + ( � Pℓ+1ϕ)((ℓ + 1) θ1/θ2) e1 ℓθ1/θ2,θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5) The proof of this result is similar to those of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8) and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Expressions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5) form the basis of the quasi one-dimensional or quasi-1D strategy, which consists in approximating the solutions ej s,θ as well as the functions tjk θ and finally the local DtN operators T jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then once the propagation operator is computed by solving the constrained Riccati equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14), the solution u+ θ may be constructed directly cell by cell using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 Discretization The quasi-1D approach requires two distinct approximate spaces associated to the transverse and the θ–oriented directions (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 32 Transverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We begin with a one-dimensional mesh Th(0, 1) of Σ♯ 2,0 ≡ (0, 1) with a mesh step h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let Vh(0, 1) be the approximation space of H1(0, 1) by Lagrange finite elements of order d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We denote by (ϕp)0≤p≤N the usual nodal basis, which satisfies in particular ϕp(sq) = δp,q, where (sp)0≤p≤N are points (including the mesh vertices) such that 0 = s0 < · · · < sN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then an internal approximation of L2(0, 1) is Vh,per(0, 1) := Span{ϕ0 + ϕN, ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' , ϕN−1}, which is chosen so that Vh,per(0, 1) ⊂ Cper(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In particular, from the definition of the basis functions ϕi, one has the following decomposition ∀ϕh ∈ Vh,per(0, 1), ϕh = N � p=0 ϕh(sp) ϕp, with ϕh(s0) = ϕh(sN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6) θ–oriented direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Let Thθ(Iθ) denote a mesh of the line segment Iθ with a mesh step hθ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Set Vhθ(Iθ) as the approximation space of H1(Iθ) by Lagrange finite elements of order dθ > 0 and define Vhθ,0(Iθ) := Vhθ(Iθ) ∩ H1 0(Iθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The approximation of e0 s,θ and e1 s,θ can be seen as a two-step process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' First, for any s ∈ R, consider the solution ej s,θ,hθ of the discrete variational formulation associated to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In practice, the solution ej s,θ,hθ can only be computed for a finite number of s ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This is where the discretization in the transverse direction comes into play: given x ∈ Iθ, the function s �→ ej s,θ,hθ(x) may be interpolated in Vh,per(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The interpolation process requires to compute the discrete solution ej s,θ,hθ only for s = sp, p ∈ �0, N − 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then, using the decomposition formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6), ej s,θ shall be approximated by ∀ (s, x) ∈ (0, 1) × Iθ, ej s,θ,h(x) = N � p=0 ej sp,θ,hθ(x) ϕp(s), with h = (h, hθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7) where ej 0,θ,hθ = ej 1,θ,hθ (because ej s,θ is 1–periodic with respect to s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' From the solutions ej s,θ,h, we introduce the discrete local DtN functions ∀ s ∈ (0, 1), tjk θ,h(s) = θn � 1/θn 0 � µs,θ dej s,θ,h dx dek s,θ,h dx − ρs,θ ω2 ej s,θ,h ek s,θ,h � , which are inspired from the weak expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3) of the local DtN functions tjk θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then, by analogy with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4), we define the discrete DtN operators T jk h ∈ L(Vh,per(0, 1)) for any ϕh, ψh ∈ Vh,per(0, 1) as follows: � T jk h ϕh, ψh � = � 1 0 tjk θ,h(s − k θ1/θ2) ϕh(s + (j − k) θ1/θ2) ψh(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8) These discrete DtN operators, when computed for ϕh, ψh being the basis functions of Vh,per(0, 1), are represented as N × N matrices, where N = dim Vh,per(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The integrals 33 which appear in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8) are evaluated in practice using a specifically designed quadrature rule whose description is omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Finally, let ϕh ∈ Vh,per(0, 1) ⊂ Cper(R) such that ϕh(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5), the solution of the half-line problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) can be approximated with the function defined cell by cell by ∀ ℓ ∈ N, u+ θ,h(· + ℓ/θ2)|Iθ = (Pℓ hϕh)(ℓ θ1/θ2) e0 ℓθ1/θ2,θ,h + (Pℓ+1 h ϕh)((ℓ + 1) θ1/θ2) e1 ℓθ1/θ2,θ,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' where Ph ∈ L(Vh,per(0, 1)) corresponds to a suitable discrete RN×N approximation of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The computation of such an operator is the subject of the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3 Approximation of the propagation operator In order to find a suitable approximation Ph ∈ L(Vh,per(0, 1)) of the propagation operator P, it is natural to introduce the discrete constrained Riccati equation ������ Find Ph ∈ L(Vh,per(0, 1)) such that ρ(Ph) < 1 and Th(Ph) = 0, where Th(Ph) := T 10 h P2 h + (T 00 h + T 11 h ) Ph + T 01 h , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9) and where (T 00 h , T 01 h , T 10 h , T 11 h ) are obtained via one of the methods described in Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Using the same arguments as for the elliptic Helmholtz equation [10], it can be proved that this discrete equation admits a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In order to solve (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9), two methods have been proposed in [19]: a spectral decomposition method, and a modified Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Here, we only describe the spectral approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The spectral decomposition method consists in characterizing Ph by means of its eigenpairs (λi, ψi) of Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Doing so however raises an important question: is Ph completely defined by its eigenpairs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This is equivalent to wondering if Ph is diagonalizable or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The diagonaliz- ability of Ph is an open question, but for the sake of simplicity, we will assume in the sequel that this is the case, namely The family of eigenfunctions (ψi)1≤i≤N forms a basis of Vh,per(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In practice, this is the situation that we always have encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Moreover, in the case where this assumption fails to be true, one can still adapt the method, and recover Ph through a Jordan decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (See [10, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=') The spectral approach relies on the results presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4, which remain true for the discrete equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In particular, by analogy with Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7, (λh, ψh) ∈ C × Vh,per(0, 1) is an eigenpair of Ph if and only if it satisfies Th(λh) ψh = 0, with ψh ̸= 0 and |λh| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Solving the Riccati equation hence comes down to solving a quadratic eigenvalue problem: ������ Find (λh, ψh) ∈ C × Vh,per(0, 1) such that ψh ̸= 0, |λh| < 1 and λ2 h T 10 h ψh + λh (T 00 h + T 11 h )ψh + T 01 h ψh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10) 34 If one sets N = dim Vh,per(0, 1), then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10) can be reduced to a 2N × 2N linear eigenvalue problem, thus yielding 2N eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In order to pick the N eigenvalues of the propagation operator, we need a criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' To do so, note that with the 2D or the quasi-1D method, the properties of the local DtN operators (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2) remain preserved for the discrete operators T jk h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Hence Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9 admits the following discrete version: Ker Th(λh) ̸= {0} ⇐⇒ Ker Th(1/λh) ̸= {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Therefore, as already expected with Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10, the solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10) can be grouped into pairs (λh, 1/λh), where 0 < |λh| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Consequently, in order to compute Ph, one can solve (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10) (using for instance linearization techniques), and choose the N eigenpairs (λh, ψh) which satisfy |λh| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4 The DtN coefficient Finally, consider a function ϕh ∈ Vh,per(0, 1) ⊂ Cper(R) which satisfies ϕh(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Then by analogy with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='16), and in the spirit of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5, we define the discrete DtN operator and the discrete DtN coefficient as follows: Λh = T 10 h Ph + T 00 h and λ+ h = (Λhϕh)(0) θ2 , where T 10 h and T 00 h are computed using one of the methods presented in Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2, and where Ph is the solution of the discrete Riccati equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 Numerical results We present some numerical results to validate the method, to illustrate its efficiency, and to compare the multi-dimensional and the quasi one-dimensional methods in the case where the order of quasiperiodicity is set to n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Simulations will be carried out with the periodic coefficients µp and ρp, defined for y = (y1, y2) ∈ R2 by µp(y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 + cos(2πy1) cos(2πy2) and ρp(y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 sin(2πy1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 sin(2πy2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We set θ = (cos π/3, sin π/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' As the ratio θ2/θ1 = √ 3 is irrational, θ is an irrational vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For a = 1, the source term f is the cut-off function ∀ x ∈ R, f(x) = exp � 100 �1 − 1/(1 − x2) �� χ(−1,1), and the local perturbations µi and ρi are defined as piecewise constants, so that the coefficients µ and ρ of the model problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) are represented in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 35 −6 −4 −2 0 2 4 6 1 2 µ −6 −4 −2 0 2 4 6 1 2 ρ −6 −4 −2 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 f Figure 8: The locally perturbed quasiperiodic coefficients µ and ρ, and the source term f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 The half-line and the half-guide solutions The model problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) is solved by computing the solutions of the half-line problems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8), as well as the DtN coefficients λ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In this part, only results regarding the numerical resolution of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) are going to be presented, as the problem set on (−∞, −a) provides the same overall results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Error analysis In order to validate the method, we introduce for L > 0 the unique function u+ θ,L in H1(0, L) that satisfies Problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) on the truncated domain (0, L), with u+ θ,L(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Similarly, define ΩL := (0, 1)n−1 × (0, L), and for any ϕ ∈ L2(Σ♯ n,0), let U+ θ,L(ϕ) ∈ H1 θ(ΩL) denote the unique function that satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54) on ΩL, with U+ θ,L(ϕ)|y2=L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In presence of absorption, the solutions u+ θ and U+ θ (ϕ) decay exponentially at infinity (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='58) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4)), and by studying the problems satisfied by u+ θ,L − u+ θ and U+ θ,L(ϕ) − U+ θ (ϕ), it can be proved as in [11] that there exist constants α, c > 0 such that for any L > 0, ∥u+ θ,L − u+ θ ∥H1(0,L) ≤ c e−α Im ωL ∥u+ θ ∥H1(0,L) ∥U+ θ,L(ϕ) − U+ θ (ϕ)∥H1 θ(ΩL) ≤ c e−α Im ωL ∥U+ θ (ϕ)∥H1 θ(ΩL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11) with α = � ρ−/µ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In particular, if L is chosen large enough, then u+ θ,L and U+ θ,L(ϕ) can be viewed as suitable approximations of u+ θ and U+ θ (ϕ), and thus can serve as reference solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In the upcoming results, to make the truncation errors in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11) negligible with respect to the errors induced by the numerical method, we choose L so that exp � − � ρ−/µ+ Im ω L � ≤ 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='12) The corresponding solutions u+ θ,L and U+ θ,L(ϕ), which will be denoted by u+ ref and U+ ref(ϕ) respectively, are computed via P1 Lagrange finite elements, with a mesh step h = 5 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 36 In the following, the boundary data is fixed to ϕ = 1, and is omitted in the notation of U+ θ and U+ ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Also, we only plot relative errors corresponding to the 1D solution, as the errors for the 2D solution behave similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In Figure 9, the relative error ε(u+ θ ) := ∥u+ θ,h − u+ ref ∥H1(0,4/θ2) ∥u+ ref ∥H1(0,4/θ2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='13) is represented with respect to the mesh step h, and for both the 2D and the quasi-1D method (with hθ = h for the quasi-1D method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The solutions are computed using Lagrange finite elements of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' One sees that the errors tend to 0 as h at least, as expected for P1 Lagrange finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' With the quasi-1D method however, ε(u+ θ ) behaves as h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This is a special superconvergence phenomenon, which is probably due to the fact that the problems solved in practice with the quasi-1D method are one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Note also that in general, the quasi-1D method appears to be more accurate than the 2D method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='03 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='46 32 64 128 256 512 10−3 10−2 10−1 100 Relative errors −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='98 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='64 32 64 128 256 512 Discretization parameter 1/h 2D method Quasi-1D method (a) ω = 8 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='25 i (b) ω = 20 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='25 i Figure 9: Relative error in H1 norm of the half-line solution for different values of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For a fixed mesh step, the relative error increases with the real frequency Re ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This is a well- known particularity of the Helmholtz equation: since Re ω represents the spatial frequency of the time-harmonic waves, the discretization parameter h has to be adapted in order to take their oscillations into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Representation of the half-guide solution The half-guide solution is represented in Figure 10 for different values of ω, when ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 37 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 1 0 1 2 3 4 −1 0 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 1 0 1 2 3 4 −1 0 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 1 0 1 2 3 4 −1 0 1 (a) ω = 8 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='25 i (b) ω = 20 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='25 i (c) ω = 20 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='05 i Figure 10: Real part of the half-guide solution computed using the quasi-1D approach, with P1 Lagrange finite elements and h = 2 × 10−3, and for different values of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Dependence with respect to the boundary data The goal of this part is to see how the half-line and the half-guide solutions depend on the boundary data ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' To do so, we choose three different datas: ϕ1(s) = 1, ϕ2(s) = cos(2πs), and ϕ3(s) = 1 − 1[1/3,2/3](s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='14) We set ω = 8 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='25 i, and we display results obtained with the quasi-1D method, knowing that the 2D method yields the same conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The computations are carried out using P1 Lagrange finite elements, with mesh steps h = hθ = 2 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Size of periodicity cell 0 1 2 3 4 −1 0 1 ϕ1 ϕ2 ϕ3 Figure 11: Real part of the half-line solution computed using the quasi-1D approach, with P1 Lagrange finite elements and h = 2 × 10−3, and for different values of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 38 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 1 0 1 2 3 4 −1 0 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 1 0 1 2 3 4 −1 0 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 1 0 1 2 3 4 −1 0 1 (a) ϕ1 (b) ϕ2 (c) ϕ3 Figure 12: Real part of the half-guide solution computed using the quasi-1D approach, with P1 Lagrange finite elements and h = 2 × 10−3, and for different values of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' As expected, and as Figures 11 and 12a–12c show, the aspect of half-guide solution changes extensively with respect to the boundary data, whereas the half-line solution looks invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 The whole line problem The solutions u± θ of the half-line problems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='8) allow one to compute the DtN coefficients λ±, to solve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='9), and then to compute the solution u of Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Recall that the coefficients µ, ρ, and the source term f are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) is represented in Figure 13 for different values of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (a) ω = 8 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='25 i −6 −4 −2 0 2 4 6 −1 0 1 39 (b) ω = 20 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='25 i −6 −4 −2 0 2 4 6 −1 0 1 (c) ω = 20 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='05 i −6 −4 −2 0 2 4 6 −1 0 1 Figure 13: Real part of the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1) computed using the quasi-1D approach, with P1 Lagrange finite elements and h = 2 × 10−3, and for different values of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3 About the dependence with respect to the absorption We come back to the numerical resolution of Problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1), and we study the convergence of the 2D and quasi-1D methods depending on the absorption, especially when it tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' As in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1, the solutions are computed with Lagrange finite elements of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The relative error ε(u+ θ ) defined (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='13) is represented in Figure 14 for both the 2D and the quasi-1D method, and for different values of Im ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='03 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='46 32 64 128 256 512 10−3 10−2 10−1 100 Relative errors −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='07 32 64 128 256 512 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3 32 64 128 256 512 Discretization parameter 1/h 2D method Quasi-1D method (a) ω = 8 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='25 i (b) ω = 8 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='01 i (c) ω = 8 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='001 i Figure 14: Relative error in H1 norm of the half-line solution for different values of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 40 As Figure 14 shows, the error deteriorates with Im ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' It would mean that the numerical method becomes less efficient as the absorption decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This issue is closely related to the well-posedness of the local cell problems with Dirichlet boundary conditions when Im ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In fact, for the elliptic Helmholtz equation, it is known (see [10, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1] for instance) that the local cell problems are well-posed except for a countable set of frequencies which correspond to the eigenvalues of the associated differential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In our case however, as the differential operator has a non-elliptic principal part, it also has a continuous spectrum, and one can show that when µp and ρp are non-constant, the local cell problems are well- posed only for frequencies in a bounded set (that can even be empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' An alternative to avoid this problem is to use a Robin-to-Robin operator instead of the DtN operator, which would involve solving well-posed local cell problems with Robin boundary conditions, as it is done in [12] for periodic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This will be done in a forthcoming paper for quasiperiodic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4 About the spectral approximation of the propagation operator As explained in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3, the discrete propagation operator Ph is computed by means of its eigenpairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In this section, the eigenvalues of Ph are compared with the spectrum of the exact propagation operator which, according to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11, is a circle of radius Mlog(pθ) = exp � � 1 0 log |pθ(s)| ds � , with pθ(s) = u+ s−θ1/θ2,θ(1/ sin θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' To compute this radius, u+ s,θ is approximated by the unique function u+ s,θ,L that satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='57) on a truncated domain (0, L), with u+ s,θ,L(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' One can show similar estimates to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11), and if L is chosen large enough (for instance, if L satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='12)), then u+ s,θ,L can be used as a reference solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In practice, u+ s,θ,L is computed for several s, and finally the integral that defines Mlog(pθ) is evaluated using a rectangular quadrature rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The spectra of Ph and P are shown in Figure 16 for ω = 8 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='25 i, and for different values of the discretization parameter h (with hθ = h for the quasi-1D method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Figure 15 represents the number Nh of eigenvalues of Ph that are close by 5% to σ(P), namely Nh = # � λh ∈ σ(Ph) � ���� |λh| − Mlog(pθ) Mlog(pθ) ���� ≤ 5% � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='15) In Figure 15, one sees that Nh increases with 1/h, which means that more and more eigen- values of Ph are close to σ(P) when h decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In other words, a finer discretization leads to a better approximation of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The number Nh of such eigenvalues also seems to increase linearly with 1/h (up to a subsequence for the quasi-1D method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Finally, note that Nh is higher with the quasi-1D method than with the 2D method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' As Figure 16 shows, the eigenvalues of Ph are all included in the disk of radius ρ(P), but one observes some spectral pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This is a classical phenomenon when one approximates the spectrum of an operator which is neither compact nor self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' What is striking however, is that the pollution behaviours are very different depending on the method used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' On one hand, the eigenvalues obtained with the 2D approach tend to accumulate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' A likely explanation for this phenomenon is that solving the local cell problems on 2D meshes does not take their directional structure into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Since the location of the eigenvalues 41 20 40 60 80 100 120 140 160 180 200 220 240 50 100 150 200 Discretization parameter 1/h Nh 2D Quasi-1D Figure 15: Number of eigenvalues of Ph that are close by 5% to σ(P) with respect to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' −1 0 1 −1 0 1 2D method 1/h = 32 −1 0 1 1/h = 64 −1 0 1 1/h = 129 −1 0 1 1/h = 258 −1 0 1 −1 0 1 Quasi-1D method −1 0 1 −1 0 1 −1 0 1 Figure 16: Eigenvalues of the discrete propagation operator (circle-shaped markers) compared to the spectrum of the exact propagation operator (circle in dashed line) for ω = 8 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='25 i, and for different values of the discretization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 42 of Ph is similar to the one obtained in the elliptic case, for which P is compact (see [19, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1]), we believe the 2D method somehow regularizes the half-guide problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='54) by introducing an elliptic (discrete) approximation of the corresponding differential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' On the other hand, with the quasi-1D approach, the spectrum of Ph “oscillates” as the discretization parameter h tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This phenomenon has to do with the particular nature of P which is a weighted translation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' We strongly suspect that one can extract a subsequence (Ph′) whose spectrum converges towards σ(P) in a sense to be defined precisely, as it is suggested by the peaks in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The investigation of this assumption as well as the construction of such a subsequence are subject to ongoing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' With both approaches, it has been observed numerically that the eigenfunctions associated to the spurious eigenvalues were highly oscillating functions that were badly approximated by the discretization, whereas the components of the half-guide solution with respect to these eigenfunctions are very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This might explain why the spectral pollution does not have a visible influence on the approximation of the half-guide and the half-line solutions, as the errors in Figure 9 seem to suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 6 Perspectives and ongoing works A numerical method has been proposed to solve Helmholtz equation in 1D unbounded quasiperiodic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Using the presence of absorption, we justified that this equation could be lifted onto a higher-dimensional problem which, in turn, can be solved using a Dirichlet- to-Neumann approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For the discretization, we presented a multi-dimensional method, as well as a so-called quasi one-dimensional method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' As shown by numerical simulations, both methods provide a suitable approximation of the solution as long as there is absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' However, the quasi-1D method proved to be more efficient than the 2D method, as it takes the anisotropy of the problems involved into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' The method presented opens up numerous perspectives, and raises multiple questions that are subject to ongoing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' For instance, it would be interesting to approximate efficiently the spectrum of the propagation operator, even though the spectral pollution seems to have no major impact on the efficiency of the overall method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Another key extension concerns the case where the absorption tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' This extension, which will be presented in a subsequent paper, involves replacing the DtN method by a Robin-to-Robin method as explained in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1, and finding a way to characterize the propagation operator which is no longer uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Finally, an approach which is similar to the one presented in this paper can be used to study the propagation of waves in presence of a 2D periodic half-space when the interface does not lie in any direction of periodicity, or in presence of two 2D periodic half-spaces with non-commensurable periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 43 References [1] Shmuel Agmon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Spectral properties of Schrödinger operators and scattering theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 (1975), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 151– 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [2] Anatolij Antonevich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Linear functional equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Operator approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Birkhäuser, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [3] Abram Samoïlovitch Besicovitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Almost Periodic Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 1932, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 891–921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [4] Xavier Blanc, Claude Le Bris, and Pierre-Louis Lions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Local profiles for elliptic prob- lems at different scales: defects in, and interfaces between periodic structures”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Com- munications in Partial Differential Equations (Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1080/03605302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1043464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' url: https://hal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='archives-ouvertes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='fr/hal-01143193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [5] Harald Bohr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Almost periodic functions (Translated from German).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 1947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [6] Guy Bouchitté, Sébastien Guenneau, and Frédéric Zolla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Homogenization of Dielectric Photonic Quasi Crystals”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Multiscale Modeling and Simulation: A SIAM Interdis- ciplinary Journal 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 (Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 1862–1881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1137/090770333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' url: https://hal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='archives-ouvertes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='fr/hal-00544537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [7] Jean-Michel Combes and Lyn Carey Thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Communications in Mathematical Physics 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4 (1973), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 251–270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [8] Matthias Ehrhardt, Houde Han, and Chunxiong Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Numerical simulation of waves in periodic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' WIAS, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [9] Daniel Eidus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “The limiting absorption and amplitude principles for the diffraction problem with two unbounded media”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Communications in mathematical physics 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 (1986), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 29–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [10] Sonia Fliss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Analyse mathématique et numérique de problèmes de propagation des ondes dans des milieux périodiques infinis localement perturbés”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Theses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Ecole Poly- technique X, May 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' url: https://pastel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='archives- ouvertes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='fr/pastel- 00005464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [11] Sonia Fliss and Laure Giovangigli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Time harmonic wave propagation in one dimen- sional weakly randomly perturbed periodic media”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: SN Partial Differential Equa- tions and Applications 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='40 (Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' url: https://hal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='fr/hal-02504392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [12] Sonia Fliss, Patrick Joly, and Jing-Rebecca Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Exact boundary conditions for wave propagation in periodic media containing a local perturbation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [13] Martin Gardner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “MATHEMATICAL GAMES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Extraordinary nonperiodic tiling that enriches the theory of tiles”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Scientific American 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 (1977), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 110–121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' issn: 00368733, 19467087.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='org/stable/24953856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [14] David Gérard-Varet and Nader Masmoudi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Homogenization and boundary layers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Acta mathematica 209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 (2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 133–178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [15] David Gérard-Varet and Nader Masmoudi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Homogenization in polygonal domains”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Journal of the European Mathematical society 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='5 (2011), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 1477–1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 44 [16] Godfrey Harold Hardy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' An introduction to the theory of numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 6th ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Oxford university press, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [17] Vu Hoang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “The limiting absorption principle for a periodic semi-infinite waveguide”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: SIAM Journal on Applied Mathematics 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3 (2011), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 791–810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [18] Patrick Joly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Some trace theorems in anisotropic Sobolev spaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: SIAM journal on mathematical analysis 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='3 (1992), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 799–819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [19] Patrick Joly, Jing-Rebecca Li, and Sonia Fliss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Exact boundary conditions for periodic waveguides containing a local perturbation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Phys 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='6 (2006), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 945–973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [20] Andreas Kirsch and Armin Lechleiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “A radiation condition arising from the limit- ing absorption principle for a closed full-or half-waveguide problem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Mathematical Methods in the Applied Sciences 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='10 (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 3955–3975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [21] Lauwerens Kuipers and Harald Niederreiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Uniform Distribution of Sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Dover Books on Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Dover Publications, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' isbn: 9780486149998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [22] Boris Moiseevich Levitan and Vasilii Vasilévich Zhikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Almost periodic functions and differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Cambridge University Press, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [23] Yves Meyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Quasicrystals, Diophantine approximation and algebraic numbers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Beyond quasicrystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Springer, 1995, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 3–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [24] Roger Penrose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Pentaplexity A Class of Non-Periodic Tilings of the Plane”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: The Mathematical Intelligencer 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 (Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 1979), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 32–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' issn: 0343-6993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [25] Maria Radosz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “New limiting absorption and limit amplitude principles for periodic operators”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Zeitschrift für angewandte Mathematik und Physik 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='2 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 253– 275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [26] Walter Rudin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Functional Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' International series in pure and applied mathemat- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' McGraw-Hill, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' isbn: 9780070542365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [27] Marjorie Senechal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Quasicrystals and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' CUP Archive, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [28] Dan Shechtman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Metallic Phase with Long-Range Orientational Order and No Translational Symmetry”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 53 (20 Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 1984), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 1951–1953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [29] Roger Temam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Sur la stabilité et la convergence de la méthode des pas fractionnaires”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Annali di Matematica pura ed applicata 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 (1968), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 191–379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [30] Niklas Wellander, Sébastien Guenneau, and Elena Cherkaev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Homogenization of quasiperi- odic structures and two-scale cut-and-projection convergence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: (Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [31] Calvin H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' Wilcox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Wave operators and asymptotic solutions of wave propagation prob- lems of classical physics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Archive for Rational Mechanics and Analysis 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='1 (1966), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 37–76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [32] Lijun Yuan and Ya Yan Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “A recursive-doubling Dirichlet-to-Neumann-map method for periodic waveguides”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Journal of Lightwave Technology 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='11 (2007), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 3649– 3656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' [33] Ruming Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' “Numerical methods for scattering problems in periodic waveguides”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' In: Numerische Mathematik 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content='4 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 959–996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} +page_content=' 45' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AzT4oBgHgl3EQfN_u0/content/2301.01159v1.pdf'} diff --git a/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf b/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..643404f5d822e8b08ab8316793fb3fc91122e587 Binary files /dev/null and b/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf differ diff --git a/o9E2T4oBgHgl3EQf0Aih/content/tmp_files/2301.04137v1.pdf.txt b/o9E2T4oBgHgl3EQf0Aih/content/tmp_files/2301.04137v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a3c2207f7805f25d1289d58a157cd2d19b42017f --- /dev/null +++ b/o9E2T4oBgHgl3EQf0Aih/content/tmp_files/2301.04137v1.pdf.txt @@ -0,0 +1,154 @@ +A Global Radio Remote Sensing Network for Observing Space Weather Dynamics +Ryan Volz, Philip J. Erickson | MIT Haystack Observatory +Scott E. Palo | University of Colorado Boulder +Jorge L. Chau | Leibniz Institute of Atmospheric Physics at the University of Rostock Juha Vierinen | UiT Arctic University of Norway +Thomas Y. Chen | Columbia University +Introduction: The need for more and better data is a constant refrain in geospace science. Past workshops and +decadal surveys have noted a compelling need for real-time observations and a clear realization that our current +sampling of the vast geospace environment is wholly insufficient to measure the highly variable (both in space +and time) space environment.1 Regular, densely-sampled measurements, even if they lack the detail of those +from our flagship instruments, would be a boon to scientific understanding and modeling of the geospace +system. These ideas are not new, but we posit that the technology has now arrived to make dense observations +of the upper atmosphere feasible in cost and effort. This white paper sketches out the scientific rationale for a +network of radio instruments delivering dense observations of the near-Earth space environment and the broad +steps necessary to implement wide-scale coverage in the next 30 years. +Scientific rationale: Space weather provides the background quiet time climatology and geomagnetic storm +time conditions that must be predicted and accommodated for success of both future technological systems and +human presence in the near-Earth space environment. However, our observational knowledge of the space +environment is decades behind that of lower atmosphere terrestrial weather for many reasons. Chief among +these is the difficulty in enabling sufficient remote sensing observations, due to the physics-imposed +requirement to sample vast spatial and temporal scales. Factors such as charged particle dynamics +experiencing electrodynamic forcing over very long distances, and corresponding two-way coupled influences +on the neutral upper atmosphere, pose a considerable challenge to data collection and understanding. +Our ability to predict the terrestrial weather has continuously improved and provides a useful comparative +case study of what is possible with sufficiently dense observations and appropriately driven models. Consider +that currently a 5-day forecast is as accurate as a 24-hour forecast was in 1980, and long-term forecasts of a +week or more are now useful. On the path to better forecast success, improvements in observations have been +a critical element, both in supplying real-time data for assimilation but also in amassing archival data that can be +used for post-analysis, model skill assessment, and improvement of the key physical processes included in the +models. The data from radiosondes launching every 12 hours for 365 days a year and measuring winds, +temperature, pressure, and relative humidity, along with continuous radar data on winds and precipitation, make +this forecast capability a reality. In contrast, the upper atmosphere and in particular the mesosphere-lower +thermosphere (MLT) system is far less densely sampled. This latter fact is in contrast to the vast number of +complex and interesting frontier science topics that have yet to be clarified, such as atmospheric gravity wave +breaking and momentum transfer, severe neutral wind shear effects on ionospheric E region variability, and +ionosphere-thermosphere mass and energy dynamics. Despite this fact, unlocking the secrets of the intimately +coupled physical pathways in the upper neutral and ionized atmosphere has critical importance for studying +whole atmosphere physics, and these subjects are at the frontier of understanding the space environment. +Particularly important ionosphere-thermosphere science topics in need of significant additional study include +the influence of lower atmospheric disturbances on the upper atmosphere. Specifically, estimated wave energy +flux in the lower thermosphere (“space weather from below”2) is comparable to daily average Joule power input +from above to the thermosphere3. Furthermore, this wave flux is likely underestimated by a large amount since +most model simulations do not yet capture the full wave spectrum. Additionally, little is still known about the +relative importance of planetary waves, tides, and gravity waves taken as a joint whole, and how these +influences evolve in both time and space. Appropriate coupled analysis in fact is largely absent from the +1 National Research Council, Distributed Arrays of Small Instruments for Solar-Terrestrial Research: Report of a Workshop. The National +Academies Press, 2006. +2 H.-L. Liu, “Variability and predictability of the space environment as related to lower atmosphere forcing,” Space Weather, vol. 14, no. 9, pp. +634–658, 2016, doi: 10.1002/2016SW001450. +3 D. J. Knipp, W. K. Tobiska, and B. A. Emery, “Direct and Indirect Thermospheric Heating Sources for Solar Cycles 21–23,” Sol Phys, vol. 224, no. +1, p. 495, Oct. 2004, doi: 10.1007/s11207-005-6393-4. + +literature today due to a lack of sufficiently dense observations, and so typical studies have focused only on +isolated aspects such as tidal forcing.4 A similar set of observational statements can be made for specification of +the full atmospheric wave spectrum (time periods, spatial wavelengths) in the ionosphere-thermosphere system, +with a goal of determining how these regions are structured and how and when the system selects preferential +wavelengths exhibiting high coupling efficiency. +Additionally, the current state of the art whole atmosphere coupling models remain relatively poor at +propagating input energy influences throughout the ionosphere-thermosphere system when run in an +observationally unconstrained mode. For example, Pedatella et al. (2014)5 shows that four different modeling +runs without data constraints above ~50 km altitude have fundamentally inconsistent responses in the +mesosphere-lower thermosphere system. These large discrepancies in predicted response then exhibit a further +and similarly large uncertainty in projected ionospheric influences and ionosphere-thermosphere energy +exchange.6 As one would expect, data assimilation-driven models show clear improvement in predicted vs. +observed response when mesosphere-lower thermosphere observations are used as compared to a +free-running prediction absent these data.7 +Observationally, MLT data from the TIMED satellite has been shown in the literature to be effective (when +available) at improving model performance and prediction. However, TIMED has been on orbit since 2001 and +its ~95-minute orbital period presents a challenge to wave interpretation. The satellite is only in one location at +any given time, which makes it impossible to measure the spatial/temporal variability of the MLT as this +variability is not global nor shorter than a period of 1 day. Clearly, the important mesoscale structure which +drives space weather systems is severely undersampled. By contrast, consider the lower atmosphere observing +strategy which consists of both low earth orbit and geostationary satellite observations coupled with extensive +ground based observations of the system. +There is a clear need for more modern observations of the near-Earth space environment with better local +time coverage. In particular, a ground-based sensor locked to the planet’s rotation provides a badly needed and +comprehensive view of MLT conditions and wave activity near orographic gravity wave generating features such +as mountains. Ground-based sensors also have the considerable advantage of resolving short-term temporal +variations in the MLT wave spectrum compared to e.g. TIMED data, which can only provide averaged information +over long spatial and temporal scales that obscure vitally important dynamics. If we take a page from the +terrestrial weather community, it is clear that a significant investment in a real-time mesoscale observing network +is critical to both advance community understanding of the near-Earth space region and its role in the whole +atmosphere system and to unlock the development of a robust space weather forecast system. +Supporting measurement techniques: A dense and scalable network of ground-based remote sensing +instruments can be composed of a variety of measurement techniques. Each observes a slice of the +fundamental parameters of interest, including neutral winds and ionospheric density, to collectively describe the +physical processes in the near-Earth space environment. We highlight two of the key enabling technologies +here, although we note that other techniques and instruments can play a large role in providing complementary +measurements (e.g. HF amateur radio beacons used for tomography of ionospheric irregularities). +MIMO meteor radar [MLT wind field]: Recent developments in multiple-input multiple-output (MIMO) meteor +radar networks have made higher-resolution wind measurements of the upper atmosphere possible8. These +networks operate over coded continuous-wave links between separately-located transmitter and receiver sites to +increase the density of specular meteor trail observations and provide diversity in sensing Doppler-derived wind +4 S. L. England, “A Review of the Effects of Non-migrating Atmospheric Tides on the Earth’s Low-Latitude Ionosphere,” Space Sci Rev, vol. 168, no. +1, pp. 211–236, Jun. 2012, doi: 10.1007/s11214-011-9842-4. +5 N. M. Pedatella et al., “The neutral dynamics during the 2009 sudden stratosphere warming simulated by different whole atmosphere models,” +J. Geophys. Res. Space Physics, vol. 119, no. 2, pp. 1306–1324, Feb. 2014, doi: 10.1002/2013JA019421. 6 N. M. Pedatella et al., “Multimodel +comparison of the ionosphere variability during the 2009 sudden stratosphere warming,” Journal of Geophysical Research: Space Physics, vol. +121, no. 7, pp. 7204–7225, Jul. 2016, doi: 10.1002/2016JA022859. +7 M. Jones et al., “Evaluating Different Techniques for Constraining Lower Atmospheric Variability in an Upper Atmosphere General Circulation Model: +A Case Study During the 2010 Sudden Stratospheric Warming,” Journal of Advances in Modeling Earth Systems, vol. 10, no. 12, pp. 3076–3102, +2018, doi: 10.1029/2018MS001440. +8 J. L. Chau et al., “Novel specular meteor radar systems using coherent MIMO techniques to study the mesosphere and lower +thermosphere,” Atmospheric Measurement Techniques, vol. 12, no. 4, pp. 2113–2127, Apr. 2019, doi: 10.5194/amt-12-2113-2019. + +projections. Such datasets contain enough information to estimate the three-dimensional wind field within the +observation volume to a resolution limited only by the measurement density in space and time. Low-power +ionosonde [bottomside ionospheric density]: Coded continuous-wave transmissions also enable a cross-linked +network of low-power ionosondes operating in much the same manner as the above-described MIMO meteor +radar network. Each transmitter and receiver pair can be used to produce an oblique ionogram. With +combinatorial scaling, this provides a cost-effective way to densely sample the ionospheric density along each +TX-RX link. Innovations such as the electro-magnetic vector sensor (6 orthogonal antenna elements with a +common phase center) could help to increase each individual instrument's degrees of freedom further. This +could lead to volumetric imaging of the bottom side ionosphere within the regional network. +Network composition: The envisioned network would consist of nodes consisting of at least the above +instruments. A key to feasibility is that required infrastructure, consisting of site power, high-bandwidth internet, +licensing, and/or building and ground space, can be co-located at relatively few sites. These anchor sites would +thus encompass meteor radar and ionosonde transmitters, meteor radar interferometric antenna arrays, and +computational resources. Receiver sites would require comparatively little infrastructure and can be operated +off-grid with solar power and/or wireless internet, so they have much greater freedom to be located where +necessary and in greater numbers to fill out the network. The cross-linked radar nodes form the backbone of the +network, and other instruments (including but not limited to GNSS receivers) are envisioned to piggy-back onto +that infrastructure investment with relatively low add-on cost. By focusing on low cost and modest power, space, +and connectivity requirements for the majority of the network sites, it becomes possible to engage with educators +and amateur enthusiasts to expand and support the network. Community-focused receiver sites can be built with +commodity equipment and located on rooftops or in backyards, and feeding data back into the network can be +enabled with open source software. HamSCI9 has similarly engaged the amateur radio community to much +success, and this would be a way to expand and strengthen those connections. +Progression timeline: Since the network of radio instruments described above can be expanded simply by +adding more nodes, we suggest a staged deployment beginning with targeted dense regional deployments and +expanding with maturing technology to eventually span continents or even the globe. +Current state (2020): Efforts are already underway to deploy regional MIMO meteor radar networks and +develop the low-power ionosonde into an operational instrument. GNSS TEC receivers already cover much of +the globe, but density can always be improved. These instruments/deployments are being developed and +funded separately, and near-term progress will be through developing the supporting technologies and investing +in dense regional networks that complement existing flagship instruments. +2025: By this time, we expect the currently-planned regional networks to be operational and planning for the +next stage of expansion to be well underway. Expansion will necessarily involve overlapping the coverage areas +of the supporting instruments and combining future deployments. These regional networks will individually allow +for novel study of mesoscale features with regional connections (e.g. the influence of orography or overhead +convection) and collectively add coverage for global studies (e.g. tides or planetary waves). Even at this regional +scale, the scientific return would be significant. +2030-2035: The primary objective for this timeframe is to achieve large-region or continental-scale dense +coverage with a network composed of all of the complementary instruments. Such an effort will require +leadership from a core group with significant community backing and investment. Long-term planning will have +to start now to reach that goal. At this stage scientifically, the network will unlock temporal and spatial variations +in mesoscale features and provide enough regular observations for data assimilation models to drive significant +improvement over the current state of the art. +2050: The ultimate goal is to span the globe with a network of radio instruments that can observe the +physical processes in the near-Earth space environment with enough density in time and space so as to +revolutionize our scientific understanding. The supporting technologies will be mature enough at this point that +expanding coverage to new regions will be a question only of scientific utility. +9 N. A. Frissell et al., “High-Frequency Communications Response to Solar Activity in September 2017 as Observed by Amateur Radio +Networks,” Space Weather, vol. 17, no. 1, pp. 118–132, 2019, doi: 10.1029/2018SW002008. + diff --git a/o9E2T4oBgHgl3EQf0Aih/content/tmp_files/load_file.txt b/o9E2T4oBgHgl3EQf0Aih/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..92efe182a4dc067adf9adffec920687d12745d3f --- /dev/null +++ b/o9E2T4oBgHgl3EQf0Aih/content/tmp_files/load_file.txt @@ -0,0 +1,157 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf,len=156 +page_content='A Global Radio Remote Sensing Network for Observing Space Weather Dynamics Ryan Volz, Philip J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Erickson | MIT Haystack Observatory Scott E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Palo | University of Colorado Boulder Jorge L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Chau | Leibniz Institute of Atmospheric Physics at the University of Rostock Juha Vierinen | UiT Arctic University of Norway Thomas Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Chen | Columbia University Introduction: The need for more and better data is a constant refrain in geospace science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Past workshops and decadal surveys have noted a compelling need for real-time observations and a clear realization that our current sampling of the vast geospace environment is wholly insufficient to measure the highly variable (both in space and time) space environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content='1 Regular, densely-sampled measurements, even if they lack the detail of those from our flagship instruments, would be a boon to scientific understanding and modeling of the geospace system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' These ideas are not new, but we posit that the technology has now arrived to make dense observations of the upper atmosphere feasible in cost and effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' This white paper sketches out the scientific rationale for a network of radio instruments delivering dense observations of the near-Earth space environment and the broad steps necessary to implement wide-scale coverage in the next 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Scientific rationale: Space weather provides the background quiet time climatology and geomagnetic storm time conditions that must be predicted and accommodated for success of both future technological systems and human presence in the near-Earth space environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' However, our observational knowledge of the space environment is decades behind that of lower atmosphere terrestrial weather for many reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Chief among these is the difficulty in enabling sufficient remote sensing observations, due to the physics-imposed requirement to sample vast spatial and temporal scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Factors such as charged particle dynamics experiencing electrodynamic forcing over very long distances, and corresponding two-way coupled influences on the neutral upper atmosphere, pose a considerable challenge to data collection and understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Our ability to predict the terrestrial weather has continuously improved and provides a useful comparative case study of what is possible with sufficiently dense observations and appropriately driven models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Consider that currently a 5-day forecast is as accurate as a 24-hour forecast was in 1980, and long-term forecasts of a week or more are now useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' On the path to better forecast success, improvements in observations have been a critical element, both in supplying real-time data for assimilation but also in amassing archival data that can be used for post-analysis, model skill assessment, and improvement of the key physical processes included in the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' The data from radiosondes launching every 12 hours for 365 days a year and measuring winds, temperature, pressure, and relative humidity, along with continuous radar data on winds and precipitation, make this forecast capability a reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' In contrast, the upper atmosphere and in particular the mesosphere-lower thermosphere (MLT) system is far less densely sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' This latter fact is in contrast to the vast number of complex and interesting frontier science topics that have yet to be clarified, such as atmospheric gravity wave breaking and momentum transfer, severe neutral wind shear effects on ionospheric E region variability, and ionosphere-thermosphere mass and energy dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Despite this fact, unlocking the secrets of the intimately coupled physical pathways in the upper neutral and ionized atmosphere has critical importance for studying whole atmosphere physics, and these subjects are at the frontier of understanding the space environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Particularly important ionosphere-thermosphere science topics in need of significant additional study include the influence of lower atmospheric disturbances on the upper atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Specifically, estimated wave energy flux in the lower thermosphere (“space weather from below”2) is comparable to daily average Joule power input from above to the thermosphere3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Furthermore, this wave flux is likely underestimated by a large amount since most model simulations do not yet capture the full wave spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Additionally, little is still known about the relative importance of planetary waves, tides, and gravity waves taken as a joint whole, and how these influences evolve in both time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Appropriate coupled analysis in fact is largely absent from the 1 National Research Council, Distributed Arrays of Small Instruments for Solar-Terrestrial Research: Report of a Workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' The National Academies Press, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Liu, “Variability and predictability of the space environment as related to lower atmosphere forcing,” Space Weather, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 634–658, 2016, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content='1002/2016SW001450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 3 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Knipp, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Tobiska, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Emery, “Direct and Indirect Thermospheric Heating Sources for Solar Cycles 21–23,” Sol Phys, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 224, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 495, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 2004, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content='1007/s11207-005-6393-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' literature today due to a lack of sufficiently dense observations, and so typical studies have focused only on isolated aspects such as tidal forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content='4 A similar set of observational statements can be made for specification of the full atmospheric wave spectrum (time periods, spatial wavelengths) in the ionosphere-thermosphere system, with a goal of determining how these regions are structured and how and when the system selects preferential wavelengths exhibiting high coupling efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Additionally, the current state of the art whole atmosphere coupling models remain relatively poor at propagating input energy influences throughout the ionosphere-thermosphere system when run in an observationally unconstrained mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' For example, Pedatella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' (2014)5 shows that four different modeling runs without data constraints above ~50 km altitude have fundamentally inconsistent responses in the mesosphere-lower thermosphere system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' These large discrepancies in predicted response then exhibit a further and similarly large uncertainty in projected ionospheric influences and ionosphere-thermosphere energy exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content='6 As one would expect, data assimilation-driven models show clear improvement in predicted vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' observed response when mesosphere-lower thermosphere observations are used as compared to a free-running prediction absent these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content='7 Observationally, MLT data from the TIMED satellite has been shown in the literature to be effective (when available) at improving model performance and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' However, TIMED has been on orbit since 2001 and its ~95-minute orbital period presents a challenge to wave interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' The satellite is only in one location at any given time, which makes it impossible to measure the spatial/temporal variability of the MLT as this variability is not global nor shorter than a period of 1 day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Clearly, the important mesoscale structure which drives space weather systems is severely undersampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' By contrast, consider the lower atmosphere observing strategy which consists of both low earth orbit and geostationary satellite observations coupled with extensive ground based observations of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' There is a clear need for more modern observations of the near-Earth space environment with better local time coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' In particular, a ground-based sensor locked to the planet’s rotation provides a badly needed and comprehensive view of MLT conditions and wave activity near orographic gravity wave generating features such as mountains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Ground-based sensors also have the considerable advantage of resolving short-term temporal variations in the MLT wave spectrum compared to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' TIMED data, which can only provide averaged information over long spatial and temporal scales that obscure vitally important dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' If we take a page from the terrestrial weather community, it is clear that a significant investment in a real-time mesoscale observing network is critical to both advance community understanding of the near-Earth space region and its role in the whole atmosphere system and to unlock the development of a robust space weather forecast system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Supporting measurement techniques: A dense and scalable network of ground-based remote sensing instruments can be composed of a variety of measurement techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Each observes a slice of the fundamental parameters of interest, including neutral winds and ionospheric density, to collectively describe the physical processes in the near-Earth space environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' We highlight two of the key enabling technologies here, although we note that other techniques and instruments can play a large role in providing complementary measurements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' HF amateur radio beacons used for tomography of ionospheric irregularities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' MIMO meteor radar [MLT wind field]: Recent 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Such datasets contain enough information to estimate the three-dimensional wind field within the observation volume to a resolution limited only by the measurement density in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Low-power ionosonde [bottomside ionospheric density]: Coded continuous-wave transmissions also enable a cross-linked network of low-power ionosondes operating in much the same manner as the above-described MIMO meteor radar network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Each transmitter and receiver pair can be used to produce an oblique ionogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' With combinatorial scaling, this provides a cost-effective way to densely sample the ionospheric density along each TX-RX link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=" Innovations such as the electro-magnetic vector sensor (6 orthogonal antenna elements with a common phase center) could help to increase each individual instrument's degrees of freedom further." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' This could lead to volumetric imaging of the bottom side ionosphere within the regional network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Network composition: The envisioned network would consist of nodes consisting of at least the above instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' A key to feasibility is that required infrastructure, consisting of site power, high-bandwidth internet, licensing, and/or building and ground space, can be co-located at relatively few sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' These anchor sites would thus encompass meteor radar and ionosonde transmitters, meteor radar interferometric antenna arrays, and computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Receiver sites would require comparatively little infrastructure and can be operated off-grid with solar power and/or wireless internet, so they have much greater freedom to be located where necessary and in greater numbers to fill out the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' The cross-linked radar nodes form the backbone of the network, and other instruments (including but not limited to GNSS receivers) are envisioned to piggy-back onto that infrastructure investment with relatively low add-on cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' By focusing on low cost and modest power, space, and connectivity requirements for the majority of the network sites, it becomes possible to engage with educators and amateur enthusiasts to expand and support the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Community-focused receiver sites can be built with commodity equipment and located on rooftops or in backyards, and feeding data back into the network can be enabled with open source software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' HamSCI9 has similarly engaged the amateur radio community to much success, and this would be a way to expand and strengthen those connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Progression timeline: Since the network of radio instruments described above can be expanded simply by adding more nodes, we suggest a staged deployment beginning with targeted dense regional deployments and expanding with maturing technology to eventually span continents or even the globe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Current state (2020): Efforts are already underway to deploy regional MIMO meteor radar networks and develop the low-power ionosonde into an operational instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' GNSS TEC receivers already cover much of the globe, but density can always be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' These instruments/deployments are being developed and funded separately, and near-term progress will be through developing the supporting technologies and investing in dense regional networks that complement existing flagship instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 2025: By this time, we expect the currently-planned regional networks to be operational and planning for the next stage of expansion to be well underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Expansion will necessarily involve overlapping the coverage areas of the supporting instruments and combining future deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' These regional networks will individually allow for novel study of mesoscale features with regional connections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' the influence of orography or overhead convection) and collectively add coverage for global studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' tides or planetary waves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Even at this regional scale, the scientific return would be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 2030-2035: The primary objective for this timeframe is to achieve large-region or continental-scale dense coverage with a network composed of all of the complementary instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Such an effort will require leadership from a core group with significant community backing and investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Long-term planning will have to start now to reach that goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' At this stage scientifically, the network will unlock temporal and spatial variations in mesoscale features and provide enough regular observations for data assimilation models to drive significant improvement over the current state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 2050: The ultimate goal is to span the globe with a network of radio instruments that can observe the physical processes in the near-Earth space environment with enough density in time and space so as to revolutionize our scientific understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' The supporting technologies will be mature enough at this point that expanding coverage to new regions will be a question only of scientific utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 9 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' Frissell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=', “High-Frequency Communications Response to Solar Activity in September 2017 as Observed by Amateur Radio Networks,” Space Weather, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content=' 118–132, 2019, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} +page_content='1029/2018SW002008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E2T4oBgHgl3EQf0Aih/content/2301.04137v1.pdf'} diff --git a/oNFQT4oBgHgl3EQfqzYC/content/tmp_files/2301.13381v1.pdf.txt b/oNFQT4oBgHgl3EQfqzYC/content/tmp_files/2301.13381v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fd36b37a4fc8ae36d829a3231d2a9478ed819a85 --- /dev/null +++ b/oNFQT4oBgHgl3EQfqzYC/content/tmp_files/2301.13381v1.pdf.txt @@ -0,0 +1,3717 @@ +Published as a conference paper at ICLR 2023 +WHEN SOURCE-FREE DOMAIN ADAPTATION MEETS +LEARNING WITH NOISY LABELS +Li Yi1,∗ +Gezheng Xu2,∗ Pengcheng Xu2 +Jiaqi Li2 +Ruizhi Pu2 +Charles Ling2 +A. Ian McLeod1 +Boyu Wang1,2,† +1Department of Statistical and Actuarial Sciences +2Department of Computer Science +University of Western Ontario +{lyi7,gxu86,pxu67,jli3779,rpu2,charles.ling,aimcleod}@uwo.ca +bwang@csd.uwo.ca +ABSTRACT +Recent state-of-the-art source-free domain adaptation (SFDA) methods have fo- +cused on learning meaningful cluster structures in the feature space, which have +succeeded in adapting the knowledge from source domain to unlabeled target +domain without accessing the private source data. However, existing methods +rely on the pseudo-labels generated by source models that can be noisy due to +domain shift. In this paper, we study SFDA from the perspective of learning with +label noise (LLN). Unlike the label noise in the conventional LLN scenario, we +prove that the label noise in SFDA follows a different distribution assumption. We +also prove that such a difference makes existing LLN methods that rely on their +distribution assumptions unable to address the label noise in SFDA. Empirical +evidence suggests that only marginal improvements are achieved when applying +the existing LLN methods to solve the SFDA problem. On the other hand, although +there exists a fundamental difference between the label noise in the two scenar- +ios, we demonstrate theoretically that the early-time training phenomenon (ETP), +which has been previously observed in conventional label noise settings, can also +be observed in the SFDA problem. Extensive experiments demonstrate significant +improvements to existing SFDA algorithms by leveraging ETP to address the label +noise in SFDA. +1 +INTRODUCTION +Deep learning demonstrates strong performance on various tasks across different fields. However, +it is limited by the requirement of large-scale labeled and independent, and identically distributed +(i.i.d.) data. Unsupervised domain adaptation (UDA) is thus proposed to mitigate the distribution shift +between the labeled source and unlabeled target domain. In view of the importance of data privacy, +it is crucial to be able to adapt a pre-trained source model to the unlabeled target domain without +accessing the private source data, which is known as Source Free Domain Adaptation (SFDA). +The current state-of-the-art SFDA methods (Liang et al., 2020; Yang et al., 2021a;b) mainly focus +on learning meaningful cluster structures in the feature space, and the quality of the learned cluster +structures hinges on the reliability of pseudo labels generated by the source model. Among these +methods, SHOT (Liang et al., 2020) purifies pseudo labels of target data based on nearest centroids, +and then the purified pseudo labels are used to guide the self-training. G-SFDA (Yang et al., 2021b) +and NRC (Yang et al., 2021a) further refine pseudo labels by encouraging similar predictions to the +data point and its neighbors. For a single target data point, when most of its neighbors are correctly +predicted, these methods can provide an accurate pseudo label to the data point. However, as we +illustrate the problem in Figure 1i(a-b), when the majority of its neighbors are incorrectly predicted +to a category, it will be assigned with an incorrect pseudo label, misleading the learning of cluster +structures. The experimental result on VisDA (Peng et al., 2017), shown in Figure 1ii, further verifies +this phenomenon. By directly applying the pre-trained source model on each target domain instance +∗Equal contribution +†Corresponding author +1 +arXiv:2301.13381v1 [cs.LG] 31 Jan 2023 + +Published as a conference paper at ICLR 2023 +Source/Unlabeled target data +Mislabeled target data +Correctly predicted data +(a) +(b): existing +SFDA +(c): ours +AB7HicbVBNS8 +NAEJ3Ur1q/qh69LBbBU0mkqBeh6MVjBdMW2lA2027dLMJuxOhlP4GLx4U8eoP8ua/cdvmoK0PBh7vzTAzL0ylMOi6305hbX1jc6u4XdrZ3ds/KB8eNU2SacZ9lshEt0NquB +SK+yhQ8naqOY1DyVvh6G7mt564NiJRjzhOeRDTgRKRYBSt5OMN9rxeueJW3TnIKvFyUoEcjV75q9tPWBZzhUxSYzqem2IwoRoFk3xa6maGp5SN6IB3LFU05iaYzI+dkjOr9E +mUaFsKyVz9PTGhsTHjOLSdMcWhWfZm4n9eJ8PoOpgIlWbIFVsijJMCGz0lfaM5Qji2hTAt7K2FDqilDm0/JhuAtv7xKmhdV7Jae6hV6rd5HEU4gVM4Bw+uoA730AfGA +h4hld4c5Tz4rw7H4vWgpPHMfOJ8/YTKOag=t = t1 +Label Noise +AB6nicbVBNS8NAE +J3Ur1q/qh69LBbBU0lE1ItQ9OKxov2ANpTNdtMu3WzC7kQoT/BiwdFvPqLvPlv3LY5aOuDgcd7M8zMCxIpDLrut1NYWV1b3yhulra2d3b3yvsHTROnmvEGi2Ws2wE1XArFGyhQ8n +aiOY0CyVvB6Hbqt564NiJWjzhOuB/RgRKhYBSt9IDXbq9cavuDGSZeDmpQI56r/zV7csjbhCJqkxHc9N0M+oRsEkn5S6qeEJZSM64B1LFY248bPZqRNyYpU+CWNtSyGZqb8nMho +ZM4C2xlRHJpFbyr+53VSDK/8TKgkRa7YfFGYSoIxmf5N+kJzhnJsCWVa2FsJG1JNGdp0SjYEb/HlZdI8q3oX1fP780rtJo+jCEdwDKfgwSXU4A7q0AGA3iGV3hzpPivDsf89aCk +8cwh84nz/TSY2Ct = 0 +Source Model +AB9XicbVBNS8NAEN3Ur1q/qh69BIvgqSQi6kUoevFYwX5AG8tmM2mXbjZhd6KW0P/hxYMiXv0v3vw +3btsctPXBwO9GWbm+YngGh3n2yosLa+srhXSxubW9s75d29po5TxaDBYhGrtk81C6hgRwFtBMFNPIFtPzh9cRvPYDSPJZ3OErAi2hf8pAzika6x0vsdRGeMAMZjHvlilN1prAXiZuTCslR75W/ukHM0gkMkG17rhOgl5GFXImYFzqphoSyoa0Dx1DJY1Ae9n06rF9ZJTADmNlSqI9VX9PZDTSehT5pjOiONDz3kT8z+ukGF54GZdJiDZbFGYCht +jexKBHXAFDMXIEMoUN7fabEAVZWiCKpkQ3PmXF0nzpOqeVU9vTyu1qzyOIjkgh+SYuOSc1MgNqZMGYUSRZ/JK3qxH68V6tz5mrQUrn9knf2B9/gAN7JLjt = tend +(i) Overview of the SFDA problem and our method +Plane Bcycl Bus +Car Horse Knife McyclPersonPlant Sktbrd Train Truck +0.00 +0.20 +0.40 +0.60 +0.80 +1.00 +Misleading Neighbors Ratio +Over-Confident Misleading +Neighbors Ratio +(ii) Neighbors Label Noise Analysis On VisDA +Figure 1: (i) (a) The SFDA problem can be formulated as an LLN problem. (b) The existing SFDA +algorithms using the local cluster information cannot address label noise due to the unbounded label +noise (Section 3). (c) We prove that ETP exists in SFDA, which can be leveraged to address the +unbounded label noise (Section 4). (ii) Observed Label Noise Phenomena on VisDA dataset. +(central instance), we collect its neighbors and evaluate their quality. We observed that for each class +a large proportion of the neighbors are misleading (i.e., the neighbors’ pseudo labels are different +from the central instance’s true label), some even with high confidence (e.g., the over-confident +misleading neighbors whose prediction score is larger than 0.75). Based on this observation, we can +conclude that: (1) the pseudo labels leveraged in current SFDA methods can be heavily noisy; (2) +some pseudo-label purification methods utilized in SFDA, which severely rely on the quality of the +pseudo label itself, will be affected by such label noise, and the prediction error will accumulate as +the training progresses. More details can be found in Appendix A. +In this paper, we address the aforementioned problem by formulating SFDA as learning with label +noise (LLN). Unlike existing studies that heuristically rely on cluster structures or neighbors, we +investigate the properties of label noise in SFDA and show that there is an intrinsic discrepancy +between the SFDA and the LLN problems. Specifically, in conventional LLN scenarios, the label +noise is generated by human annotators or image search engines (Patrini et al., 2017; Xiao et al., +2015; Xia et al., 2020a), where the underlying distribution assumption is that the mislabeling rate for +a sample is bounded. However, in the SFDA scenarios, the label noise is generated by the source +model due to the distribution shift, where we prove that the mislabeling rate for a sample is much +higher, and can approach 1. We term the former label noise in LLN as bounded label noise and the +latter label noise in SFDA as unbounded label noise. Moreover, we theoretically show that most +existing LLN methods, which rely on bounded label noise assumption, are unable to address the label +noise in SFDA due to the fundamental difference (Section 3). +To this end, we leverage early-time training phenomenon (ETP) in LLN to address the unbounded +label noise and to improve the efficiency of existing SFDA algorithms. Specifically, ETP indicates +that classifiers can predict mislabeled samples with relatively high accuracy during the early learning +phase before they start to memorize the mislabeled data (Liu et al., 2020). Although ETP has been +previously observed in, it has only been studied in the bounded random label noise in the conventional +LLN scenarios. In this work, we theoretically and empirically show that ETP still exists in the +unbounded label noise scenario of SFDA. Moreover, we also empirically justify that existing SFDA +algorithms can be substantially improved by leveraging ETP, which opens up a new avenue for SFDA. +As an instantiation, we incorporate a simple early learning regularization (ELR) term (Liu et al., +2020) with existing SFDA objective functions, achieving consistent improvements on four different +SFDA benchmark datasets. As a comparison, we also apply other existing LLN methods, including +Generalized Cross Entropy (GCE) (Zhang & Sabuncu, 2018), Symmetric Cross Entropy Learning +(SL) (Wang et al., 2019b), Generalized Jensen-Shannon Divergence (GJS) (Englesson & Azizpour, +2021) and Progressive Label Correction (PLC) (Zhang et al., 2021), to SFDA. Our empirical evidence +shows that they are inappropriate for addressing the label noise in SFDA. This is also consistent with +our theoretical results (Section 4). +Our main contribution can be summarized as: (1) We establish the connection between the SFDA +and the LLN. Compared with the conventional LLN problem that assumes bounded label noise, +the problem in SFDA can be viewed as the problem of LLN with the unbounded label noise. (2) +2 + +Published as a conference paper at ICLR 2023 +We theoretically and empirically justify that ETP exists in the unbounded label noise scenario. On +the algorithmic side, we instantiate our analysis by simply adding a regularization term into the +SFDA objective functions. (3) We conduct extensive experiments to show that ETP can be utilized to +improve many existing SFDA algorithms by a large margin across multiple SFDA benchmarks. +2 +RELATED WORK +Source-free domain adaptation. Recently, SFDA are studied for data privacy. The first branch +of research is to leverage the target pseudo labels to conduct self-training to implicitly achieve +adaptation (Liang et al., 2021; Tanwisuth et al., 2021; Ahmed et al., 2021; Yang et al., 2021b). SHOT +(Liang et al., 2020) introduces k-means clustering and mutual information maximization strategy for +self-training. NRC (Yang et al., 2021a) further investigates the neighbors of target clusters to improve +the accuracy of pseudo labels. These studies more or less involve pseudo-label purification processes, +but they are primarily heuristic algorithms and suffer from the previously mentioned label noise +accumulation problem. The other branch is to utilize the generative model to synthesize target-style +training data (Qiu et al., 2021; Liu et al., 2021b). Some methods also explore the SFDA algorithms +in various settings. USFDA (Kundu et al., 2020a) and FS (Kundu et al., 2020b) design methods +for universal and open-set UDA. In this paper, we regard SFDA as the LLN problem. We aim to +explore what category of noisy labels exists in SFDA and to ameliorate such label noise to improve +the performance of current SFDA algorithms. +Learning with label noise. Existing methods for training neural networks with label noise focus +on symmetric, asymmetric, and instance-dependent label noise. For example, a branch of research +focuses on leveraging noise-robust loss functions to cope with the symmetric and asymmetric noise, +including GCE (Zhang & Sabuncu, 2018), SL (Wang et al., 2019b), NCE (Ma et al., 2020), and GJS +(Englesson & Azizpour, 2021), which have been proven effective in bounded label noise. On the other +hand, CORES (Cheng et al., 2020) and CAL (Zhu et al., 2021) are shown useful in mitigating instance- +dependent label noise. These methods are only tailed to conventional LLN settings. Recently, Liu +et al. (2020) has studied early-time training phenomenon (ETP) in conventional label noise scenarios +and proposes a regularization term ELR to exploit the benefits of ETP. PCL (Zhang et al., 2021) is +another conventional LLN algorithm utilizing ETP, but it cannot maintain the exploit of ETP in SFDA +as memorizing noisy labels is much faster in SFDA. Our contributions are: (1) We theoretically and +empirically study ETP in the SFDA scenario. (2) Based on an in depth analysis of many existing +LLN methods (Zhang & Sabuncu, 2018; Wang et al., 2019b; Englesson & Azizpour, 2021; Zhang +et al., 2021), we demonstrate that ELR is useful for many SFDA problems. +3 +LABEL NOISE IN SFDA +The presence of label noise on training datasets has been shown to degrade the model performance +(Malach & Shalev-Shwartz, 2017; Han et al., 2018). In SFDA, existing algorithms rely on pseudo- +labels produced by the source model, which are inevitably noisy due to the domain shift. The SFDA +methods such as Liang et al. (2020); Yang et al. (2021a;b) cannot tackle the situation when some +target samples and their neighbors are all incorrectly predicted by the source model. In this section, +we formulate the SFDA as the problem of LLN to address this issue. We assume that the source +domain DS and the target domain DT follow two different underlying distributions over X × Y, +where X and Y are respectively the input and label spaces. In the SFDA setting, we aim to learn a +target classifier f(x; θ) : X → Y only with a pre-trained model fS(x) on DS and a set of unlabeled +target domain observations drawn from DT . We regard the incorrectly assigned pseudo-labels as +noisy labels. Unlike the “bounded label noise” assumption in the conventional LLN domain, we will +show that the label noise in SFDA is unbounded. We further prove that most existing LLN methods +that rely on the bounded assumption cannot address the label noise in SFDA due to the difference. +Label noise in conventional LLN settings: In conventional label noise settings, the injected noisy +labels are collected by either human annotators or image search engines (Lee et al., 2018; Li et al., +2017; Xiao et al., 2015). The label noise is usually assumed to be either independent of instances (i.e., +symmetric label noise or asymmetric label noise) (Patrini et al., 2017; Liu & Tao, 2015; Xu et al., +2019b) or dependent of instances (i.e., instance-dependent label noise) (Berthon et al., 2021; Xia et al., +2020b). The underling assumption for them is that a sample x has the highest probability of being +in the correct class y, i.e., Pr[ ˜Y = i|Y = i, X = x] > Pr[ ˜Y = j|Y = i, X = x], ∀x ∈ X, i ̸= j, +3 + +Published as a conference paper at ICLR 2023 +where ˜Y is the noisy label and Y is the ground-truth label for input X. Equivalently, it assumes a +bounded noise rate. For example, given an image to annotate, the mislabeling rate for the image is +bounded by a small number, which is realistic in conventional LLN settings (Xia et al., 2020b; Cheng +et al., 2020). When the label noise is generated by the source model, the underlying assumption of +these types of label noise does not hold. +Label noise in SFDA: As for the label noise generated by the source model, mislabeling rate for +an image can approach 1, that is, Pr[ ˜Y = j|Y = i, X = x] → 1, ∃S ⊂ X, ∀x ∈ S, i ̸= j. To +understand that the label noise in SFDA is unbounded, we consider a two-component Multivariate +Gaussian mixture distribution with equal priors for both domains. Let the first component (y = 1) +of the source domain distribution DS be N(µ1, σ2Id), and the second component (y = −1) of DS +be N(µ2, σ2Id), where µ1, µ2 ∈ Rd and Id ∈ Rd×d. For the target domain distribution DT , let +the first component (y = 1) of DT be N(µ1 + ∆, σ2Id), and the second component (y = −1) of +DT be N(µ2 + ∆, σ2Id), where ∆ ∈ Rd is the shift of the two domains. Notice that the domain +shift considered is a general shift and it has been studied in Stojanov et al. (2021); Zhao et al. (2019), +where we also illustrate the domain shift in Figure 9 in supplementary material. +Let fS be the optimal source classifier. First, we build the relationship between the mislabeling rate +for target data and the domain shift: +Pr +(x,y)∼DT[fS(x) ̸= y] = 1 +2Φ(−d1 +σ ) + 1 +2Φ(−d2 +σ ), +(1) +where d1 = +�� µ2−µ1 +2 +− c +�� sign( +�� µ2−µ1 +2 +�� − ∥c∥), d2 = +�� µ2−µ1 +2 ++ c +��, c = α(µ2 − µ1), α = +∆⊤(µ2−µ1) +∥µ2−µ1∥2 is the magnitude of domain shift, and Φ is the standard normal cumulative distribution +function. Eq. (1) shows that the magnitude of the domain shift inherently controls the mislabeling +error for target data. This mislabeling rate increases as the magnitude of the domain shift increases. +We defer the proof and details to Appendix B. +More importantly, we characterize that the label noise is unbounded among these mislabeled samples. +Theorem 3.1. Without loss of generality, we assume that the ∆ is positively correlated with the +vector µ2 − µ1, i.e., ∆⊤(µ2 − µ1) > 0. For (x, y) ∼ DT , if x ∈ R, then +Pr[fS(x) ̸= y] ≥ 1 − δ, +(2) +where δ ∈ (0, 1) (i.e., δ = 0.01), R = R1 +� R2, R1 = {x : ∥x − µ1 − ∆∥ ≤ σ( +√ +d +2 − log 1−δ +δ +√ +d +)}, +and R2 = {x : x⊤1d > (σd + 2µ⊤ +1 1d)/2}. Meanwhile, R is non-empty when α > (log 1−δ +δ )/d, +where α = ∆⊤(µ2−µ1) +∥µ2−µ1∥2 +> 0 is the magnitude of the domain shift along the direction µ2 − µ1. +Conventional LLN methods assume that the label noise is bounded: Pr[fH(x) ̸= y] < m, ∀(x, y) ∼ +DT , where fH is the labeling function, and m = 0.5 if the number of clean samples of each +component are the same (Cheng et al., 2020). However, Theorem 3.1 indicates that the label +noise generated by the source model is unbounded for any x ∈ R. In practice, region R is +non-empty as neural networks are usually trained on high dimensional data such that d ≫ 1, +so α > (log 1−δ +δ )/d → 0 is easy to satisfy. The probability measure on R = R1 +� R2 (i.e., +Pr(x,y)∼DT [x ∈ R]) increases as the magnitude of the domain shift α increases, meaning more data +points contradict the conventional LLN assumption. More details can be found in Appendix C. +Given that the unbounded label noise exists in SFDA, the following Lemma establishes that many +existing LLN methods (Wang et al., 2019b; Ghosh et al., 2017; Englesson & Azizpour, 2021; Ma +et al., 2020), which rely on the bounded assumption, are not noise tolerant in SFDA. +Lemma 3.2. Let the risk of the function h : X +→ Y under the clean data be R(h) = +Ex,y[ℓLLN(h(x), y)], and the risk of h under the noisy data be �R(h) = Ex,˜y[ℓLLN(h(x), ˜y)], where +the noisy data follows the unbounded assumption, i.e., Pr[˜y ̸= y|x ∈ R] = 1 − δ for a subset +R ⊂ X and δ ∈ (0, 1). Then the global minimizer ˜h⋆ of �R(h) disagrees with the global minimizer +h⋆ of R(h) on data points x ∈ R with a high probability at least 1 − δ. +We denote ℓLLN by the existing noise-robust loss based LLN methods in Wang et al. (2019b); Ghosh +et al. (2017); Englesson & Azizpour (2021); Ma et al. (2020). When the noisy data follows the +bounded assumption, these methods are noise tolerant as the minimizer ˜h⋆ converges to the minimizer +h⋆ with a high probability. We defer the details and proof of the related LLN methods to Appendix D. +4 + +Published as a conference paper at ICLR 2023 +4 +LEARNING WITH LABEL NOISE IN SFDA +Given a fundamental difference between the label noise in SFDA and the label noise in conventional +LLN scenarios, existing LLN methods, whose underlying assumption is bounded label noise, cannot +be applied to solve the label noise in SFDA. This section focuses on investigating how to address the +unbounded label noise in SFDA. +Motivated by the recent studies Liu et al. (2020); Arpit et al. (2017), which observed an early-time +training phenomenon (ETP) on noisy datasets with bounded random label noise, we find that ETP does +not rely on the bounded random label noise assumption, and it can be generalized to the unbounded +label noise in SFDA. ETP describes the training dynamics of the classifier that preferentially fits +the clean samples and therefore has higher prediction accuracy for mislabeled samples during the +early-training stage. Such training characteristics can be very beneficial for SFDA problems in which +we only have access to the source model and the highly noisy target data. To theoretically prove ETP +in the presence of unbounded label noise, we first describe the problem setup. +We still consider a two-component Gaussian mixture distribution with equal priors. We denote y by the +true label for x, and assume it is a balanced sample from {−1, +1}. The instance x is sampled from +the distribution N(yµ, σ1d), where ∥µ∥ = 1. We denote ˜y by the noisy label for x. We observe that +the label noise generated by the source model is close to the decision boundary revealed in Theorem +3.1. So, to assign the noisy labels, we let ˜y = yβ(x, y), where β(x, y) = sign(1{yx⊤µ > r} − 0.5) +is the label flipping function, and r controls the mislabeling rate. If β(x, y) < 1, then the data point +x is mislabeled. Meanwhile, the label noise is unbounded by adopting the label flipping function +β(x, y): Pr[˜y ̸= y|yx⊤µ ≤ r] = 1, where R = {x : yx⊤µ ≤ r}. +We study the early-time training dynamics of gradient descent on the linear classifier. The parameter +θ is learned over the unbounded label noise data {xi, ˜yi}n +i=1 with the following logistic loss function: +L(θt+1) = 1 +n +n +� +i=1 +log +� +1 + exp +� +−˜yiθ⊤ +t+1xi +�� +, +where θt+1 = θt − η∇θL(θt), and η is the learning rate. Then the following theorem builds the +connection between the prediction accuracy for mislabeled samples at an early-training time T. +Theorem 4.1. Let B = {x : ˜y ̸= y} be a set of mislabeled samples. Let κ(B; θ) be the prediction +accuracy calculated by the ground-truth labels and the predicted labels by the classifier with parame- +ter θ for mislabeled samples. If at most half of the samples are mislabeled (r < 1), then there exists a +proper time T and a constant c0 > 0 such that for any 0 < σ < c0 and n → ∞, with probability +1 − op(1): +κ(B; θT ) ≥ 1 − exp{− 1 +200g(σ)2}, +(3) +where g(σ) = +Erf[ 1−r +√ +2σ ] +2(1+2σ)σ + +exp (− (r−1)2 +2σ2 +) +√ +2π(1+2σ) +> 0 is a monotone decreasing function that g(σ) → ∞ as +σ → 0, and Erf[x] = +2 +√π +� x +0 e−t2 dt. +The proof is provided in Appendix E. Compared to ETP found in Liu et al. (2020), where the label +noise is assumed to be bounded, Theorem 4.1 presents that ETP also exists even though the label +noise is unbounded. At a proper time T, the classifier trained by the gradient descent algorithm +can provide accurate predictions for mislabeled samples, where its accuracy is lower bounded by +a function of the variance of clusters σ. When σ → 0, the predictions of all mislabeled samples +equal to their ground-truth labels (i.e., κ(B; θT ) → 1). When the classifier is trained for a sufficiently +long time, it will gradually memorize mislabeled data. The predictions of mislabeled samples are +equivalent to their incorrect labels instead of their ground-truth labels (Liu et al., 2020; Maennel et al., +2020). Based on these insights, the memorization of mislabeled data can be alleviated by leveraging +their predicted labels during the early-training time. +To leverage the predictions during the early-training time, we adopt a recently established method, +early learning regularization (ELR) (Liu et al., 2020), which encourages model predictions to stick to +the early-time predictions for x. Since ETP exists in the scenarios of the unbounded label noise, ELR +can be applied to solve the label noise in SFDA. The regularization is given by: +LELR(θt) = log(1 − ¯y⊤ +t f(x; θt)), +(4) +5 + +Published as a conference paper at ICLR 2023 +(i) VisDA-C +(ii) DomainNet +(iii) Office-Home +(iv) Office-31 +Figure 2: Training accuracy on various target domains. The source models initialize the classifiers +and annotate unlabeled target data. As the classifiers memorize the unbounded label noise very fast, +for the first 90 steps, we evaluate the prediction accuracy on target data every batch, and one step +represents one training batch. After the 90 steps, we evaluate the prediction accuracy for every 0.3 +epoch, shown as one step. We use the CE, GCE, and ELR to train the classifiers on the labeled target +data, shown in solid green lines, solid orange lines, and solid blue lines, respectively. The dotted red +line represents the accuracy of labeling target data. Eventually, the classifiers memorize the label +noise, and the prediction accuracy equals the labeling accuracy (shown in (iii-iv)). Additional results +on transfer pairs can be found in Appendix F. +where we overload f(x; θt) to be the probabilistic output for the sample x, and ¯yt = β¯yt−1 + (1 − +β)f(x; θt) is the moving average prediction for x, where β is a hyperparameter. To see how ELR +prevents the model from memorizing the label noise, we calculate the gradient of Eq. (4) with respect +to f(x; θt), which is given by: +dLELR(θt) +df(x; θt) = − +¯yt +1 − ¯y⊤ +t f(x; θt). +Note that minimizing Eq. (4) forces f(x; θt) to close to ¯yt. When ¯yt is aligned better with f(x; θt), +the magnitude of the gradient becomes larger. It makes the gradient of aligning f(x; θt) with ¯yt +overwhelm the gradient of other loss terms that align f(x; θt) with noisy labels. As the training +progresses, the moving averaged predictions ¯yt for target samples gradually approach their ground- +truth labels till the time T. Therefore, Eq. (4) prevents the model from memorizing the label noise by +forcing the model predictions to stay close to these moving averaged predictions ¯yt, which are very +likely to be ground-truth labels. +Some existing LLN methods propose to assign pseudo labels to data or require two-stage training for +label noise (Cheng et al., 2020; Zhu et al., 2021; Zhang et al., 2021). Unlike these LLN methods, +Eq. (4) can be easily embedded into any existing SFDA algorithms without conflict. The overall +objective function is given by: +L = LSFDA + λLELR, +(5) +where LSFDA is any SFDA objective function, and λ is a hyperparameter. +Empirical Observations on Real-World Datasets. +We empirically verify that target classifiers +have higher prediction accuracy for target data during the early training and adaptation stage. We +propose leveraging this benefit to prevent the classifier from memorizing the noisy labels. The +observations are shown in Figure 2. The parameters of classifiers are initialized by source models. +Labels of target data are annotated by the initialized classifiers. We train the target classifiers on +target data with the standard cross-entropy (CE) loss and the generalized cross-entropy (GCE) loss, a +well-known noise-robust loss widely leveraged in bounded LLN scenarios. The solid green, orange +and blue lines represent the training accuracy of optimizing the classifiers with CE loss, GCE loss, +and ELR loss, respectively. The dotted red lines represent the labeling accuracy of the initialized +classifiers. Considering that the classifiers memorize the unbounded label noise very fast, we evaluate +the prediction accuracy on target data every batch for the first 90 steps. After 90 steps, we evaluate +the prediction accuracy for every 0.33 epoch. The green lines show that ETP exists in SFDA, which +is consistent with our theoretical result. Meanwhile, in all scenarios, green and orange lines show +that classifiers provide higher prediction accuracy during the first a few iterations. After a few +iterations, they start to memorize the label noise even with noise-robust loss (e.g., GCE). Eventually, +the classifiers are expected to memorize the whole datasets. For conventional LLN settings, it has +been empirically verified that it takes a much longer time before classifiers start memorizing the label +noise (Liu et al., 2020; Xia et al., 2020a). We provide further analysis in Appendix H. We highlight +6 + +70 +Training Acc. +60 +CE +ELR +50 +GCE +0 +20 +40 +60 +80 +100 +Synthetic → Real78 +Acc. +76 +Training +74 +CE +72 +ELR +GCE +70 +0 +20 +40 +60 +80 100 120 140 +C→R62 +CE + AcC. +60 +ELR +58 +GCE +Training +56 +54 +52 +0 +25 +50 +75 +100 125 150 175 +CI → Ar90 +CE +ACC. +ELR +85 +GCE +Training +80 +75 +0 +20 +40 +60 +80 100 120 140 +amazon -→ webcamPublished as a conference paper at ICLR 2023 +Table 1: Accuracies (%) on Office-Home for ResNet50-based methods. +Method +SFAr→ClAr→PrAr→RwCl→ArCl→PrCl→RwPr→ArPr→ClPr→RwRw→ArRw→ClRw→Pr Avg +MCD (Saito et al., 2018b) + +48.9 +68.3 +74.6 +61.3 +67.6 +68.8 +57.0 +47.1 +75.1 +69.1 +52.2 +79.6 +64.1 +CDAN (Long et al., 2018) + +50.7 +70.6 +76.0 +57.6 +70.0 +70.0 +57.4 +50.9 +77.3 +70.9 +56.7 +81.6 +65.8 +SAFN (Xu et al., 2019a) + +52.0 +71.7 +76.3 +64.2 +69.9 +71.9 +63.7 +51.4 +77.1 +70.9 +57.1 +81.5 +67.3 +Symnets (Zhang et al., 2019a) + +47.7 +72.9 +78.5 +64.2 +71.3 +74.2 +64.2 +48.8 +79.5 +74.5 +52.6 +82.7 +67.6 +MDD (Zhang et al., 2019b) + +54.9 +73.7 +77.8 +60.0 +71.4 +71.8 +61.2 +53.6 +78.1 +72.5 +60.2 +82.3 +68.1 +TADA (Wang et al., 2019a) + +53.1 +72.3 +77.2 +59.1 +71.2 +72.1 +59.7 +53.1 +78.4 +72.4 +60.0 +82.9 +67.6 +BNM (Cui et al., 2020) + +52.3 +73.9 +80.0 +63.3 +72.9 +74.9 +61.7 +49.5 +79.7 +70.5 +53.6 +82.2 +67.9 +BDG (Yang et al., 2020) + +51.5 +73.4 +78.7 +65.3 +71.5 +73.7 +65.1 +49.7 +81.1 +74.6 +55.1 +84.8 +68.7 +SRDC (Tang et al., 2020) + +52.3 +76.3 +81.0 +69.5 +76.2 +78.0 +68.7 +53.8 +81.7 +76.3 +57.1 +85.0 +71.3 +RSDA-MSTN (Gu et al., 2020)  +53.2 +77.7 +81.3 +66.4 +74.0 +76.5 +67.9 +53.0 +82.0 +75.8 +57.8 +85.4 +70.9 +Source Only + +44.6 +67.3 +74.8 +52.7 +62.7 +64.8 +53.0 +40.6 +73.2 +65.3 +45.4 +78.0 +60.2 ++ELR + +52.4 +73.5 +77.3 +62.5 +70.6 +71.0 +61.1 +50.8 +78.9 +71.7 +56.7 +81.6 +67.3 +SHOT (Liang et al., 2020) + +57.1 +78.1 +81.5 +68.0 +78.2 +78.1 +67.4 +54.9 +82.2 +73.3 +58.8 +84.3 +71.8 ++ELR + +58.7 +78.9 +82.1 +68.5 +79.0 +77.5 +68.2 +57.1 +81.9 +74.2 +59.5 +84.9 +72.6 +G-SFDA (Yang et al., 2021b) + +55.8 +77.1 +80.5 +66.4 +74.9 +77.3 +66.5 +53.9 +80.8 +72.4 +59.7 +83.2 +70.7 ++ELR + +56.4 +77.6 +81.1 +67.1 +75.2 +77.9 +65.9 +55.0 +81.2 +72.1 +60.0 +83.6 +71.1 +NRC (Yang et al., 2021a) + +56.3 +77.6 +81.0 +65.3 +78.3 +77.5 +64.5 +56.0 +82.4 +70.0 +57.1 +82.9 +70.8 ++ELR + +58.4 +78.7 +81.5 +69.2 +79.5 +79.3 +66.3 +58.0 +82.6 +73.4 +59.8 +85.1 +72.6 +that PCL (Zhang et al., 2021) leverages ETP at every epoch, so it cannot capture the benefits of +ETP and is inappropriate for unbounded label noise due to the fast memorization speed in SFDA. +As a comparison, we choose ELR since it leverages ETP at every batch. The blue lines show that +leveraging ETP via ELR can address the memorization of noisy labels in SFDA. +5 +EXPERIMENTS +We aim to improve the efficiency of existing SFDA algorithms by using ELR to leverage ETP. We +evaluate the performance on four different SFDA benchmark datasets: Office-31 (Saenko et al., 2010), +Office-Home (Venkateswara et al., 2017), VisDA (Peng et al., 2017) and DomainNet (Peng et al., +2019). Due to the limited space, the results on the dataset Office-31 and additional experimental +details are provided in Appendix G. +Evaluation. We incorporate ELR into three existing baseline methods: SHOT (Liang et al., 2020), +G-SFDA (Zhang & Sabuncu, 2018), and NRC (Yang et al., 2021a). SHOT uses k-means clustering +and mutual information maximization strategy to train the representation network while freezing the +final linear layer. G-SFDA aims to cluster target data with similar neighbors and attempts to maintain +the source domain performance. NRC also explores the neighbors of target data by graph-based +methods. ELR can be easily embedded into these methods by simply adding the regularization term +into the loss function to optimize without affecting existing SFDA frameworks. We average the +results based on three random runs. +Results. Tables 1-4 show the results before/after leveraging the early-time training phenomenon, +where Table 4 is shown in Appendix G. Among these tables, the top part shows the results of +conventional UDA methods, and the bottom part shows the results of SFDA methods. In the tables, +we use SF to indicate whether the method is source free or not. We use Source Only + ELR to +indicate ELR with self-training. The results show that ELR itself can boost the performances. As +existing SFDA methods are not able to address unbounded label noise, incorporating ELR into these +SFDA methods can further boost the performance. The four datasets, including all 31 pairs (e.g., +A → D) of tasks, show better performance after solving the unbounded label noise problem using the +early-time training phenomenon. Meanwhile, solving the unbounded label noise on existing SFDA +methods achieves state-of-the-art on all benchmark datasets. These SFDA methods also outperform +most methods that need to access source data. +Analysis about hyperparameters β and λ. +The hyperparameter β is chosen from {0.5, 0.6, +0.7, 0.8, 0.9, 0.99}, and λ is chosen from {1, 3, 7, 12, 25}. We conduct the sensitivity study on +hyperparameters of ELR on the DomainNet dataset, which is shown in Figure 3(a-b). In each Figure, +the study is conducted by fixing the other hyperparameter to the optimal one. The performance is +robust to the hyperparameter β except β = 0.99. When β = 0.99, classifiers are sensitive to changes +in learning curves. Thus, the performance degrades since the learning curves change quickly in the +unbounded label noise scenarios. Meanwhile, the performance is also robust to the hyperparameter λ +except when λ becomes too large. The hyperparameter λ is to balance the effects of existing SFDA +7 + +Published as a conference paper at ICLR 2023 +algorithms and the effects of ELR. As we indicated in Tables 1-4, barely using ELR to address the +SFDA problem is not comparable to these SFDA methods. Hence, a large value of λ makes neural +networks neglect the effects of these SFDA methods, leading to degraded performance. +Table 2: Accuracies (%) on DomainNet for ResNet50-based methods. +Method +SFR→CR→PR→SC→RC→PC→SP→RP→CP→SS→RS→CS→P Avg +MCD (Saito et al., 2018b) + 61.9 69.3 56.2 79.7 56.6 53.6 83.3 58.3 60.9 81.7 56.2 66.7 65.4 +DANN (Ganin et al., 2016) + 63.4 73.6 72.6 86.5 65.7 70.6 86.9 73.2 70.2 85.7 75.2 70.0 74.5 +DAN (Long et al., 2015) + 64.3 70.6 58.4 79.4 56.7 60.0 84.5 61.6 62.2 79.7 65.0 62.0 67.0 +COAL (Tan et al., 2020) + 73.9 75.4 70.5 89.6 70.0 71.3 89.8 68.0 70.5 88.0 73.2 70.5 75.9 +MDD (Zhang et al., 2019b) + 77.6 75.7 74.2 89.5 74.2 75.6 90.2 76.0 74.6 86.7 72.9 73.2 78.4 +Source Only + 53.7 71.6 52.9 70.8 49.5 58.3 85.2 59.6 59.1 30.6 74.8 65.7 61.0 ++ELR + 70.2 81.7 61.7 79.9 63.8 67.0 90.0 72.1 66.8 85.1 78.5 68.8 73.8 +SHOT (Liang et al., 2020) + 73.3 80.1 65.8 91.4 74.3 69.2 91.9 77.0 66.2 87.4 81.3 75.0 77.7 ++ELR + 78.0 81.9 67.4 91.1 75.9 71.0 92.6 79.3 68.0 88.7 84.8 77.0 79.7 +G-SFDA (Yang et al., 2021b)  65.8 78.9 60.2 80.5 64.7 64.6 89.3 69.9 63.6 86.4 78.8 71.1 72.8 ++ELR + 69.4 80.9 60.6 81.3 67.2 66.4 90.2 73.2 64.9 87.6 82.1 71.0 74.6 +NRC (Yang et al., 2021a) + 69.8 81.1 62.9 83.4 74.4 66.3 90.3 73.4 65.2 88.2 82.2 75.8 76.4 ++ELR + 75.6 82.2 65.7 91.2 77.2 68.5 92.7 79.8 67.5 89.3 85.1 77.6 79.4 +(a) +ACBHicbVA9SwNBEN2LXzF+nVqmWQyCVbgTUcugjWUE8wHJEfY2c8mSvQ9258RwpLDxr9hYKGLrj7Dz37hJ +rtDEBwOP92aYmecnUmh0nG+rsLK6tr5R3Cxtbe/s7tn7B0dp4pDg8cyVm2faZAigYKlNBOFLDQl9DyR9dTv3UPSo +s4usNxAl7IBpEIBGdopJ5d7iI8YDY0lkqYiEgKDrp+oCsZ1ecqjMDXSZuTiokR71nf3X7MU9DiJBLpnXHdRL0MqZQ +cAmTUjfVkDA+YgPoGBqZbdrLZk9M6LFR+jSIlakI6Uz9PZGxUOtx6JvOkOFQL3pT8T+vk2Jw6WUiSlKEiM8XBamkGN +NpIrQvFHCUY0MYV8LcSvnQRMFNErpkQnAX14mzdOqe149uz2r1K7yOIqkTI7ICXHJBamRG1InDcLJI3kmr+TNerJe +rHfrY95asPKZQ/IH1ucP1sSY3A=hyperparameter � +(b) +(c) embedding different LLN +methods into SFDA algorithms +ACBnicbVC7SgNBFJ31GeMrainCYBCswq4E +tQzaWEYwD0hCmJ29mwyZnV1m7ophSWXjr9hYKGLrN9j5N04ehSYeGDicx9zj59IYdB1v52l5ZXVtfXcRn5za3tnt7C3XzdxqjnUeCxj3fSZASkU1FCghGaigUW+hIY/uB7jXvQRsTqDocJdCLWUyIUnKGVuoWjNsIDZn1r6YRpFgG +CpqO2tCMC1i0U3ZI7AV0k3owUyQzVbuGrHcQ8jUAhl8yYlucm2MmYRsEljPLt1EDC+ID1oGWpsvtMJ5ucMaInVgloGv7FNKJ+rsjY5Exw8i3lRHDvpn3xuJ/XivF8LKTCZWkCIpPF4WpBjTcSY0EBo4yqEljGth/0p534bBbRYmb +0Pw5k9eJPWzkndeKt+Wi5WrWRw5ckiOySnxyAWpkBtSJTXCySN5Jq/kzXlyXpx352NauTMeg7IHzifP2KNmbU=hyperparameter � +Figure 3: (a)-(b) show the test accuracy on the DomainNet dataset with respect to hyperparameters +of ELR. (c) shows the test accuracy of incorporating various existing LLN methods into the SFDA +methods on the DomainNet dataset. +5.1 +DISCUSSION ON EXISTING LLN METHODS +As we formulate the SFDA as the problem of LLN, it is of interest to discuss some existing LLN +methods. We mainly discuss existing LLN methods that can be easily embedded into the current +SFDA algorithms. Based on this principle, we choose GCE (Zhang & Sabuncu, 2018), SL (Wang +et al., 2019b) and GJS (Englesson & Azizpour, 2021) that have been theoretically proved to be robust +to symmetric and asymmetric label noise, which are bounded label noise. We highlight that a more +recent method GJS outperforms ELR in real-world noisy datasets. However, we will show that GJS +is inferior to ELR in SFDA scenarios, because the underlying assumption for GJS does not hold in +SFDA. Besides ELR, which leverages ETP, PCL is another method to leverage the same phenomenon, +but we will show that it is also inappropriate for SFDA. +To show the effects of the existing LLN methods under the unbounded label noise, we test these LLN +methods on various SFDA datasets with target data whose labels are generated by source models. +As shown in Figure 4, GCE, SL, GJS, and PCL are better than CE but still not comparable to ELR. +Our analysis indicates that ELR follows the principle of ETP, which is theoretically justified in +SFDA scenarios by our Theorem 3.1. Methods GCE, SL, and GJS follow the bounded label noise +assumption, which does not hold in SFDA. Hence, they perform worse than ELR in SFDA, even +though GJS outperforms ELR in conventional LLN scenarios. PCL (Zhang et al., 2021) utilizes +ETP to purify noisy labels of target data, but it performs significantly worse than ELR. As the +memorization speed of the unbounded label noise is very fast, and classifiers memorize noisy labels +within a few iterations (shown in Figure 2), purifying noisy labels every epoch is inappropriate for +SFDA. However, we notice that PCL performs relatively better on DomainNet than on other datasets. +The reason behind it is that the memorization speed in the DomainNet dataset is relatively slow than +8 + +08 +上 +78 +77 +NRC +76 +SHOT +0.5 +0.6 +0.7 +0.8 +0.9 +0.9908 +79 +P +78 +Test +77 +NRC +1 +76 +SHOT +1 +3 +7 +12 +2580 +SHOT +NRC +Test Accuracy +779 +78 +77 +Vanilla +GCE +SL +GJS +ELRPublished as a conference paper at ICLR 2023 +Table 3: Accuracies (%) on VisDA-C (Synthesis → Real) for ResNet101-based methods. +Method +SFplanebcycl bus car horseknifemcyclpersonplantsktbrdtraintruckPer-class +DANN (Ganin et al., 2016) + 81.9 77.7 82.844.3 81.2 29.5 65.1 +28.6 51.9 54.6 82.8 7.8 +57.4 +DAN (Long et al., 2015) + 87.1 63.0 76.542.0 90.3 42.9 85.9 +53.1 49.7 36.3 85.8 20.7 +61.1 +ADR (Saito et al., 2018a) + 94.2 48.5 84.072.9 90.1 74.2 92.6 +72.5 80.8 61.8 82.2 28.8 +73.5 +CDAN (Long et al., 2018) + 85.2 66.9 83.050.8 84.2 74.9 88.1 +74.5 83.4 76.0 81.9 38.0 +73.9 +SAFN (Xu et al., 2019a) + 93.6 61.3 84.170.6 94.1 79.0 91.8 +79.6 89.9 55.6 89.0 24.4 +76.1 +SWD (Lee et al., 2019) + 90.8 82.5 81.770.5 91.7 69.5 86.3 +77.5 87.4 63.6 85.6 29.2 +76.4 +MDD (Zhang et al., 2019b) + +- +- +- +- +- +- +- +- +- +- +- +- +74.6 +MCC (Jin et al., 2020) + 88.7 80.3 80.571.5 90.1 93.2 85.0 +71.6 89.4 73.8 85.0 36.9 +78.8 +STAR (Lu et al., 2020) + 95.0 84.0 84.673.0 91.6 91.8 85.9 +78.4 94.4 84.7 87.0 42.2 +82.7 +RWOT (Xu et al., 2020) + 95.1 80.3 83.790.0 92.4 68.0 92.5 +82.2 87.9 78.4 90.4 68.2 +84.0 +Source Only + 60.9 21.6 50.967.6 65.8 6.3 +82.2 +23.2 57.3 30.6 84.6 8.0 +46.6 ++ELR + 95.4 45.7 89.769.8 94.1 97.1 92.9 +80.1 89.7 52.8 83.3 4.3 +74.6 +SHOT (Liang et al., 2020) + 94.3 88.5 80.157.3 93.1 94.9 80.7 +80.3 91.5 89.1 86.3 58.2 +82.9 ++ELR + 95.8 84.1 83.367.9 93.9 97.6 89.2 +80.1 90.6 90.4 87.2 48.2 +84.1 +G-SFDA (Yang et al., 2021b)  96.0 87.6 85.372.8 95.9 94.7 88.4 +79.0 92.7 93.9 87.2 43.7 +84.8 ++ELR + 97.3 89.1 89.879.2 96.9 97.5 92.2 +82.5 95.8 94.5 87.3 34.5 +86.4 +NRC (Yang et al., 2021a) + 96.9 89.7 84.059.8 95.9 96.6 86.5 +80.9 92.8 92.6 90.2 60.2 +85.4 ++ELR + 97.1 89.7 82.762.0 96.2 97.0 87.6 +81.2 93.7 94.1 90.2 58.6 +85.8 +(i) Office-31 +(ii) Office-Home +(iii) VisDA +(iv) DomainNet +Figure 4: Evaluation of label noise methods on SFDA problems. We use source models as an +initialization of classifiers trained on target data and also use source models to annotate unlabeled +target data. Then we treat the target datasets as noisy datasets and use different label noise methods +to solve the memorization issue. +other datasets, which is shown in Figure 2. In conventional LLN scenarios, PCL does not suffer from +the issue since the memorization speed is much lower than the conventional LLN scenarios. +In Figure 3(c), we also evaluate the performance by incorporating the existing LLN methods into the +SFDA algorithms SHOT and NRC. Since PCL and SHOT assign pseudo labels to target data, PCL is +incompatible with some existing SFDA methods and cannot be easily embedded into some SFDA +algorithms. Hence, we only embed GCE, SL, GJS, and ELR into the SFDA algorithms. The figure +illustrates that ELR still performs better than other LLN methods when incorporated into SHOT and +NRC. We also notice that GCE, SL, and GJS provide marginal improvement to the vanilla SHOT +and NRC methods. We think the label noise in SFDA datasets is the hybrid noise that consists of +both bounded label noise and unbounded label noise due to the non-linearity of neural networks. The +GCE, SL, and GJS can address the bounded label noise, while ELR can address both bounded and +unbounded label noise. Therefore, these experiments demonstrate that using ELR to leverage ETP +can successfully address the unbounded label noise in SFDA. +6 +CONCLUSION +In this paper, we study SFDA from a new perspective of LLN by theoretically showing that SFDA +can be viewed as the problem of LLN with the unbounded label noise. Under this assumption, +we rigorously justify that robust loss functions are not able to address the memorization issues of +unbounded label noise. Meanwhile, based on this assumption, we further theoretically and empirically +analyze the learning behavior of models during the early-time training stage and find that ETP can +benifit the SFDA problems. Through extensive experiments across multiple datasets, we show that +ETP can be exploited by ELR to improve prediction performance, and it can also be used to enhance +existing SFDA algorithms. +9 + +75 +70 +Test Accuracy +65 +60 +55 +50 +CE +GCE +SL +GJSPCL E +ELR73 +72 +71 +70 +69 +68 +67 +66 +65 +CE +GCE +SL +GJS +PCL +ELR85.0 +84.5 +2 + Accurae +84.0 +83.5 +Test +83.0 +82.5 +82.0 +CE +GCE +SL +GJS +PCL +ELR68 +67 +Test Accuracy +66 +65 +64 +63 +62 +61 +60 +CE +GCE +SL +GJSPCL E +ELRPublished as a conference paper at ICLR 2023 +REFERENCES +Sk Miraj Ahmed, Dripta S Raychaudhuri, Sujoy Paul, Samet Oymak, and Amit K Roy-Chowdhury. +Unsupervised multi-source domain adaptation without access to source data. 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In Proceedings of the IEEE/CVF Conference on Computer Vision and +Pattern Recognition, pp. 10113–10123, 2021. +13 + +Published as a conference paper at ICLR 2023 +A +NEIGHBORS LABEL NOISE OBSERVATIONS ON REAL-WORLD DATASETS +This section provides more observed results and explanations of Neighbors’ label noise during the +Source-Free Domain Adaptation process on real-world datasets. +1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465 +Class Index +0.00 +0.10 +0.20 +0.30 +0.40 +0.50 +0.60 +0.70 +0.80 +True Neighbors Ratio +False Neighbors Ratio +Figure 5: True/False Neighbors on Office-Home +Plane Bcycl Bus +Car Horse Knife McyclPersonPlant Sktbrd Train Truck +0.00 +0.10 +0.20 +0.30 +0.40 +0.50 +0.60 +0.70 +0.80 +True Neighbors Ratio +False Neighbors Ratio +Figure 6: True/False Neighbors on VisDA +1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 +Class Index +0.00 +0.20 +0.40 +0.60 +0.80 +1.00 +Misleading Neighbors Ratio +Over-Confident Misleading +Neighbors Ratio +Figure 7: Neighbors Label Noise Analysis on Office-Home +Currently, most SFDA methods inevitably leverage the pseudo-labels for self-supervised learning +or to learn the cluster structure of the target data in the feature space, in order to realize the domain +adaptation goal. However, the pseudo labels generated by the source domain are usually noisy and +of poor quality due to the domain distribution shift. Some neighborhood-based heuristic methods +(Yang et al., 2021a;b) have been proposed to purify these target domain pseudo labels, which use +the pseudo label of neighbors in the feature space to correct and reassign the central data’s pseudo +label. In fact, such methods rely on a strong assumption: a relatively high quality of the neighbors’ +pseudo label. However, in our experimental observations, we find that at the very beginning of the +adaptation process, the similarity of two data points in the feature space can not fully represent their +label space’s connection. Furthermore, such methods are easy to provide useless and noisy prediction +information for the central data. We will show some statistical results on VisDA and Office-Home, +these two real-world datasets. +Following the neighborhood construction method in Yang et al. (2021a;b), we use the pre-trained +source model to infer the target data, extract the feature space outputs and get the prediction results. +We use the cosine similarity on the feature space to find the top k similar neighbors (e.g., k = 2) +for each data point (named as the central data point). Then, we collect the neighbors regarding the +ground truth label of central data points and study the neighbor’s quality for each class. +14 + +Published as a conference paper at ICLR 2023 +Neighbors who do not belong to the correct category We define the neighbors who do not belong +to the same category as its central data point as False Neighbor, which means their ground-truth +labels are not the same: Yneighbor ̸= Ycentral. And the results of VisDA (train → validation) and +Office-Home (Pr → Cl) datasets are shown in Figure 6 and Figure 5. +Neighbors who can not provide useful prediction information We further study the prediction +information provided by such neighbors. Regardless of their true category properties, we consider +neighbors whose Predicted Label is the same as the Ground Truth Label of the central data point to +be Useful Neighbors; otherwise, they are Misleading Neighbors, as they can not provide the expected +useful prediction information. We denote the Misleading Neighbors Ratio as the proportion of noisy +neighbors among all neighbors for each class. Besides, as some methods heuristically utilize the +predicted logits as the predicted probability or confidence score in the pseudo label purification +process, we further study the Over-Confident Misleading Neighbors Ratio for each class. We defined +the over-confident misleading neighbors ratio as the number of over-confident misleading neighbors +(misleading neighbors with a high predicted logit, larger than 0.75) divided by the number of all +neighbors per class. The results on VisDA and Office-Home are shown in Figure 1ii and Figure 7. +We want to clarify that the above exploratory experiment results can only reflect the phenomenon +of unbounded noise in SFDA to some extent: the set of over-confidence misleading neighbors is +non-empty can correspond, to some extent, to the fact that R is non-empty proved in Theorem 3.1; +but the definition of misleading neighbors does not rigorously satisfies the definition of unbounded +label noise. +B +RELATIONSHIP BETWEEN MISLABELING ERROR AND DOMAIN SHIFT +In this part, we focus on explaining the relationship between the label noise and the domain shift, as +illustrated in Figure 9. The following theorem characterizes the relationship between the labeling +error and the domain shift. +Theorem B.1. Without loss of generality, we assume that the ∆ is positively correlated with the +vector µ2 − µ1, i.e., ∆⊤(µ2 − µ1) > 0. Let fS be the Bayes optimal classifier for the source +domain S. Then +Pr +(x,y)∼DT[fS(x) ̸= y] = 1 +2Φ(−d1 +σ ) + 1 +2Φ(−d2 +σ ), +(6) +where d1 = +�� µ2−µ1 +2 +− c +�� sign( +�� µ2−µ1 +2 +�� − ∥c∥), d2 = +�� µ2−µ1 +2 ++ c +��, c = (µ2 − µ1) ∆⊤(µ2−µ1) +∥µ2−µ1∥2 , +and Φ is the standard normal cumulative distribution function. +Theorem B.1 indicates that the labeling error for the target domain can be represented by a function +of the domain shift ∆, which can be shown numerically in Figure 8. The projection of the domain +shift ∆ on the vector µ2 − µ1 is given by c. Since c is on the direction of µ2 − µ1, c can also be +represented by α(µ2 − µ1), where α ∈ R characterizes the magnitude of the domain shift. More +specifically, in Figure 8, we present the relationship between the mislabeling rate and α for all possible +∆. When ∆ is positively correlated with µ2 − µ1 (assumption in Theorem B.1), we have α > 0, and +when ∆ is negatively correlated with µ2 − µ1, we obtain α < 0. In both situations, we can observe +that the labeling error increases with the absolute value of α increasing, which implies that the more +severe the domain shift is, the greater the mislabeling error will be obtained. Besides, we note that +when the source and target domains are the same, the mislabeling error in Eq. (6) is minimized and +degraded to the Bayes error, which cannot be reduced (Fukunaga, 2013). This corresponds to the +situation when ∆ is perpendicular to µ2 − µ1, c = 0, and α = 0 shown in Figure 8. +B.1 +PROOFS FOR THEOREM B.1 +Proof. The Bayes classifier fS predicts x to the first component when +log Pr[y = 1|X = x] +Pr[y = −1|X = x] > 0. +(7) +Since the distributions of the two components with the same priors for the source domain are given +by N(µ1, σ2Id) and N(µ2, σ2Id), respectively. Based on Bayes’ rule, Eq. (7) is equivalent to +log Pr[X = x|y = 1] +Pr[X = x|y = −1] > 0 +(8) +15 + +Published as a conference paper at ICLR 2023 +−1.5 +−1.0 +−0.5 +0.0 +0.5 +1.0 +1.5 +α = ∆T (µ2−µ1) +||µ2−µ1||2 = sign(c) +||c|| +||µ2−µ1|| +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Mislabelling Rate on All Target Data +Figure 8: Plot of Mislabeling Rate with different α. We define c as the projection of the domain shift +∆ on the vector µ2 − µ1, and α represents the magnitude of domain shift projected on µ2 − µ1. +Solving the left hand side of Eq. (8) by using the knowledge of two multivariate Gaussian distributions, +we get +hS(x) := log Pr[X = x|y = 1] +Pr[X = x|y = −1] = x⊤(µ1 − µ2) +σ2 +− ∥µ1∥2 − ∥µ2∥2 +2σ2 +. +(9) +So fS predicts x to the first component when hS(x) > 0 and fS predicts x to the second component +when hS(x) ≤ 0 The decision boundary is z such that hS(z) = 0. When there is no domain shift +∆ = 0, we have DS = DT , and the mislabeling rate is the Bayes error, which is given by: +Pr +(x,y)∼DS[fS(x) ̸= y] = 1 +2 +Pr +x∼N (µ1,σ2Id)[hS(x) < 0|y = 1] + 1 +2 +Pr +x∼N (µ2,σ2Id)[hS(x) > 0|y = −1] +(10) +We first study the first term in Eq. (10): +Pr +x∼N (µ1,σ2Id)[hS(x) < 0|y = 1] += +� +· · · +� +{x|x⊤(µ1−µ2)< ∥µ1∥2−∥µ2∥2 +2 +} +1 +(2πσ2) +d +2 exp +� +−∥x − µ1∥2 +2σ2 +� +dx1dx2 · · · dxd += +� +· · · +� +{x|−∞AB9HicbVBNS8 +NAEJ3Ur1q/qh69LBbBU0lE1GPRi8cK9gPaUDababt0s4m7m0IJ/R1ePCji1R/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19 +Y3NreJ2aWd3b/+gfHjU1HGqGDZYLGLVDqhGwSU2DcC24lCG +gUCW8Hobua3xqg0j+WjmSToR3QgeZ8zaqzkd1MZogoUZhNe+WKW +3XnIKvEy0kFctR75a9uGLM0QmYoFp3PDcxfkaV4UzgtNRNSaUjegAO5ZKGqH2s/nRU3JmlZD0Y2VLGjJXf09kNJ6EgW2 +M6JmqJe9mfif10 +lN/8bPuExSg5ItFvVTQUxMZgmQkCtkRkwsoUxeythQ2ojMDankg3BW35lTQvqt5V9fLhslK7zeMowgmcwjl4 +cA01uIc6NIDBEzDK7w5Y+fFeXc+Fq0FJ585hj9wPn8ASzKSdA=|{z} +Mislabeling Area: +with a very high probability +Figure 9: Illustration of noisy labels generated by the domain shift. +verify that µ2 − µ1 is perpendicular to the hyperplane. Thus, we project the domain shift ∆ onto the +vector µ2 − µ1 to get the component of ∆ that is perpendicular to the hyperplane, which is given by: +c = (µ2 − µ1)∆⊤(µ2 − µ1) +∥µ2 − µ1∥2 . +(12) +Since we assume ∆ is positively correlated to the vector µ2 − µ1, α = ∆⊤(µ2−µ1) +∥µ2−µ1∥2 +can be regarded +as the magnitude of the domain shift along the direction µ2 − µ1. Note that the results also hold for +the case where ∆ is negatively correlated to µ2 − µ1. The whole proof can be obtained by following +the very similar proof steps for the positively correlated case. +The mislabeling rate of the optimal source classifier fS on target data is: +Pr +(x,y)∼DT[fS(x) ̸= y] = 1 +2 +Pr +N (µ1+∆,σ2Id)[hS(x) < 0|y = 1]+ 1 +2 +Pr +N (µ2+∆,σ2Id)[hS(x) > 0|y = −1] +(13) +We first calculate the first term of Eq. (13). Following the same tricks discussed above: +Pr +x∼N (µ1+∆,σ2Id)[hS(x) < 0|y = 1] += +Pr +x∼N (µ1+c,σ2Id)[hS(x) < 0|y = 1] += +� +· · · +� +{x|x⊤(µ1−µ2)< ∥µ1∥2−∥µ2∥2 +2 +} +1 +(2πσ2) +d +2 exp +� +−∥x − µ1 − ∆∥2 +2σ2 +� +dx1dx2 · · · dxd += +� +· · · +� +{x|−∞ 0|y = −1] =Φ(−d2 +σ ), +(15) +where d2 = +�� µ2−µ1 +2 ++ c +��. +Taking Eq. (14) and Eq. (15) into Eq. (13), we have +Pr +(x,y)∼DT[fS(x) ̸= y] = 1 +2Φ(−d1 +σ ) + 1 +2Φ(−d2 +σ ). +(16) +C +PROOFS FOR THEOREM 3.1 +Proof. Without loss of generality, we choose to assume µ2 = µ1 + σ1d as the convenient way to +present our results. From the proof for Theorem B.1, we know that x0 = µ1+µ2 +2 ++ ∆ is at the +decision boundary such that hT (x0) = 0, where +hT (x) = x⊤(µ1 − µ2) +σ2 +− ∥µ1 + ∆∥2 − ∥µ2 + ∆∥2 +2σ2 +. +Let fT be the optimal Bayes classifier for the target domain, which can be obtained the same way as +fS mentioned in B.1. The equation hT (x0) = 0 implies that +Pr +(x,y)∼DT[y = 1|X = x0] = +Pr +(x,y)∼DT[y = −1|X = x0]. +Note that x0 is on the affine hyperplane z where hT (z) = 0. Any data points on this hyperplane will +have the equal probabilities to be correctly classified. We start from this hyperplane and calculate +another point x1, where Pr(x,y)∼DT [y = 1|X = x1] is at least 1−δ +δ +Pr(x,y)∼DT [y = −1|X = x1]. +Thus, for any points that are mislabeled and far away from x1 will result in Pr(x,y)∼DT [y = 1|X = +x1] ≥ 1 − δ. We first aim to find such a data point x1. Let x1 = x0 − m0σ1d, where m0 is the scalar +measures the distance between the point x1 to the hyperplane z. We need to find m0 such that +PT (x1|y = 1) +PT (x1|y = −1) ≥1 − δ, +(17) +where +PT (x1|y = 1) +PT (x1|y = −1) += exp +� +− ∥x1 − µ1 − ∆∥2 +2σ2 ++ ∥x1 − µ2 − ∆∥2 +2σ2 +� += exp +� +− +�� µ2−µ1 +2 +− m0σ1d +��2 +2σ2 ++ +�� µ2−µ1 +2 ++ m0σ1d +��2 +2σ2 +� += exp (m0d) +(18) +Taking Eq. (18) into Eq. (17), we get m0 ≥ (log 1−δ +δ )/d. Since the isotropic Gaussian random +vectors has the rotationally symmetric property, we can transform the integration of multivariate +normal distribution to standard normal distribution with different intervals of integration. Then any +data points from a region that have at most ∥x1 − µ1 − ∆∥ distance to its mean µ1 + ∆ will have +at least 0.99 probability coming from the first component. Let the region R1 be: +R1 = {x : ∥x − µ1 − ∆∥ ≤ ∥x1 − µ1 − ∆∥} +Equivalently, taking R1 can be simplified: +R1 = {x : ∥x − µ1 − ∆∥ ≤ σ( +√ +d +2 − log 1−δ +δ +√ +d +)} +18 + +Published as a conference paper at ICLR 2023 +The region R1 is valid when data dimension d is large. This is realistic in practice. Since neural +networks are usually dealing with high dimension data, for example d ≫ (1), the region R1 is valid. +On the other hand, we aim to find a region R2 where all data points are mislabeled. From the proof +for Theorem 1, the source classifier hS is given by +hS(x) = x⊤(µ1 − µ2) +σ2 +− ∥µ1∥2 − ∥µ2∥2 +2σ2 +. +(19) +Any data points are classified to the second component if hS(x) < 0. Hence +R2 = {x : x⊤1d > σd + 2µ⊤ +1 1d +2 +} +We take the intersection of R1 and R2, all data points from this intersection are (1) having at least +1 − δ probability coming from the first component, and (2) being classified to the second component. +Formally, for (x, y) ∼ DT , if x ∈ R1 +� R2, then +Pr[fS(x) ̸= y] ≥ 1 − δ, +(20) +We note that x ∈ R1 +� R2 is non-empty when (log 1−δ +δ )/d < α, where α = ∆⊤(µ2−µ1) +∥µ2−µ1∥2 +is the +magnitude of the domain shift along with the direction µ2 − µ1. Since x1 is chosen from R1, to +verify that R1 +� R2 is non-empty, we only need to verify that x1 also belongs to R2. +x1 ∈ R2 if and only if: +x⊤ +1 1d >σd + 2µ⊤ +1 1d +2 +(µ1 + c + σ +2 1d − m0σ1d)⊤1d >σd + 2µ⊤ +1 1d +2 +(µ1 + ασ1d + σ +2 1d − m0σ1d)⊤1d >σd + 2µ⊤ +1 1d +2 +(α − m0)σd >0, +where c = (µ2 − µ1) ∆⊤(µ2−µ1) +∥µ2−µ1∥2 . +Therefore, if α > m0 ≥ (log 1−δ +δ )/d, R1 +� R2 is non-empty. +Next, we show Pr(x,y)∼DT [x ∈ R] increases as α increases. +Let event A0 be a set of x such that they are mislabeled by fS (i.e. fS(x) ̸= y). Let event A1 be a +set of x such that they are from the first component but are mislabeled to the second component with +a probability Pr[fS(x ̸= y)] < 1 − δ. Let event A2 be a set of x such that they are from the second +component but are mislabeled to the first component with a probability Pr[fS(x ̸= y)] < 1 − δ. Thus +Pr +(x,y)∼DT[x ∈ R] = +Pr +(x,y)∼DT[A0] − +Pr +(x,y)∼DT[A1] − +Pr +(x,y)∼DT[A2] +(21) +Let event A3 be a set of x such that they are from the first component such that Pr[fS(x ̸= y)] < 1−δ +or Pr[fS(x = y)] < 1 − δ. Let event A4 be a set of x such that they are from the second component +but are mislabeled to the first component. For Pr[A3], +Pr +(x,y)∼N (µ1+∆,σ2Id)[A3] = +Pr +(x,y)∼N (µ1+∆,σ2Id)[R∁ +1], +which does not change as the domain shift ∆ varies. Meanwhile, +Pr +(x,y)∼N (µ2+∆,σ2Id)[A4] = Φ(− +�� µ2−µ1 +2 ++ c +�� +σ +), +which is given by Eq. (15). By our assumption, the domain shift ∆ is positively correlated with the +vector µ2 − µ1. So when α increases, Pr(x,y)∼N (µ2+∆,σ2Id)[A4] decreases. +19 + +Published as a conference paper at ICLR 2023 +Since A1 ⊆ A3 and A2A4, the probability measure on R is given by: +Pr +(x,y)∼DT[x ∈ R] = +Pr +(x,y)∼DT[A0] − +Pr +(x,y)∼DT[A1] − +Pr +(x,y)∼DT[A2] +≥ +Pr +(x,y)∼DT[A0] − +Pr +(x,y)∼DT[A3] − +Pr +(x,y)∼DT[A4], +(22) +where the first term is the mislabeling rate that increases as α increases (given by Theorem B.1); the +second term is a constant; the third term decreases as as α increases. The equality in Eq. (22) holds +when α → ∞. Therefore, when the magnitude of the domain shift α increases, the lower bound of +Pr(x,y)∼DT [x ∈ R] increases, which forces more points to break the conventional LLN assumption. +D +BACKGROUND INTRODUCTION AND PROOFS FOR LEMMA 3.2 +Learning with label noise is an important task and topic in deep learning and modern artificial +intelligence research. The main idea behind it is robust training, which can be further divided into +fine-grained categories, such as robust architecture, robust regularization, robust loss design, and +simple selection (Song et al., 2022). For example, for the robust architecture-based methods, they +propose to modify the deep model’s architecture, including adding an adaptation layer or leveraging +a dedicated module, to learn the label transition process and to tackle the noisy label. In addition, the +robust regularization approaches usually enforce the DNN to overfit less to false-labeled examples +by adopting a regularizer, explicitly or implicitly. For instance, Yi et al. (2022) proposed to utilize +a contrastive regularization term to learn a noisy-label robust representation. In this paper, we will, +however, develop our discussion based on the robust loss methods in LLN. +In this section, we will first introduce the concepts and technical details of some noise-robust loss +based LLN methods, including GCE (Zhang & Sabuncu, 2018), SL (Wang et al., 2019b), NCE (Ma +et al., 2020), and GJS (Englesson & Azizpour, 2021). Then, we will present the proof details of +Lemma 3.2. +D.1 +NOISE-ROBUST LOSS FUNCTIONS IN LLN METHODS +Among the numerous studies of LLN methods, loss correction is a major branch of research. The +main idea of loss correction is to modify the loss function and make it robust to noisy labels. +As indicated in Ma et al. (2020), the loss function ℓ is defined to be noise robust if �K +k=1 ℓ(h(x), k) = +C, where C is a positive constant and K is the overall class number of label space. For example, +the most widely utilized Cross-Entropy (CE) loss is unbounded and therefore is not robust to the +label noise. Some LLN studies show that existing loss functions such as mean absolute error (MAE) +(Ghosh et al., 2017), reverse cross entropy (RCE) (Wang et al., 2019b), normalized cross entropy +(NCE) (Ma et al., 2020), and normalized focal loss (NFL) are noise-robust and that combining them +with CE can help mitigate the sensitivity of the model to noisy labels. +More specifically, for a given data (x, y) and a classifier h(x), GCE (Zhang & Sabuncu, 2018) +leverages the negative Box-Cox transformation as a loss function, which can exploit the benefits of +both the noise-robustness provided by MAE and the implicit weighting scheme of CE: +ℓGCE(h(x), ek) = (1 − hk(x)q) +q +where q ∈ (0, 1] is a hyperparameter to be decided. +Another noise-robust loss based method SL (Wang et al., 2019b) proposes combining the reverse +cross entropy (RCE) loss, which is noise tolerant, with CE loss and obtain the LSL: +ℓSL = αℓCE + βℓRCE += −(α +K +� +k=1 +q(k|x)logp(k|x) + β +K +� +k=1 +p(k|x)logq(k|x)) +where p(k|x) is the predicted distribution over labels by classifier h(x) and q(k|x) is the ground +truth class distribution conditioned on sample x. +20 + +Published as a conference paper at ICLR 2023 +GJS (Zhang & Sabuncu, 2018) utilizes the multi-distribution generalization of Jensen-Shannon +Divergence as loss function, which has been proven noise-robust and is in fact a generalization of CE +and MAE. Concretely, the generalized JS divergence and GJS loss are defined as: +DGJSπ = +M +� +i=1 +πiDKL +� +p(i)��� +��� +M +� +j=1 +πjp(j)� +ℓGJS(x, y, h) = DGJSπ(y, h(˜x(2)), ..., h(˜x(M))) +Z +where π, p(i) are categorical distributions over K classes, ˜x(i) ∼ A(x), a random perturbation of +sample x, and Z = −(1 − π1)log(1 − π1) +Further, Ma et al. (2020) shows a simple loss normalization scheme which can be applied for any +loss L: +ℓNORM = +ℓ(h(x), y) +�K +k=1 ℓ(h(x), k) +The study found that the normalized loss can indeed satisfy the robustness condition. However, it +will also cause an underfitting problem in some situations. +Note that generalized cross entropy (GCE (Zhang & Sabuncu, 2018)) extends MAE and symmetric +loss (SL (Wang et al., 2019b)) extends RCE. So we study GCE and SL in our experiments instead +studying MAE and RCE. Besides, GJS (Englesson & Azizpour, 2021) is shown to be tightly bounded +around �K +k=1 ℓ(h(x), k). All these methods have shown to be noise tolerant under either bounded +random label noise or bounded class-conditional label noise with additional assumption that R(h⋆) = +0. We show that under the same assumption with unbounded label noise datasets, these methods are +not noise tolerant in section D.2. +D.2 +PROOFS FOR LEMMA 3.2 +Proof. Let ηyk(x) be the Pr[ ˜Y = k|Y = y, X = x] probability of observing a noisy label k given +the ground-truth label y and a sample x. Let ηy(x) = � +k̸=y ηyk(x). The risk of h under noisy data +is given by +�R(h) =Ex,˜y[ℓLLN(h(x), ˜y)] +=ExEy|xE˜y|x,y[ℓLLN(h(x), ˜y)] +=Ex,y +� +(1 − ηy(x))ℓLLN(h(x), y) + +� +k̸=y +ηyk(x)ℓLLN(h(x), k) +� +=Ex,y +� +(1 − ηy(x)) +� K +� +k=1 +ℓLLN(h(x), k) − +� +k̸=y +ℓLLN(h(x), k) +� ++ +� +k̸=y +ηyk(x)ℓLLN(h(x), k) +� +=Ex,y +� +(1 − ηy(x)) +� +C − +� +k̸=y +ℓLLN(h(x), k) +� ++ +� +k̸=y +ηyk(x)ℓLLN(h(x), k) +� +=Ex,y +� +(1 − ηy(x))C +� +− Ex,y +� � +k̸=y +� +1 − ηy(x) − ηyk(x) +� +ℓLLN(h(x), k) +� +. +(23) +Since Eq. (23) holds for both ˜h⋆ and h⋆, we have +�R(˜h⋆) = Ex,y +� +(1 − ηy(x))C +� +− Ex,y +� � +k̸=y +� +1 − ηy(x) − ηyk(x) +� +ℓLLN(˜h⋆(x), k) +� +(24) +and +�R(h⋆) = Ex,y +� +(1 − ηy(x))C +� +− Ex,y +� � +k̸=y +� +1 − ηy(x) − ηyk(x) +� +ℓLLN(h⋆(x), k) +� +. +(25) +21 + +Published as a conference paper at ICLR 2023 +As ˜h⋆ is the minimizer of �R(h), �R(˜h⋆) ≤ �R(h⋆). Then we combine Eq. (24) and Eq. (25), we have +Ex,y +� � +k̸=y +� +1 − ηy(x) − ηyk(x) +�� +ℓLLN(h⋆(x), k) − ℓLLN(˜h⋆(x), k) +�� +≤ 0. +(26) +We note that ℓLLN(˜h⋆(x), k) ≥ ℓLLN(h⋆(x), k) implies pk(x) = 0 and py(x) = 1 for k ̸= y, where +pk(x) is the probability output by ˜h⋆ for predicting the sample x to be the class k. This argument +is proved given by Wang et al. (2019b); Ghosh et al. (2017); Yang et al. (2021b); Ma et al. (2020) +(Theorem 1&2 in Ghosh et al. (2017), Theorem 1 in Wang et al. (2019b), Lemma 1&2 in Ma et al. +(2020) and Theorem 1&2 in Englesson & Azizpour (2021)). +To let ℓLLN(˜h⋆(x), k) ≥ ℓLLN(h⋆(x), k) holds for all inputs x, previous studies assume the bounded +label noise, which is given by +1 − ηy(x) − ηyk(x) > 0 ∀x s.t. P(X = x) > 0. +(27) +For random label noise which assumes that the mislabeling probability from the ground-truth label +to any other label is the same for all inputs, i.e. ηji(x) = a0 ∀i ̸= j, where a0 is a constant. Let +η = (K − 1)a0, then Eq. (27) is degraded to +1 − η − +η +K − 1 > 0 +1 > +K +K − 1η +η < 1 − 1 +K . +This bounded assumption is commonly assumed by Wang et al. (2019b); Ghosh et al. (2017); Yang +et al. (2021b); Ma et al. (2020) (Theorem 1 in Ghosh et al. (2017), Theorem 1 in Wang et al. (2019b), +Lemma 1 in Ma et al. (2020) and Theorem 1 in Englesson & Azizpour (2021)). +For class-conditional label noise, which assumes the ηji(x1) = ηji(x2) for any inputs x1 and x2. +Let ηji(x) = ηji, Then the bounded assumption Eq. (27) is degraded to +ηyk < 1 − ηy. +This bounded assumption is also commonly assumed, and it can be found in Theorem 2 in Ghosh +et al. (2017), Theorem 1 in Wang et al. (2019b), 2 in Ma et al. (2020) and Theorem 2 in Englesson & +Azizpour (2021). +However, in SFDA, we proved that the following event B holds with a probability at least 1 − δ: +1 − ηy(x) − ηyk(x) < 0 ∀x ∈ R. +(28) +Indeed, we first denote B1 = {˜y ̸= y|x ∈ R} by the event that x ∈ R is mislabeled. Then +Pr[B] = Pr[B|B1] + Pr[B|B∁ +1] Pr[B∁ +1] +≥ Pr[B|B1] Pr[B1] +≥ 1 − δ +Given the result in Eq. (28), and combined it with the Eq. (26), we have +ℓLLN(˜h⋆(x), k) ≤ ℓLLN(h⋆(x), k). +When the event B holds, the condition ℓLLN(˜h⋆(x), k) ≤ ℓLLN(h⋆(x), k) holds. +Note that only ℓLLN(˜h⋆(x), k) ≥ ℓLLN(h⋆(x), k) means pk(x) = 0 for k ̸= y and py(x) = 1 for +k ̸= y. It means that the optimal classifier ˜h⋆ from noisy data can make correct predictions on any +inputs, which is consistent with the optimal classifier h⋆ obtained from clean data. +As for the condition ℓLLN(˜h⋆(x), k) ≤ ℓLLN(h⋆(x), k), we can get pk(x) = 1 for a k ̸= y, which +means that the optimal classifier ˜h⋆ from noisy data cannot make correct predictions on samples +22 + +Published as a conference paper at ICLR 2023 +x ∈ R. To verify this, we use the robust loss function RCE ℓRCE as an example, and it can be easily +generalized to other robust los functions mentioned above. Based on the definition of the RCE loss +(Wang et al., 2019b), we have +ℓRCE(˜h⋆(x), k) =CRCE(1 − pk(x)) +ℓRCE(h⋆(x), k) =CRCE, +where CRCE > 0 is a constant. The above equations show that any 0 ≤ pk(x) ≤ 1 can make the +condition ℓLLN(˜h⋆(x), k) ≤ ℓLLN(h⋆(x), k) hold. Meanwhile, ˜h⋆ is the global minimizer of the risk +over the noisy data, which makes ˜h⋆ memorize the noisy dataset. +Therefore, ˜h⋆ makes incorrect predictions for x ∈ R such that pk(x) = 1 for a k ̸= y, and h⋆ is the +global optimal over clean data, which gives correct predictions for x ∈ R such that pk(x) = 1 for a +k = y. That completes the proof as h⋆ makes different predictions on x ∈ R compared to ˜h⋆. +E +PROOFS FOR THEOREM 4.1 +The proof for Theorem 4.1 is partially adopted from Liu et al. (2020). Note that we are dealing +with unbounded label noise, whereas the bounded label noise is considered in Liu et al. (2020). As +indicated in Liu et al. (2020), T is set as the smallest positive integer such that θ⊤ +T µ ≥ 0.1, and +T = Ω(1/η) with high probability. Parameters θ is initialized by Kaiming initialization (He et al., +2015) that θ0 ∼ N(0, 2 +dId), and |θ⊤ +0 µ| converges in probability to 0. For simplicity, we assume +θ0 = 0 without loss of generality. The proof consists of two parts. The first part is to show that θT −1 +is highly positively correlated with the ground truth classifier. The second part is to show that the +prediction accuracy on mislabeled samples can be represented as the correlation between the learned +classifier and the ground truth classifier. +Proof. To begin with, we show the first part. Let samples xi = yi(µ − σzi), where z ∼ N(0, Id). +The gradient of the logistic loss function with respect to the parameter θ is given by: +∇θL(θt) = 1 +2n +n +� +i=1 +xi +� +tanh(θ⊤ +t xi) − ˜yi +� += − 1 +2n +n +� +i=1 +˜yixi +� +�� +� +x ++ 1 +2n +n +� +i=1 +xitanh(θ⊤ +t xi) +� +�� +� +y +(29) +Then we will show that −µ⊤∇θL(θt) is lower bounded by a positive number. We first show the +bound on xin Eq. (29). Since xi is sampled from standard normal distribution, 1 +n +�n +i=1 ˜yiµ⊤xi has +limited variance. By the law of large number, 1 +n +�n +i=1 ˜yiµ⊤xi converges in probability to its mean. +Therefore, +E[˜yx⊤µ] =E[˜yµ⊤x1{yx⊤µ ≤ r}] + E[˜yµ⊤x1{yx⊤µ > r}] +=E[E[˜yµ⊤x1{yx⊤µ ≤ r}]|y] ++ E[E[˜yµ⊤x1{yx⊤µ > r}]|y] +=E[−µ⊤x1{x⊤µ ≤ r}|y = 1] + E[µ⊤x1{x⊤µ > r}|y = 1] +23 + +Published as a conference paper at ICLR 2023 +Note that x|y = 1 is a Gaussian random vector with independent entries, we have x⊤µ +d= w + 1, +where w ∼ N(0, σ2). Therefore, the above expectation is equivalent to +E[˜yx⊤µ] = − +� r−1 +−∞ +(w + 1) dPw + +� ∞ +r−1 +(w + 1) dPw += − +� r−1 +−∞ +w dPw + +� +∞ +r−1 +w dPw − +� r−1 +−∞ +dPw + +� +∞ +r−1 +dPw += +� 1−r +r−1 +dPw − +� r−1 +−∞ +w dPw + +� +∞ +r−1 +w dPw +=Erf[1 − r +√ +2σ ] + 2 σ +√ +2π exp +� +− (r − 1)2 +2σ2 +� +, +(30) +where Erf[x] = +2 +√π +� x +0 e−t2 dt. Note that r < 1, which means that most half of samples are +mislabeled. Thus +1 +2E[˜yiµ⊤xi] = 1 +2Erf[1 − r +√ +2σ ] + +σ +√ +2π exp +� +− (r − 1)2 +2σ2 +� +> 0. +Now we deal with the yin in Eq. (29). +1 +2n|µ⊤� +n +� +i=1 +tanh(θ⊤ +t xi) +� +| = 1 +2n|q⊤p| +≤ 1 +2n ∥q∥ ∥p∥ , +(31) +q = (µ⊤x1, µ⊤x2, . . . , µ⊤xn) ∈ Rn, and p = (tanh(θ⊤ +t x1), tanh(θ⊤ +t x2), . . . , tanh(θ⊤ +t xn)) ∈ +Rn. +By triangle inequality of the norm, +∥q∥ = ∥q − 1 + 1∥ ≤ ∥q − 1∥ + ∥1∥ = √n + ∥q − 1∥ , +where q − 1 is a random vector with Gaussian coordinates. By Lemma E.1, +∥q − 1∥ /σ ≤ 2σ√n +(32) +with probability 1 − δ when n ≥ c1 log 1/δ, where c1 is a constant. +On the other hand, +��p − tanh(θ⊤ +t µ)1n + tanh(θ⊤ +t µ)1n +�� ≤ +��tanh(θ⊤ +t µ)1n +�� + +��p − tanh(θ⊤ +t µ)1n +�� +≤ +��tanh(θ⊤ +t µ)1n +�� + ∥θt∥ ∥q − 1∥ +=tanh(θ⊤ +t µ)√n + 2σ√n ∥θt∥ , +(33) +where the second inequality is by Lemma 9 from Liu et al. (2020), the last inequality by Lemma E.1. +Then we take Eq. (31) and Eq.(33) together, and then take them and Eq.(30) into −µ⊤∇θL(θt), +which gives us: +− ∇θL(θt)⊤µ ≥ 1 +2Erf[1 − r +√ +2σ ] + +σ +√ +2π exp +� +− (r − 1)2 +2σ2 +� +− σ(tanh(θ⊤ +t µ) + 2σ ∥θt∥) +(34) +By Lemma 8 from Liu et al. (2020), we have supθ∈Rd ∥∇θL(θ)∥ ≤ 1 + 2σ. Therefore, Eq. (34) can +be rewritten as: +−∇θL(θt)⊤µ +∥∇θL(θt)∥ +≥ +Erf[ 1−r +√ +2σ] + 2 +σ +√ +2π exp +� +− (r−1)2 +2σ2 +� +1 + 2σ +− σ(tanh(θ⊤ +t µ) + 2σ ∥θt∥) +1 + 2σ +≥ +b0 +1 + 2σ − σ(tanh(θ⊤ +t µ) + 2σ ∥θt∥) +1 + 2σ +, +(35) +24 + +Published as a conference paper at ICLR 2023 +where we let b0 = 1 +2Erf[ 1−r +√ +2σ] + +σ +√ +2π exp +� +− (r−1)2 +2σ2 +� +. +Then we prove −∇θL(θt)⊤µ +∥∇θL(θt)∥ +≥ +1 +10 +b0 +1+2σ by mathematical induction, which can help us get rid of the +dependence on θt for the lower bound in Eq. (35). +For t += +0, the inequality holds trivially. +By the gradient descent algorithm, θt+1 += +−η �t +i=0 ∇θL(θi), where −µ⊤∇θL(θi)/ ∥∇θL(θi)∥ ≥ +1 +10 +b0 +1+2σ. +θ⊤ +t+1µ +∥θt+1∥ ≥−η �t +i=0 µ⊤∇θL(θi) +η +����t +i=0 ∇θL(θi) +��� +≥ +1 +10 +b0 +1+2σ(�t +i=0 ∥∇θL(θi)∥) +�t +i=0 ∥∇θL(θi)∥ +≥ 1 +10 +b0 +1 + 2σ +As t + 1 < T, we have ∥θt+1∥ ≤ 10 1+2σ +b0 θ⊤ +t+1µ ≤ 1+2σ +b0 . Taking it into Eq. (35), we have +−∇θL(θt)⊤µ +∥∇θL(θt)∥ +≥ +b0 +1 + 2σ − +σ(0.1 + 1+2σ +b0 ) +1 + 2σ +To show −∇θL(θt)⊤µ +∥∇θL(θt)∥ +is lower bounded by +1 +10 +b0 +1+2σ, we need to have +h(σ) = 9 +10 +b0 +1 + 2σ − σ(0.1 + 1 + 2σ +b0 +) > 0 +It is straightforward to verify that h(σ = 0) > 0 and it can be verified that when 0 < σ < c0, we +have h′(σ) > 0. Therefore, for 0 < σ < c0 and any t < T − 1 +−∇θL(θt)⊤µ +∥∇θL(θt)∥ +≥ 1 +10 +b0 +1 + 2σ +Hence by gradient descent algorithm θT = −η �T −1 +i=0 ∇θL(θi) and the same proof above, we have +θ⊤ +T µ +∥θT ∥ ≥ 1 +10 +b0 +1 + 2σ +(36) +For the second part: the prediction accuracy on mislabeled sample set B converges in probability +to its mean. Therefore, the expectation of the prediction accuracy on mislabeled samples is given by +E[1{sign(θ⊤ +T x) = y}] =E[1{sign(yθ⊤ +T (µ − σz)) = y}] +=E[1{sign(θ⊤ +T (µ − σz)) = 1}] += Pr[σθ⊤ +T z > θ⊤ +T µ] +(37) +Note that z is a standard Gaussian vector, θ⊤ +T z is distributed as N(0, ∥θT ∥2) Thus, Eq. (37) is +equivalent to Φ( θ⊤ +T µ +σ∥θT ∥). +By the inequality 1 − Φ(x) ≤ exp{−x2/2} for x > 0, then we have +Φ( θ⊤ +T µ +σ ∥θT ∥) ≥ 1 − exp{− +( θ⊤ +T µ +σ∥θT ∥)2 +2 +} ≥ 1 − exp{− 1 +200 +� +b0 +(1 + 2σ)σ +�2} +We denote g(σ) by: +g(σ) = +Erf[ 1−r +√ +2σ] +2(1 + 2σ)σ + exp (− (r−1)2 +2σ2 ) +√ +2π(1 + 2σ) , +where g(σ) > 0 for any σ > 0. Note that g(σ) → ∞ when σ → 0, and g(σ) is monotone decreasing +as σ increases since g′(σ) < 0 for σ > 0. +25 + +Published as a conference paper at ICLR 2023 +Lemma E.1. Let X = (X1, X2, . . . , Xn) ∈ Rn be a random vector with independent, Gaussian +coordinates Xi with E[Xi] = 0 and E[X2 +i ] = 1 < ∞. Then +Pr[| ∥X∥2 − √n| ≥ √n] ≤ 2 exp +� +− an +� +, +where a > 0 is a constant. +Proof. The Gaussian concentration result is taken from Proposition 5.34 in Vershynin (2018), which +will be used here for proving Theorem 4.1. +F +ADDITIONAL LEARNING CURVES +We provide additional learning curves on DomainNet dataset, shown in Figure 10. The dataset +contains 12 pairs of tasks showing: (1) target classifiers have higher prediction accuracy during the +early-training time; (2) leverage ETP by using ELR can alleviate the memorization of unbounded +noisy labels generated by source models. +Figure 10: The source models are used to initialize the classifiers and annotate unlabeled target data. +As the classifiers memorize the unbounded label noise very fast, we evaluate the prediction accuracy +on target data every batch for the first 90 steps. After the 90 steps, we evaluate the prediction accuracy +for every 0.3 epoch. We use the CE and ELR to train the classifiers on the labeled target data, shown +in solid green lines and solid blue lines, respectively. +G +EXPERIMENTAL DETAILS +In this section, we additionally show the overall training process of our method, illustrated in Figure +11 and in Algorithm 1. Besides, we provide more experimental information of our paper in details. +Datasets. We use four benchmark datasets, which have been widely utilized in the Unsupervised +Domain Adaptation (UDA) (Long et al., 2015; Tan et al., 2020; Wang et al., 2022) and Source-Free +26 + +64 - +CE +CE +64 - +CE +ELR +ELR +ELR +Accuracy +76 + +58 +74 +56 +2 +58 - +54 +52 +56 +0 +21 +120 +14D +0 +21 +120 +14D +0 +21 +120 +140 +c→ +C→r +Stu +72 - +CE + 6 +CE +66 - +CE +ELR +ELR +好 +ELR +66 +86 +58 - +58 +21 +85 +0 +4 +120 +140 +0 +21 +4 +140 +120 +140 +0 +21 +41 +120 +140 +P→R +P+s +CE +CE +CE +67.5 +ELR +81 +ELR +58 +ELR +.0 +Ω2.5 +64.0 +T6 - +57.5 +74 - +55.0 +Manl +52 +52.5 +F2 +0 +24 +140 +120 +14D +0 +21 +4 +140 +120 +14D +0 +21 +120 +140 +R→P +R→S +CE +CE +76 +ELR +ELR +66 +好 +CE +- +ELR + 9L +66 - +58 +0 +21 +4 +iD +120 +14D +0 +21 +120 +140 +0 +21 +10 +120 +140 +s+c +s→P +S-→RPublished as a conference paper at ICLR 2023 +Data +\ ++ +Target Classifier +Figure 11: Overview of the SFDA problem and our method. +Algorithm 1: SFDA ELR - Source Free Domain Adaptation with ELR +Input: Source Pre-Trained Model: f(x; θ0), Target Data: Xt(xt), Training Epochs: T +1 Initialize a prediction bank Y with ¯y0 = 0 +2 for epoch=1 to T do +3 +for iterations t=1,2,3,... do +4 +Compute the SFDA objective LSFDA (depends on concrete SFDA algorithms) +5 +Update the prediction bank Y: ¯yt = β¯yt−1 + (1 − β)f(xt; θt) +6 +Compute the ELR regularization LELR: LELR = log(1 − ¯y⊤ +t f(xt)) +7 +Compute the total loss: L = LSFDA + λLELR +8 +Update the parameters of f(θt) via L +9 +end +10 end +Output: Target Adapted Model f(x; θT ) +27 + +Published as a conference paper at ICLR 2023 +Table 4: Accuracies (%) on Office-31 for ResNet50-based methods. +Method +SF +A→D +A→W +D→W +W→D +D→A +W→A +Avg +MCD (Saito et al., 2018b) + +92.2 +88.6 +98.5 +100.0 +69.5 +69.7 +86.5 +CDAN (Long et al., 2018) + +92.9 +94.1 +98.6 +100.0 +71.0 +69.3 +87.7 +MDD (Zhang et al., 2019b) + +90.4 +90.4 +98.7 +99.9 +75.0 +73.7 +88.0 +BNM (Cui et al., 2020) + +90.3 +91.5 +98.5 +100.0 +70.9 +71.6 +87.1 +DMRL (Wu et al., 2020) + +93.4 +90.8 +99.0 +100.0 +73.0 +71.2 +87.9 +BDG (Yang et al., 2020) + +93.6 +93.6 +99.0 +100.0 +73.2 +72.0 +88.5 +MCC (Jin et al., 2020) + +95.6 +95.4 +98.6 +100.0 +72.6 +73.9 +89.4 +SRDC (Tang et al., 2020) + +95.8 +95.7 +99.2 +100.0 +76.7 +77.1 +90.8 +RWOT (Xu et al., 2020) + +94.5 +95.1 +99.5 +100.0 +77.5 +77.9 +90.8 +RSDA-MSTN (Gu et al., 2020) + +95.8 +96.1 +99.3 +100.0 +77.4 +78.9 +91.1 +Source Only + +80.8 +76.9 +95.3 +98.7 +60.3 +63.6 +79.3 ++ELR + +90.9 +89.0 +98.2 +100.0 +67.1 +64.1 +84.9 +SHOT (Liang et al., 2020) + +94.0 +90.1 +98.4 +99.9 +74.7 +74.3 +88.6 ++ELR + +94.9 +91.6 +98.7 +100.0 +75.2 +74.5 +89.3 +G-SFDA (Yang et al., 2021b) + +85.9 +87.3 +98.6 +99.8 +71.4 +72.1 +85.8 ++ELR + +86.9 +87.8 +98.7 +99.8 +71.4 +72.9 +86.2 +NRC (Yang et al., 2021a) + +93.7 +93.8 +97.8 +100.0 +75.5 +75.6 +89.4 ++ELR + +93.8 +93.3 +98.0 +100.0 +76.2 +76.9 +89.6 +Table 5: Optimal Hypermaraters (β/λ) on various datasets. +Hyperparameters: β/λ +Office-31 +Office-Home +VisDA +DomainNet +ELR only +0.9/− +0.99/− +0.99/− +0.9/− +ELR + SHOT +0.7/1.0 +0.6/3.0 +0.6/25 +0.7/7.0 +ELR + G-SFDA +0.8/1.0 +0.9/1.0 +0.5/7.0 +0.8/12.0 +ELR + NRC +0.5/1.0 +0.6/3.0 +0.5/3.0 +0.8/3.0 +Domain Adaptation (SFDA) (Liang et al., 2020) scenarios, to verify the effectiveness of leveraging +the early-time training phenomenon to address unbounded label noise. Office-31 (Saenko et al., 2010) +contains 4, 652 images in three domains (Amazon, DSLR, and Webcam), and each domain consists of +31 classes. Office-Home (Venkateswara et al., 2017) contains 15, 550 images in four domains (Real, +Clipart, Art, and Product), and each domain consists of 65 classes. VisDA (Peng et al., 2017) contains +152K synthetic images and 55K real object images with 12 classes. DomainNet (Peng et al., 2019) +contains around 600K images in six different domains (Clipart, Infograph, Painting, Quickdraw, Real +and Sketch). Following previous work Tan et al. (2020); Liu et al. (2021a), we select 40 the most +commonly-seen classes from four domains: Real, Clipart, Painting, and Sketch. +Implementation. We use ResNet-50 (He et al., 2016) for Office-31, Office-Home and DomainNet, +and ResNet-101 (He et al., 2016) for VisDA as backbones. We adopt a fully connected (FC) +layer as the feature extractor on the backbone and another FC layer as the classifier head. The +batch normalization layer is put between the two FC layers and the weight normalization layer is +implemented on the last FC layer. We set the learning rate to 1e-4 for all layers except for the last two +FC layers, where we apply 1e-3 for the learning rate for all datasets. The training for source models +are set to be consistent with the SHOT (Liang et al., 2020). The hyperparameters for ELR with +self-training, ELR with SHOT, ELR with G-SFDA, and ELR with NRC on four different datasets are +shown in Table 5. We note that for ELR with self-training, there is only one hyperparameter β to +tune. The hyperparameters for existing SFDA algorithms are set to be consistent with their reported +values for Office-31, Office-Home, and VisDA datasets. As these SFDA algorithms have not reported +their performance for DomainNet dataset, We follow the hyperparameter search strategy from their +work (Liang et al., 2020; Yang et al., 2021a;b), and choose the optimal hyperparameters β = 0.3 for +SHOT, K = 5 and M = 5 for NRC, and k = 5 for G-SFDA. +28 + +Published as a conference paper at ICLR 2023 +H +MEMORIZATION SPEED BETWEEN LABEL NOISE IN SFDA AND IN +CONVENTIONAL LLN SETTINGS +Although ETP exists in both SFDA and conventional LLN scenarios, the memorization speed for them +is still different. Specifically, the target classifiers memorize noisy labels much faster in the SFDA +scenario. It has already been shown that it takes many epochs before classifiers start memorizing +noisy labels in conventional LLN scenario (Liu et al., 2020; Xia et al., 2020a). We highlight that +the main factor causing the difference is the label noise. To show it, we replace the unbounded +label noise in SFDA with bounded random label noise, and we keep the other settings unchanged as +introduced in 4. To replace the unbounded label noise with bounded random label noise, we use the +source model to identify mislabeled target samples, then we assign random labels to these mislabeled +samples. Figure 13 and Figure 14 show the learning curves on Office-Home and Office-31 datasets +with unbounded label noise and random bounded label noise. To better visualize the learning curves +with unbounded label noise, we re-plot Figures 13-14 with different y scale in Figures 15-16. These +figures demonstrate that target classifiers memorizing noisy labels with unbounded label noise is +much faster than noisy labels with random bounded label noise. The classifiers with bounded label +noise (colored in red) are expected to memorize all noisy labels eventually. As illustrated in Figures +15-16, the classifiers with unbounded label noise (colored in green) show that the noisy labels are +already memorized. We note that for the first 90 steps, the prediction accuracy is evaluated every +batch, while the prediction accuracy is evaluated every 0.3 epoch after that time. Therefore, for +unbounded label noise, target classifiers start memorizing the noisy labels within the first epoch +(consisting of more than 90 batches). +There are some existing LLN methods such as PCL (Zhang et al., 2021) to purify noisy labels every +epoch based on ETP. Due to this difference, these LLN methods are not helpful to solving label noise +in SFDA as they are not able to capture the benefits of ETP. Our empirical results in Section 5.1 can +support this argument. We also note that PCL does not suffer from the fast memorization speed and +is able to capture the benefits of ETP in conventional LLN settings. As we indicated in Figures 13-14, +it takes much longer time (more than a few epochs) for target classifiers to start memorizing bounded +noisy labels. We hope these insights can motivate the researcher to consider memorization speed and +design algorithms better for SFDA. +I +ADDITIONAL ANALYSIS OF ELR AND A STANDARD SFDA METHOD - NRC +In this section, we will theoretically and empirically compare ELR and NRC in detail. Specifically, +NRC (Yang et al., 2021a) is a well-known SFDA method that explores the neighbors of target +data by graph-based methods and utilizes these neighbors’ information to correct the target data’s +pseudo-label, in order to boost the SFDA performance. The proposed NRC loss has the following +form: +ℓNRC = Ldiv + LN + LE + Lself +with: +Ldiv = +K +� +k=1 +KL( ¯pk||qk) +the diversity loss where ¯pk is the empirical label distribution and q is a uniform distribution; and +LN = − 1 +nt +� +i +� +m∈N i +M +AimST +mh(xi) +the neighbors loss, where m is the index of the m-th nearest neighbors of xi, Sm is the m-th item in +memory bank S, Aim is the affinity value of m-th nearest neighbors of input xi in the feature space. +LE = − 1 +nt +� +i +� +m∈N i +M +� +j∈Em +N +rST +nh(xi) +the expanded neighbors loss, where Em +N contain the N-nearest neighbors of neighbor m in NM. +Lself = − 1 +nt +nt +� +i +ST +i h(xi) +29 + +Published as a conference paper at ICLR 2023 +the self-regularization loss, where Si means the stored prediction in the memory bank, a constant +vector and is identical to the h(xi) as in NRC they update the memory banks before the training. +I.1 +THEORETICAL ANALYSIS OF NRC’S SELF-REGULARIZATION TERM COMPARED TO ELR +To emphasize the novelty of our proposed ELR in SFDA problems, we will compare the original +formulas and also the gradients of ELR and NRC’s self-regularization (SR) term in detail. And +then, we will explain why NRC can not benefit from the ETP only by adopting the SR term. As we +formulate in the main paper, we can represent the ELR loss and the SR loss as follows: +LELR(θt) = log(1 − ¯y⊤ +t f(x; θt)) +(38) +and +LSR(θt) = −ˆy⊤ +t f(x; θt) +(39) +where ¯yt = β¯yt−1+(1−β)f(x; θt) in ELR is the moving average prediction for x, and ˆyt = f(x; θt) +in SR is the constant vector copied from the current training step’s prediction. Besides, the gradients +of ELR and SR are: +dLELR(θt) +df(x; θt) = − +¯yt +1 − ¯y⊤ +t f(x; θt) +(40) +and +dLSR(θt) +df(x; θt) = − ˆyt +(41) +The motivation of the SR term is to emphasize the ego feature of current prediction and, therefore, to +reduce the potential impact of noisy neighbors, whereas the ELR proposed in this paper considers the +changes of prediction quality during the training process and aims to encourage the model prediction +to stick to the early-time predictions for each data point. +As shown in Eq. ( 38) and Eq. ( 39), we can directly observe that ELR involves the previous training +step’s prediction information in loss (included in ¯yt), however, SR leverages only the prediction result +of current step. +Besides, if we further look at the gradient formulas of these two losses and analyze the back- +propagation process, we can find that the gradient dLELR(θt) +df(x;θt) increases as the model prediction closes +to the target ¯yt, which will further force the prediction f(x; θt) close to ¯y thanks to the large magnitude +of the gradient. And this will help with the utilization of early-time predictions and ETP. However, +the gradient of LSR is a constant vector with values of prediction logits, which could be very small. +So when dLSR(θt) +df(x;θt) is small, SR term can be easily overwhelmed by the other loss terms that favour +fitting incorrect pseudo labels, leading to poor performance. +The above analysis shows a fundamentally different difference between SR and ELR. Specifically, +SR does not utilize ETP and cannot handle the unbounded label noise either. +I.2 +EMPIRICAL ANALYSIS OF ELR AND NRC IN TERMS OF THE UTILIZATION OF ETP +In addition to the above theoretical analysis for the loss functions, we also observed the same +conclusion through experiments. As shown in Figure 12, we observe that thanks to the update of +the pseudo-label with the process of adaptation in the SFDA method, overall, NRC can obtain a +model with relatively high accuracy on the target domain. However, the performance drop still exists +when using the NRC method alone, which can be effectively avoided by adding the ELR term. This +confirms that ELR can effectively leverage ETP and avoid the problem of noisy label memorization. +J +ADDITIONAL DISCUSSION OF PSEUDO-LABEL PURIFICATIONS IN SFDA +AND LLN APPROACHES +In this section, we will further discuss the similarities and the differences between the LLN approaches +and the pseudo-label purification processes proposed in current SFDA methods. +The main similarity between the existing SFDA approaches and the LLN methods is that both research +fields have to deal with data with noise, aiming to get a model with promising performance. As +30 + +Published as a conference paper at ICLR 2023 +Figure 12: Fine-Grained Training Accuracy of NRC and NRC + ELR on Office-Home dataset. The +solid green lines represent the training process of NRC, whereas the solid blue lines represent the +training process of NRC with ELR term. The colored bands represent the performance drop. +for the differences, they can be mainly divided into the following aspects. From the perspective +of motivations, most of the existing SFDA approaches are developed under the domain adaptation +setting. They study how to best exploit the distribution relationship between the source and target +domains in the absence of source data, so as to achieve domain adaptation better. Their motivation +is to investigate how to better assign the pseudo-label. In contrast, LLN is an independent field +that mainly studies, given a set of noisy data, how to deal with the label noise, conduct the model +training, and obtain a noise-robust model with better performance. Traditionally, the study of LLN +does not involve assumptions about the data domain or source model. Meanwhile, there are more +in-depth and rigorous studies (theoretical and methodological) on the types of noises, and how to +handle and exploit them. From the perspective of the methodology, in order to obtain a higher quality +pseudo-label, many SFDA methods heuristically use clustering or neighbor features to correct the +pseudo-labels, and use the corrected labels to perform a normal supervised learning. The current +SFDA methods focus on the explicit pseudo-label purification process, which can be summarized as +noisy label correction. However, for LLN, the noisy label correction is just a research sub-branch. +LLN also includes many other research directions, such as studies of different label noise types, +research about how to utilize and even benefit from label noise in the training process, and how to +train the model more robustly. Many noise-robust loss functions and related theoretical analyses have +been developed. +We would like to emphasize that the motivation of our paper is to investigate how to study SFDA +from the perspective of learning with label noise. We combine the characteristics of SFDA with the +LLN approaches and discover the unbounded nature of label noise in SFDA. Further, we rigorously +distinguish which LLN methods can help SFDA problems and which approaches are limited in their +use in SFDA. We believe that the studies of LLN can open new avenues for the research of SFDA +and bring more ideas and inspiration to the design of the SFDA algorithm. +31 + +0.78 +0.58 +0.56 +0.54 +0.74 +0.78 +0.52 +0.50 - +0.70 +NRC +0.76 +NRC +0.4B +NRC +0.68 +NRC+ELR +NRC+ELR +0.46 +NRC+ELR +0.66 +0.74 +0 +21 +0 +120 +14D +0 +21 +120 +14D +0 +21 +120 +140 +d+e +a→r +0.675 +0.B2 +0.575 +0.65 +Accuracy +0.55 +0.B0 +0.625 +0.525 +0.600 +0.78 +0.500 +Training +Training i +0.575 +0.76 +0.475 +0.55 +NRC +NRC +0.450 +NRC +NRC+ELR +0.74 +NRC+ELR +0.425 +0.525 +NRC+ELR +0.400 +0 +24 +140 +120 +14D +0 +140 +120 +14D +0 +24 +4 +8 +140 +120 +140 +p→a +p→r +p→c +0.B00 +0.B00 +0.675 +0.775 +0.775 +0.75 +0.750 +0.625 +0.725 +0.725 +Training , +009'0 +DO0 +0.700 +0.575 +0.675 +0.675 +0.55 +NRC +NRC +NRC +0.65 +99'0 +0.525 +NRC+ELR +NRC+ELR +NRC+ELR +0.625 +0.625 +21 +120 +14D +21 +4 +140 +120 +14D +24 +120 +140 +e+> +c→r +c→p +0.60 +Accuracy +0.B4 +NRC +0.56 +0.70 +0.B2 +Training +0.54 +NRC+ELR +EuIl +0.68 +0.52 +Train +0.BO +0.50 +NRC +NRC +0.66 +NRC+ELR +0.78 +NRC+ELR +0.4B +0 +21 +140 +120 +140 +0 +24 +140 +120 +14D +0 +24 +140 +120 +140 +r→a +r→c +r-→pPublished as a conference paper at ICLR 2023 +Figure 13: Training accuracy on Office-Home dataset. The solid green lines represent the unbounded +label noise in SFDA, whereas the solid red lines represent the bounded label noise. +Figure 14: Training accuracy on Office-31 dataset. The solid green lines represent the unbounded +label noise in SFDA, whereas the solid red lines represent the bounded label noise. +32 + +0 +10 +140 +Training Accuracy +Training Accuracy +90 +95 +Training Accuracy +90 +unbounded +unbounded +unbounded +bounded +T0 +bounded +bounded +85 +8 +50 +恬 +0 +25 +50 +125 +150 +175 +0 +25 +50 +140 +125 +150 +175 +25 +50 +5 +125 +150 +175 +Rw -+ Pr +RW-+ CI +Rw-+ Ar +100 +40 +unbounded +Training Accuracy +Training Acuracy +90 +Training Accuracy +95 +bounded +unbounded +8 +unbounded +bounded +T +bounded +41 +75 +50 +0 +25 +50 +125 +150 +175 +0 +25 +50 +190 +125 +150 +175 +0 +25 +50 +125 +150 +175 +Pr-→+ Cl +Pr-+ Ar +Pr-+ Rw +40 +0 +0 +unbounded +Accuracy +90 +Training Accuracy +95 +Training Accuracy +95 +90 +bounded +unbounded +unbounded +90 +Training i +bounded +bounded +& +5 +8 +50 +75 +0 +25 +50 +125 +150 +175 +0 +25 +50 +125 +150 +175 +0 +25 +50 +125 +150 +175 +Ar-+ Cl +Ar-+ Pr +Ar-+ Rw +100 +140 +Training Accuracy +90 +Training Accuracy +90 +Training Accuracy +90 +81 +unbounded +unbounded +unbounded +bounded +8 +bounded +bounded +50 +25 +50 +14iD +125 +150 +0 +25 +50 +125 +150 +175 +175 +0 +25 +50 +125 +150 +175 +Cl -→+ Ar +Cl -→ Rw0 +140 +unbounded +unbounded +Training Accuracy +Training Accuracy +56 +bounded +56 +bounded +90 +85 +8 +75 +0 +区 +1iD +120 +140 +0 +24 +120 +140 +C→R +unbounded +unbounded +Accuracy +Accuracy +bounded +66 +bounded +Training +81 +Training i +26 +96 . +0 +区 +& +1iD +120 +14D +0 +21 +120 +140 +C+S +10D +P+C +unbounded +140.0 +Training Accuracy +Training Acuracy +6 +bounded +99.8 +99.6 +unbounded +8 +99.4 +bounded +99.2 +WW +0'66 +0 +21 +1iD +120 +14D +0 +21 +140 +120 +140 +P-R +5tdPublished as a conference paper at ICLR 2023 +Figure 15: Figure 13 with different y-scale to better show learning details of the unbounded label +noise. +Figure 16: Figure 14 with different y-scale to better show learning details of the unbounded label +noise. +33 + +79.5 +51 +67.00 +unbounded +66.75 +unbounded +79.0 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b/oNFQT4oBgHgl3EQfqzYC/content/tmp_files/load_file.txt @@ -0,0 +1,2423 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf,len=2422 +page_content='Published as a conference paper at ICLR 2023 WHEN SOURCE-FREE DOMAIN ADAPTATION MEETS LEARNING WITH NOISY LABELS Li Yi1,∗ Gezheng Xu2,∗ Pengcheng Xu2 Jiaqi Li2 Ruizhi Pu2 Charles Ling2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Ian McLeod1 Boyu Wang1,2,† 1Department of Statistical and Actuarial Sciences 2Department of Computer Science University of Western Ontario {lyi7,gxu86,pxu67,jli3779,rpu2,charles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='ling,aimcleod}@uwo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='ca bwang@csd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='uwo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='ca ABSTRACT Recent state-of-the-art source-free domain adaptation (SFDA) methods have fo- cused on learning meaningful cluster structures in the feature space, which have succeeded in adapting the knowledge from source domain to unlabeled target domain without accessing the private source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' However, existing methods rely on the pseudo-labels generated by source models that can be noisy due to domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In this paper, we study SFDA from the perspective of learning with label noise (LLN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Unlike the label noise in the conventional LLN scenario, we prove that the label noise in SFDA follows a different distribution assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We also prove that such a difference makes existing LLN methods that rely on their distribution assumptions unable to address the label noise in SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Empirical evidence suggests that only marginal improvements are achieved when applying the existing LLN methods to solve the SFDA problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' On the other hand, although there exists a fundamental difference between the label noise in the two scenar- ios, we demonstrate theoretically that the early-time training phenomenon (ETP), which has been previously observed in conventional label noise settings, can also be observed in the SFDA problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Extensive experiments demonstrate significant improvements to existing SFDA algorithms by leveraging ETP to address the label noise in SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 1 INTRODUCTION Deep learning demonstrates strong performance on various tasks across different fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' However, it is limited by the requirement of large-scale labeled and independent, and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=') data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Unsupervised domain adaptation (UDA) is thus proposed to mitigate the distribution shift between the labeled source and unlabeled target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In view of the importance of data privacy, it is crucial to be able to adapt a pre-trained source model to the unlabeled target domain without accessing the private source data, which is known as Source Free Domain Adaptation (SFDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The current state-of-the-art SFDA methods (Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='b) mainly focus on learning meaningful cluster structures in the feature space, and the quality of the learned cluster structures hinges on the reliability of pseudo labels generated by the source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Among these methods, SHOT (Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020) purifies pseudo labels of target data based on nearest centroids, and then the purified pseudo labels are used to guide the self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' G-SFDA (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021b) and NRC (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021a) further refine pseudo labels by encouraging similar predictions to the data point and its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' For a single target data point, when most of its neighbors are correctly predicted, these methods can provide an accurate pseudo label to the data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' However, as we illustrate the problem in Figure 1i(a-b), when the majority of its neighbors are incorrectly predicted to a category, it will be assigned with an incorrect pseudo label, misleading the learning of cluster structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The experimental result on VisDA (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2017), shown in Figure 1ii, further verifies this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' By directly applying the pre-trained source model on each target domain instance ∗Equal contribution †Corresponding author 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='13381v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='LG] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='31 Jan 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='Published as a conference paper at ICLR 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='Source/Unlabeled target data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='Mislabeled target data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='Correctly predicted data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='(a) ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='AB9XicbVBNS8NAEN3Ur1q/qh69BIvgqSQi6kUoevFYwX5AG8tmM2mXbjZhd6KW0P/hxYMiXv0v3vw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='3btsctPXBwO9GWbm+YngGh3n2yosLa+srhXSxubW9s75d29po5TxaDBYhGrtk81C6hgRwFtBMFNPIFtPzh9cRvPYDSPJZ3OErAi2hf8pAzika6x0vsdRGeMAMZjHvlilN1prAXiZuTCslR75W/ukHM0gkMkG17rhOgl5GFXImYFzqphoSyoa0Dx1DJY1Ae9n06rF9ZJTADmNlSqI9VX9PZDTSehT5pjOiONDz3kT8z+ukGF54GZdJiDZbFGYCht ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='jexKBHXAFDMXIEMoUN7fabEAVZWiCKpkQ3PmXF0nzpOqeVU9vTyu1qzyOIjkgh+SYuOSc1MgNqZMGYUSRZ/JK3qxH68V6tz5mrQUrn9knf2B9/gAN7JLjt = tend ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='(i) Overview of the SFDA problem and our method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='Plane Bcycl Bus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='Car Horse Knife McyclPersonPlant Sktbrd Train Truck ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='00 Misleading Neighbors Ratio Over-Confident Misleading Neighbors Ratio (ii) Neighbors Label Noise Analysis On VisDA Figure 1: (i) (a) The SFDA problem can be formulated as an LLN problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (b) The existing SFDA algorithms using the local cluster information cannot address label noise due to the unbounded label noise (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (c) We prove that ETP exists in SFDA, which can be leveraged to address the unbounded label noise (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (ii) Observed Label Noise Phenomena on VisDA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (central instance), we collect its neighbors and evaluate their quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We observed that for each class a large proportion of the neighbors are misleading (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', the neighbors’ pseudo labels are different from the central instance’s true label), some even with high confidence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', the over-confident misleading neighbors whose prediction score is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Based on this observation, we can conclude that: (1) the pseudo labels leveraged in current SFDA methods can be heavily noisy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2) some pseudo-label purification methods utilized in SFDA, which severely rely on the quality of the pseudo label itself, will be affected by such label noise, and the prediction error will accumulate as the training progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' More details can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In this paper, we address the aforementioned problem by formulating SFDA as learning with label noise (LLN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Unlike existing studies that heuristically rely on cluster structures or neighbors, we investigate the properties of label noise in SFDA and show that there is an intrinsic discrepancy between the SFDA and the LLN problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Specifically, in conventional LLN scenarios, the label noise is generated by human annotators or image search engines (Patrini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020a), where the underlying distribution assumption is that the mislabeling rate for a sample is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' However, in the SFDA scenarios, the label noise is generated by the source model due to the distribution shift, where we prove that the mislabeling rate for a sample is much higher, and can approach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We term the former label noise in LLN as bounded label noise and the latter label noise in SFDA as unbounded label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Moreover, we theoretically show that most existing LLN methods, which rely on bounded label noise assumption, are unable to address the label noise in SFDA due to the fundamental difference (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' To this end, we leverage early-time training phenomenon (ETP) in LLN to address the unbounded label noise and to improve the efficiency of existing SFDA algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Specifically, ETP indicates that classifiers can predict mislabeled samples with relatively high accuracy during the early learning phase before they start to memorize the mislabeled data (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Although ETP has been previously observed in, it has only been studied in the bounded random label noise in the conventional LLN scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In this work, we theoretically and empirically show that ETP still exists in the unbounded label noise scenario of SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Moreover, we also empirically justify that existing SFDA algorithms can be substantially improved by leveraging ETP, which opens up a new avenue for SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' As an instantiation, we incorporate a simple early learning regularization (ELR) term (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020) with existing SFDA objective functions, achieving consistent improvements on four different SFDA benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' As a comparison, we also apply other existing LLN methods, including Generalized Cross Entropy (GCE) (Zhang & Sabuncu, 2018), Symmetric Cross Entropy Learning (SL) (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2019b), Generalized Jensen-Shannon Divergence (GJS) (Englesson & Azizpour, 2021) and Progressive Label Correction (PLC) (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021), to SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Our empirical evidence shows that they are inappropriate for addressing the label noise in SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' This is also consistent with our theoretical results (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Our main contribution can be summarized as: (1) We establish the connection between the SFDA and the LLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Compared with the conventional LLN problem that assumes bounded label noise, the problem in SFDA can be viewed as the problem of LLN with the unbounded label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2) 2 Published as a conference paper at ICLR 2023 We theoretically and empirically justify that ETP exists in the unbounded label noise scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' On the algorithmic side, we instantiate our analysis by simply adding a regularization term into the SFDA objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (3) We conduct extensive experiments to show that ETP can be utilized to improve many existing SFDA algorithms by a large margin across multiple SFDA benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 2 RELATED WORK Source-free domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Recently, SFDA are studied for data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The first branch of research is to leverage the target pseudo labels to conduct self-training to implicitly achieve adaptation (Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Tanwisuth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' SHOT (Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020) introduces k-means clustering and mutual information maximization strategy for self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' NRC (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021a) further investigates the neighbors of target clusters to improve the accuracy of pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' These studies more or less involve pseudo-label purification processes, but they are primarily heuristic algorithms and suffer from the previously mentioned label noise accumulation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The other branch is to utilize the generative model to synthesize target-style training data (Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Some methods also explore the SFDA algorithms in various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' USFDA (Kundu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020a) and FS (Kundu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020b) design methods for universal and open-set UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In this paper, we regard SFDA as the LLN problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We aim to explore what category of noisy labels exists in SFDA and to ameliorate such label noise to improve the performance of current SFDA algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Learning with label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Existing methods for training neural networks with label noise focus on symmetric, asymmetric, and instance-dependent label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' For example, a branch of research focuses on leveraging noise-robust loss functions to cope with the symmetric and asymmetric noise, including GCE (Zhang & Sabuncu, 2018), SL (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2019b), NCE (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020), and GJS (Englesson & Azizpour, 2021), which have been proven effective in bounded label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' On the other hand, CORES (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020) and CAL (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021) are shown useful in mitigating instance- dependent label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' These methods are only tailed to conventional LLN settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Recently, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2020) has studied early-time training phenomenon (ETP) in conventional label noise scenarios and proposes a regularization term ELR to exploit the benefits of ETP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' PCL (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021) is another conventional LLN algorithm utilizing ETP, but it cannot maintain the exploit of ETP in SFDA as memorizing noisy labels is much faster in SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Our contributions are: (1) We theoretically and empirically study ETP in the SFDA scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2) Based on an in depth analysis of many existing LLN methods (Zhang & Sabuncu, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Englesson & Azizpour, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021), we demonstrate that ELR is useful for many SFDA problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 3 LABEL NOISE IN SFDA The presence of label noise on training datasets has been shown to degrade the model performance (Malach & Shalev-Shwartz, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In SFDA, existing algorithms rely on pseudo- labels produced by the source model, which are inevitably noisy due to the domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The SFDA methods such as Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='b) cannot tackle the situation when some target samples and their neighbors are all incorrectly predicted by the source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In this section, we formulate the SFDA as the problem of LLN to address this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We assume that the source domain DS and the target domain DT follow two different underlying distributions over X × Y, where X and Y are respectively the input and label spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In the SFDA setting, we aim to learn a target classifier f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' θ) : X → Y only with a pre-trained model fS(x) on DS and a set of unlabeled target domain observations drawn from DT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We regard the incorrectly assigned pseudo-labels as noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Unlike the “bounded label noise” assumption in the conventional LLN domain, we will show that the label noise in SFDA is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We further prove that most existing LLN methods that rely on the bounded assumption cannot address the label noise in SFDA due to the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Label noise in conventional LLN settings: In conventional label noise settings, the injected noisy labels are collected by either human annotators or image search engines (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The label noise is usually assumed to be either independent of instances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', symmetric label noise or asymmetric label noise) (Patrini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Liu & Tao, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2019b) or dependent of instances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', instance-dependent label noise) (Berthon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The underling assumption for them is that a sample x has the highest probability of being in the correct class y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', Pr[ ˜Y = i|Y = i, X = x] > Pr[ ˜Y = j|Y = i, X = x], ∀x ∈ X, i ̸= j, 3 Published as a conference paper at ICLR 2023 where ˜Y is the noisy label and Y is the ground-truth label for input X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Equivalently, it assumes a bounded noise rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' For example, given an image to annotate, the mislabeling rate for the image is bounded by a small number, which is realistic in conventional LLN settings (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' When the label noise is generated by the source model, the underlying assumption of these types of label noise does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Label noise in SFDA: As for the label noise generated by the source model, mislabeling rate for an image can approach 1, that is, Pr[ ˜Y = j|Y = i, X = x] → 1, ∃S ⊂ X, ∀x ∈ S, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' To understand that the label noise in SFDA is unbounded, we consider a two-component Multivariate Gaussian mixture distribution with equal priors for both domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Let the first component (y = 1) of the source domain distribution DS be N(µ1, σ2Id), and the second component (y = −1) of DS be N(µ2, σ2Id), where µ1, µ2 ∈ Rd and Id ∈ Rd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' For the target domain distribution DT , let the first component (y = 1) of DT be N(µ1 + ∆, σ2Id), and the second component (y = −1) of DT be N(µ2 + ∆, σ2Id), where ∆ ∈ Rd is the shift of the two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Notice that the domain shift considered is a general shift and it has been studied in Stojanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2019), where we also illustrate the domain shift in Figure 9 in supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Let fS be the optimal source classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' First, we build the relationship between the mislabeling rate for target data and the domain shift: Pr (x,y)∼DT[fS(x) ̸= y] = 1 2Φ(−d1 σ ) + 1 2Φ(−d2 σ ), (1) where d1 = �� µ2−µ1 2 − c �� sign( �� µ2−µ1 2 �� − ∥c∥), d2 = �� µ2−µ1 2 + c ��, c = α(µ2 − µ1), α = ∆⊤(µ2−µ1) ∥µ2−µ1∥2 is the magnitude of domain shift, and Φ is the standard normal cumulative distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (1) shows that the magnitude of the domain shift inherently controls the mislabeling error for target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' This mislabeling rate increases as the magnitude of the domain shift increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We defer the proof and details to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' More importantly, we characterize that the label noise is unbounded among these mislabeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Without loss of generality, we assume that the ∆ is positively correlated with the vector µ2 − µ1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', ∆⊤(µ2 − µ1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' For (x, y) ∼ DT , if x ∈ R, then Pr[fS(x) ̸= y] ≥ 1 − δ, (2) where δ ∈ (0, 1) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='01), R = R1 � R2, R1 = {x : ∥x − µ1 − ∆∥ ≤ σ( √ d 2 − log 1−δ δ √ d )}, and R2 = {x : x⊤1d > (σd + 2µ⊤ 1 1d)/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Meanwhile, R is non-empty when α > (log 1−δ δ )/d, where α = ∆⊤(µ2−µ1) ∥µ2−µ1∥2 > 0 is the magnitude of the domain shift along the direction µ2 − µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Conventional LLN methods assume that the label noise is bounded: Pr[fH(x) ̸= y] < m, ∀(x, y) ∼ DT , where fH is the labeling function, and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5 if the number of clean samples of each component are the same (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' However, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1 indicates that the label noise generated by the source model is unbounded for any x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In practice, region R is non-empty as neural networks are usually trained on high dimensional data such that d ≫ 1, so α > (log 1−δ δ )/d → 0 is easy to satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The probability measure on R = R1 � R2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', Pr(x,y)∼DT [x ∈ R]) increases as the magnitude of the domain shift α increases, meaning more data points contradict the conventional LLN assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' More details can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Given that the unbounded label noise exists in SFDA, the following Lemma establishes that many existing LLN methods (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Englesson & Azizpour, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020), which rely on the bounded assumption, are not noise tolerant in SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Let the risk of the function h : X → Y under the clean data be R(h) = Ex,y[ℓLLN(h(x), y)], and the risk of h under the noisy data be �R(h) = Ex,˜y[ℓLLN(h(x), ˜y)], where the noisy data follows the unbounded assumption, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', Pr[˜y ̸= y|x ∈ R] = 1 − δ for a subset R ⊂ X and δ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Then the global minimizer ˜h⋆ of �R(h) disagrees with the global minimizer h⋆ of R(h) on data points x ∈ R with a high probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We denote ℓLLN by the existing noise-robust loss based LLN methods in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2019b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Englesson & Azizpour (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' When the noisy data follows the bounded assumption, these methods are noise tolerant as the minimizer ˜h⋆ converges to the minimizer h⋆ with a high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We defer the details and proof of the related LLN methods to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 4 Published as a conference paper at ICLR 2023 4 LEARNING WITH LABEL NOISE IN SFDA Given a fundamental difference between the label noise in SFDA and the label noise in conventional LLN scenarios, existing LLN methods, whose underlying assumption is bounded label noise, cannot be applied to solve the label noise in SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' This section focuses on investigating how to address the unbounded label noise in SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Motivated by the recent studies Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Arpit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2017), which observed an early-time training phenomenon (ETP) on noisy datasets with bounded random label noise, we find that ETP does not rely on the bounded random label noise assumption, and it can be generalized to the unbounded label noise in SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' ETP describes the training dynamics of the classifier that preferentially fits the clean samples and therefore has higher prediction accuracy for mislabeled samples during the early-training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Such training characteristics can be very beneficial for SFDA problems in which we only have access to the source model and the highly noisy target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' To theoretically prove ETP in the presence of unbounded label noise, we first describe the problem setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We still consider a two-component Gaussian mixture distribution with equal priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We denote y by the true label for x, and assume it is a balanced sample from {−1, +1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The instance x is sampled from the distribution N(yµ, σ1d), where ∥µ∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We denote ˜y by the noisy label for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We observe that the label noise generated by the source model is close to the decision boundary revealed in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' So, to assign the noisy labels, we let ˜y = yβ(x, y), where β(x, y) = sign(1{yx⊤µ > r} − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5) is the label flipping function, and r controls the mislabeling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' If β(x, y) < 1, then the data point x is mislabeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Meanwhile, the label noise is unbounded by adopting the label flipping function β(x, y): Pr[˜y ̸= y|yx⊤µ ≤ r] = 1, where R = {x : yx⊤µ ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We study the early-time training dynamics of gradient descent on the linear classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The parameter θ is learned over the unbounded label noise data {xi, ˜yi}n i=1 with the following logistic loss function: L(θt+1) = 1 n n � i=1 log � 1 + exp � −˜yiθ⊤ t+1xi �� , where θt+1 = θt − η∇θL(θt), and η is the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Then the following theorem builds the connection between the prediction accuracy for mislabeled samples at an early-training time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Let B = {x : ˜y ̸= y} be a set of mislabeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Let κ(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' θ) be the prediction accuracy calculated by the ground-truth labels and the predicted labels by the classifier with parame- ter θ for mislabeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' If at most half of the samples are mislabeled (r < 1), then there exists a proper time T and a constant c0 > 0 such that for any 0 < σ < c0 and n → ∞, with probability 1 − op(1): κ(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' θT ) ≥ 1 − exp{− 1 200g(σ)2}, (3) where g(σ) = Erf[ 1−r √ 2σ ] 2(1+2σ)σ + exp (− (r−1)2 2σ2 ) √ 2π(1+2σ) > 0 is a monotone decreasing function that g(σ) → ∞ as σ → 0, and Erf[x] = 2 √π � x 0 e−t2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The proof is provided in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Compared to ETP found in Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2020), where the label noise is assumed to be bounded, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1 presents that ETP also exists even though the label noise is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' At a proper time T, the classifier trained by the gradient descent algorithm can provide accurate predictions for mislabeled samples, where its accuracy is lower bounded by a function of the variance of clusters σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' When σ → 0, the predictions of all mislabeled samples equal to their ground-truth labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', κ(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' θT ) → 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' When the classifier is trained for a sufficiently long time, it will gradually memorize mislabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The predictions of mislabeled samples are equivalent to their incorrect labels instead of their ground-truth labels (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Maennel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Based on these insights, the memorization of mislabeled data can be alleviated by leveraging their predicted labels during the early-training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' To leverage the predictions during the early-training time, we adopt a recently established method, early learning regularization (ELR) (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020), which encourages model predictions to stick to the early-time predictions for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Since ETP exists in the scenarios of the unbounded label noise, ELR can be applied to solve the label noise in SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The regularization is given by: LELR(θt) = log(1 − ¯y⊤ t f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' θt)), (4) 5 Published as a conference paper at ICLR 2023 (i) VisDA-C (ii) DomainNet (iii) Office-Home (iv) Office-31 Figure 2: Training accuracy on various target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The source models initialize the classifiers and annotate unlabeled target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' As the classifiers memorize the unbounded label noise very fast, for the first 90 steps, we evaluate the prediction accuracy on target data every batch, and one step represents one training batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' After the 90 steps, we evaluate the prediction accuracy for every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='3 epoch, shown as one step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We use the CE, GCE, and ELR to train the classifiers on the labeled target data, shown in solid green lines, solid orange lines, and solid blue lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The dotted red line represents the accuracy of labeling target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Eventually, the classifiers memorize the label noise, and the prediction accuracy equals the labeling accuracy (shown in (iii-iv)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Additional results on transfer pairs can be found in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' where we overload f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' θt) to be the probabilistic output for the sample x, and ¯yt = β¯yt−1 + (1 − β)f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' θt) is the moving average prediction for x, where β is a hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' To see how ELR prevents the model from memorizing the label noise, we calculate the gradient of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (4) with respect to f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' θt), which is given by: dLELR(θt) df(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' θt) = − ¯yt 1 − ¯y⊤ t f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' θt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Note that minimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (4) forces f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' θt) to close to ¯yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' When ¯yt is aligned better with f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' θt), the magnitude of the gradient becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' It makes the gradient of aligning f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' θt) with ¯yt overwhelm the gradient of other loss terms that align f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' θt) with noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' As the training progresses, the moving averaged predictions ¯yt for target samples gradually approach their ground- truth labels till the time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (4) prevents the model from memorizing the label noise by forcing the model predictions to stay close to these moving averaged predictions ¯yt, which are very likely to be ground-truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Some existing LLN methods propose to assign pseudo labels to data or require two-stage training for label noise (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Unlike these LLN methods, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (4) can be easily embedded into any existing SFDA algorithms without conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The overall objective function is given by: L = LSFDA + λLELR, (5) where LSFDA is any SFDA objective function, and λ is a hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Empirical Observations on Real-World Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We empirically verify that target classifiers have higher prediction accuracy for target data during the early training and adaptation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We propose leveraging this benefit to prevent the classifier from memorizing the noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The observations are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The parameters of classifiers are initialized by source models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Labels of target data are annotated by the initialized classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We train the target classifiers on target data with the standard cross-entropy (CE) loss and the generalized cross-entropy (GCE) loss, a well-known noise-robust loss widely leveraged in bounded LLN scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The solid green, orange and blue lines represent the training accuracy of optimizing the classifiers with CE loss, GCE loss, and ELR loss, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The dotted red lines represent the labeling accuracy of the initialized classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Considering that the classifiers memorize the unbounded label noise very fast, we evaluate the prediction accuracy on target data every batch for the first 90 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' After 90 steps, we evaluate the prediction accuracy for every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='33 epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The green lines show that ETP exists in SFDA, which is consistent with our theoretical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Meanwhile, in all scenarios, green and orange lines show that classifiers provide higher prediction accuracy during the first a few iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' After a few iterations, they start to memorize the label noise even with noise-robust loss (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', GCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Eventually, the classifiers are expected to memorize the whole datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' For conventional LLN settings, it has been empirically verified that it takes a much longer time before classifiers start memorizing the label noise (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We provide further analysis in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We highlight 6 70 Training Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 60 CE ELR 50 GCE 0 20 40 60 80 100 Synthetic → Real78 Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 76 Training 74 CE 72 ELR GCE 70 0 20 40 60 80 100 120 140 C→R62 CE AcC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 60 ELR 58 GCE Training 56 54 52 0 25 50 75 100 125 150 175 CI → Ar90 CE ACC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' ELR 85 GCE Training 80 75 0 20 40 60 80 100 120 140 amazon -→ webcamPublished as a conference paper at ICLR 2023 Table 1: Accuracies (%) on Office-Home for ResNet50-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Method SFAr→ClAr→PrAr→RwCl→ArCl→PrCl→RwPr→ArPr→ClPr→RwRw→ArRw→ClRw→Pr Avg MCD (Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2018b) \x17 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='9 68.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='9 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='6 G-SFDA (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021b) \x13 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='8 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='4 74.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1 NRC (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021a) \x13 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='6 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='0 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='3 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='3 77.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='8 +ELR \x13 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='4 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='3 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='6 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='4 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='8 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='6 that PCL (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021) leverages ETP at every epoch, so it cannot capture the benefits of ETP and is inappropriate for unbounded label noise due to the fast memorization speed in SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' As a comparison, we choose ELR since it leverages ETP at every batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The blue lines show that leveraging ETP via ELR can address the memorization of noisy labels in SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 5 EXPERIMENTS We aim to improve the efficiency of existing SFDA algorithms by using ELR to leverage ETP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We evaluate the performance on four different SFDA benchmark datasets: Office-31 (Saenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2010), Office-Home (Venkateswara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2017), VisDA (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2017) and DomainNet (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Due to the limited space, the results on the dataset Office-31 and additional experimental details are provided in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We incorporate ELR into three existing baseline methods: SHOT (Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2020), G-SFDA (Zhang & Sabuncu, 2018), and NRC (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' SHOT uses k-means clustering and mutual information maximization strategy to train the representation network while freezing the final linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' G-SFDA aims to cluster target data with similar neighbors and attempts to maintain the source domain performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' NRC also explores the neighbors of target data by graph-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' ELR can be easily embedded into these methods by simply adding the regularization term into the loss function to optimize without affecting existing SFDA frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We average the results based on three random runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Tables 1-4 show the results before/after leveraging the early-time training phenomenon, where Table 4 is shown in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Among these tables, the top part shows the results of conventional UDA methods, and the bottom part shows the results of SFDA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In the tables, we use SF to indicate whether the method is source free or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We use Source Only + ELR to indicate ELR with self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The results show that ELR itself can boost the performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' As existing SFDA methods are not able to address unbounded label noise, incorporating ELR into these SFDA methods can further boost the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The four datasets, including all 31 pairs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', A → D) of tasks, show better performance after solving the unbounded label noise problem using the early-time training phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Meanwhile, solving the unbounded label noise on existing SFDA methods achieves state-of-the-art on all benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' These SFDA methods also outperform most methods that need to access source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Analysis about hyperparameters β and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The hyperparameter β is chosen from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='99}, and λ is chosen from {1, 3, 7, 12, 25}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We conduct the sensitivity study on hyperparameters of ELR on the DomainNet dataset, which is shown in Figure 3(a-b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In each Figure, the study is conducted by fixing the other hyperparameter to the optimal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The performance is robust to the hyperparameter β except β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' When β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='99, classifiers are sensitive to changes in learning curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Thus, the performance degrades since the learning curves change quickly in the unbounded label noise scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Meanwhile, the performance is also robust to the hyperparameter λ except when λ becomes too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The hyperparameter λ is to balance the effects of existing SFDA 7 Published as a conference paper at ICLR 2023 algorithms and the effects of ELR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' As we indicated in Tables 1-4, barely using ELR to address the SFDA problem is not comparable to these SFDA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Hence, a large value of λ makes neural networks neglect the effects of these SFDA methods, leading to degraded performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Table 2: Accuracies (%) on DomainNet for ResNet50-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Method SFR→CR→PR→SC→RC→PC→SP→RP→CP→SS→RS→CS→P Avg MCD (Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2018b) \x17 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='9 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='7 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='6 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='6 83.' metadata={'source': 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+page_content='(c) embedding different LLN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='methods into SFDA algorithms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='ACBnicbVC7SgNBFJ31GeMrainCYBCswq4E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='tQzaWEYwD0hCmJ29mwyZnV1m7ophSWXjr9hYKGLrN9j5N04ehSYeGDicx9zj59IYdB1v52l5ZXVtfXcRn5za3tnt7C3XzdxqjnUeCxj3fSZASkU1FCghGaigUW+hIY/uB7jXvQRsTqDocJdCLWUyIUnKGVuoWjNsIDZn1r6YRpFgG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='CpqO2tCMC1i0U3ZI7AV0k3owUyQzVbuGrHcQ8jUAhl8yYlucm2MmYRsEljPLt1EDC+ID1oGWpsvtMJ5ucMaInVgloGv7FNKJ+rsjY5Exw8i3lRHDvpn3xuJ/XivF8LKTCZWkCIpPF4WpBjTcSY0EBo4yqEljGth/0p534bBbRYmb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='0Pw5k9eJPWzkndeKt+Wi5WrWRw5ckiOySnxyAWpkBtSJTXCySN5Jq/kzXlyXpx352NauTMeg7IHzifP2KNmbU=hyperparameter � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='Figure 3: (a)-(b) show the test accuracy on the DomainNet dataset with respect to hyperparameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='of ELR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (c) shows the test accuracy of incorporating various existing LLN methods into the SFDA methods on the DomainNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1 DISCUSSION ON EXISTING LLN METHODS As we formulate the SFDA as the problem of LLN, it is of interest to discuss some existing LLN methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We mainly discuss existing LLN methods that can be easily embedded into the current SFDA algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Based on this principle, we choose GCE (Zhang & Sabuncu, 2018), SL (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2019b) and GJS (Englesson & Azizpour, 2021) that have been theoretically proved to be robust to symmetric and asymmetric label noise, which are bounded label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We highlight that a more recent method GJS outperforms ELR in real-world noisy datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' However, we will show that GJS is inferior to ELR in SFDA scenarios, because the underlying assumption for GJS does not hold in SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Besides ELR, which leverages ETP, PCL is another method to leverage the same phenomenon, but we will show that it is also inappropriate for SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' To show the effects of the existing LLN methods under the unbounded label noise, we test these LLN methods on various SFDA datasets with target data whose labels are generated by source models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' As shown in Figure 4, GCE, SL, GJS, and PCL are better than CE but still not comparable to ELR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Our analysis indicates that ELR follows the principle of ETP, which is theoretically justified in SFDA scenarios by our Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Methods GCE, SL, and GJS follow the bounded label noise assumption, which does not hold in SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Hence, they perform worse than ELR in SFDA, even though GJS outperforms ELR in conventional LLN scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' PCL (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021) utilizes ETP to purify noisy labels of target data, but it performs significantly worse than ELR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' As the memorization speed of the unbounded label noise is very fast, and classifiers memorize noisy labels within a few iterations (shown in Figure 2), purifying noisy labels every epoch is inappropriate for SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' However, we notice that PCL performs relatively better on DomainNet than on other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The reason behind it is that the memorization speed in the DomainNet dataset is relatively slow than 8 08 上 78 77 NRC 76 SHOT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='9908 79 P 78 Test 77 NRC 1 76 SHOT 1 3 7 12 2580 SHOT NRC Test Accuracy 779 78 77 Vanilla GCE SL GJS ELRPublished as a conference paper at ICLR 2023 Table 3: Accuracies (%) on VisDA-C (Synthesis → Real) for ResNet101-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Method SFplanebcycl bus car horseknifemcyclpersonplantsktbrdtraintruckPer-class DANN (Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2016) \x17 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='9 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='7 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='844.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='2 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1 28.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='2 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='8 (i) Office-31 (ii) Office-Home (iii) VisDA (iv) DomainNet Figure 4: Evaluation of label noise methods on SFDA problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We use source models as an initialization of classifiers trained on target data and also use source models to annotate unlabeled target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Then we treat the target datasets as noisy datasets and use different label noise methods to solve the memorization issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' other datasets, which is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In conventional LLN scenarios, PCL does not suffer from the issue since the memorization speed is much lower than the conventional LLN scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In Figure 3(c), we also evaluate the performance by incorporating the existing LLN methods into the SFDA algorithms SHOT and NRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Since PCL and SHOT assign pseudo labels to target data, PCL is incompatible with some existing SFDA methods and cannot be easily embedded into some SFDA algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Hence, we only embed GCE, SL, GJS, and ELR into the SFDA algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The figure illustrates that ELR still performs better than other LLN methods when incorporated into SHOT and NRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We also notice that GCE, SL, and GJS provide marginal improvement to the vanilla SHOT and NRC methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We think the label noise in SFDA datasets is the hybrid noise that consists of both bounded label noise and unbounded label noise due to the non-linearity of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The GCE, SL, and GJS can address the bounded label noise, while ELR can address both bounded and unbounded label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Therefore, these experiments demonstrate that using ELR to leverage ETP can successfully address the unbounded label noise in SFDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 6 CONCLUSION In this paper, we study SFDA from a new perspective of LLN by theoretically showing that SFDA can be viewed as the problem of LLN with the unbounded label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Under this assumption, we rigorously justify that robust loss functions are not able to address the memorization issues of unbounded label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Meanwhile, based on this assumption, we further theoretically and empirically analyze the learning behavior of models during the early-time training stage and find that ETP can benifit the SFDA problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Through extensive experiments across multiple datasets, we show that ETP can be exploited by ELR to improve prediction performance, and it can also be used to enhance existing SFDA algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 9 75 70 Test Accuracy 65 60 55 50 CE GCE SL GJSPCL E ELR73 72 71 70 69 68 67 66 65 CE GCE SL GJS PCL ELR85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5 2 Accurae 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='0 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5 Test 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='0 CE GCE SL GJS PCL ELR68 67 Test Accuracy 66 65 64 63 62 61 60 CE GCE SL GJSPCL E ELRPublished as a conference paper at ICLR 2023 REFERENCES Sk Miraj Ahmed, Dripta S Raychaudhuri, Sujoy Paul, Samet Oymak, and Amit K Roy-Chowdhury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Unsupervised multi-source domain adaptation without access to source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In 32nd Conference on Neural Information Processing Systems (NeurIPS), 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Han Zhao, Remi Tachet Des Combes, Kun Zhang, and Geoffrey Gordon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' On learning invariant representations for domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 7523–7532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Zhaowei Zhu, Tongliang Liu, and Yang Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' A second-order approach to learning with instance- dependent label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 10113–10123, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 13 Published as a conference paper at ICLR 2023 A NEIGHBORS LABEL NOISE OBSERVATIONS ON REAL-WORLD DATASETS This section provides more observed results and explanations of Neighbors’ label noise during the Source-Free Domain Adaptation process on real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465 Class Index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='80 True Neighbors Ratio False Neighbors Ratio Figure 5: True/False Neighbors on Office-Home Plane Bcycl Bus Car Horse Knife McyclPersonPlant Sktbrd Train Truck 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='80 True Neighbors Ratio False Neighbors Ratio Figure 6: True/False Neighbors on VisDA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Class Index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='00 Misleading Neighbors Ratio Over-Confident Misleading Neighbors Ratio Figure 7: Neighbors Label Noise Analysis on Office-Home Currently, most SFDA methods inevitably leverage the pseudo-labels for self-supervised learning or to learn the cluster structure of the target data in the feature space, in order to realize the domain adaptation goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' However, the pseudo labels generated by the source domain are usually noisy and of poor quality due to the domain distribution shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Some neighborhood-based heuristic methods (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='b) have been proposed to purify these target domain pseudo labels, which use the pseudo label of neighbors in the feature space to correct and reassign the central data’s pseudo label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In fact, such methods rely on a strong assumption: a relatively high quality of the neighbors’ pseudo label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' However, in our experimental observations, we find that at the very beginning of the adaptation process, the similarity of two data points in the feature space can not fully represent their label space’s connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Furthermore, such methods are easy to provide useless and noisy prediction information for the central data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We will show some statistical results on VisDA and Office-Home, these two real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Following the neighborhood construction method in Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='b), we use the pre-trained source model to infer the target data, extract the feature space outputs and get the prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We use the cosine similarity on the feature space to find the top k similar neighbors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', k = 2) for each data point (named as the central data point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Then, we collect the neighbors regarding the ground truth label of central data points and study the neighbor’s quality for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' 14 Published as a conference paper at ICLR 2023 Neighbors who do not belong to the correct category We define the neighbors who do not belong to the same category as its central data point as False Neighbor, which means their ground-truth labels are not the same: Yneighbor ̸= Ycentral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' And the results of VisDA (train → validation) and Office-Home (Pr → Cl) datasets are shown in Figure 6 and Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Neighbors who can not provide useful prediction information We further study the prediction information provided by such neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Regardless of their true category properties, we consider neighbors whose Predicted Label is the same as the Ground Truth Label of the central data point to be Useful Neighbors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' otherwise, they are Misleading Neighbors, as they can not provide the expected useful prediction information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We denote the Misleading Neighbors Ratio as the proportion of noisy neighbors among all neighbors for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Besides, as some methods heuristically utilize the predicted logits as the predicted probability or confidence score in the pseudo label purification process, we further study the Over-Confident Misleading Neighbors Ratio for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We defined the over-confident misleading neighbors ratio as the number of over-confident misleading neighbors (misleading neighbors with a high predicted logit, larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='75) divided by the number of all neighbors per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The results on VisDA and Office-Home are shown in Figure 1ii and Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We want to clarify that the above exploratory experiment results can only reflect the phenomenon of unbounded noise in SFDA to some extent: the set of over-confidence misleading neighbors is non-empty can correspond, to some extent, to the fact that R is non-empty proved in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' but the definition of misleading neighbors does not rigorously satisfies the definition of unbounded label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' B RELATIONSHIP BETWEEN MISLABELING ERROR AND DOMAIN SHIFT In this part, we focus on explaining the relationship between the label noise and the domain shift, as illustrated in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The following theorem characterizes the relationship between the labeling error and the domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Without loss of generality, we assume that the ∆ is positively correlated with the vector µ2 − µ1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=', ∆⊤(µ2 − µ1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Let fS be the Bayes optimal classifier for the source domain S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Then Pr (x,y)∼DT[fS(x) ̸= y] = 1 2Φ(−d1 σ ) + 1 2Φ(−d2 σ ), (6) where d1 = �� µ2−µ1 2 − c �� sign( �� µ2−µ1 2 �� − ∥c∥), d2 = �� µ2−µ1 2 + c ��, c = (µ2 − µ1) ∆⊤(µ2−µ1) ∥µ2−µ1∥2 , and Φ is the standard normal cumulative distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1 indicates that the labeling error for the target domain can be represented by a function of the domain shift ∆, which can be shown numerically in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The projection of the domain shift ∆ on the vector µ2 − µ1 is given by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Since c is on the direction of µ2 − µ1, c can also be represented by α(µ2 − µ1), where α ∈ R characterizes the magnitude of the domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' More specifically, in Figure 8, we present the relationship between the mislabeling rate and α for all possible ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' When ∆ is positively correlated with µ2 − µ1 (assumption in Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1), we have α > 0, and when ∆ is negatively correlated with µ2 − µ1, we obtain α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' In both situations, we can observe that the labeling error increases with the absolute value of α increasing, which implies that the more severe the domain shift is, the greater the mislabeling error will be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Besides, we note that when the source and target domains are the same, the mislabeling error in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (6) is minimized and degraded to the Bayes error, which cannot be reduced (Fukunaga, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' This corresponds to the situation when ∆ is perpendicular to µ2 − µ1, c = 0, and α = 0 shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1 PROOFS FOR THEOREM B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' The Bayes classifier fS predicts x to the first component when log Pr[y = 1|X = x] Pr[y = −1|X = x] > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (7) Since the distributions of the two components with the same priors for the source domain are given by N(µ1, σ2Id) and N(µ2, σ2Id), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Based on Bayes’ rule, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (7) is equivalent to log Pr[X = x|y = 1] Pr[X = x|y = −1] > 0 (8) 15 Published as a conference paper at ICLR 2023 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5 α = ∆T (µ2−µ1) ||µ2−µ1||2 = sign(c) ||c|| ||µ2−µ1|| 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content='5 Mislabelling Rate on All Target Data Figure 8: Plot of Mislabeling Rate with different α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' We define c as the projection of the domain shift ∆ on the vector µ2 − µ1, and α represents the magnitude of domain shift projected on µ2 − µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' Solving the left hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (8) by using the knowledge of two multivariate Gaussian distributions, we get hS(x) := log Pr[X = x|y = 1] Pr[X = x|y = −1] = x⊤(µ1 − µ2) σ2 − ∥µ1∥2 − ∥µ2∥2 2σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (9) So fS predicts x to the first component when hS(x) > 0 and fS predicts x to the second component when hS(x) ≤ 0 The decision boundary is z such that hS(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' When there is no domain shift ∆ = 0, we have DS = DT , and the mislabeling rate is the Bayes error, which is given by: Pr (x,y)∼DS[fS(x) ̸= y] = 1 2 Pr x∼N (µ1,σ2Id)[hS(x) < 0|y = 1] + 1 2 Pr x∼N (µ2,σ2Id)[hS(x) > 0|y = −1] (10) We first study the first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFQT4oBgHgl3EQfqzYC/content/2301.13381v1.pdf'} +page_content=' (10): Pr x∼N (µ1,σ2Id)[hS(x) < 0|y = 1] = � · · � {x|x⊤(µ1−µ2)< ∥µ1∥2−∥µ2∥2 2 } 1 (2πσ2) d 2 exp � −∥x − µ1∥2 2σ2 � dx1dx2 · · · dxd = � · · � {x|−∞-type dislocations in Ti with hexagonal symmetry. +Fourth, the microstructure of samples was additionally examined by transmission electron +microscopy (TEM). Discs with a 3 mm diameter were cut from the edge of HPT-processed samples +using electric discharge machining and mechanically thinned to a thickness of 0.10-0.15 mm using +sandpapers. The 3 mm discs were further thinned to foils with electron transparency using a twin- +jet electro-polisher for Al (solution: 15 vol% HClO4 + 15 vol% C3H5(OH)3 + 70 vol% CH3OH, +voltage: 13 V, temperature: 263 K), Cu (solution: 15 vol% HNO3 + 15 vol% C3H5(OH)3 + 70 vol% +CH3OH, voltage: 10 V, temperature: 263 K), Ti (5 vol% HClO4 + 25 vol% C3H3(CH2)2CH2OH + +70 vol% CH3OH, voltage: 13 V, temperature: 263 K), Fe (90 vol% CH3COOH + 10 vol% HClO4 +voltage: 14 V, temperature: 300 K) and Cu-Zn (solution: 20 vol% HNO3 + 80 vol% CH3OH, +voltage: 20 V, temperature: 263 K). The thin foils were examined by TEM under an acceleration +voltage of 200 keV or 300 keV using bright-field images, dark-field images and selected area +electron diffraction (SAED) patterns. + + + +4 + + + +Figure 1. (a) Photograph of high-speed HPT machine. (b) Photograph of processing unit of HPT +machine. (c) Surfaces of disc samples before and after HPT processing with 1 rpm. + + + +(a) +Processing Unit +ControlUnit +[201Y +ALWAYS +Electric +Motor +Hydraulic +Press +(b) +Anvils +Sample +Pressure +(c) +Anneal +HPT +10 mm +Al +Cu +Cu-Zn +Ti +Fe5 + +Results +Fig. 2a shows the typical variation of hardness against distance from the disc center for Cu- +Zn discs processed by HPT with different rotation speeds. The hardness values are at the steady +state, except for the center of discs which show slightly lower hardness. The variation of hardness +against the distance from the center for the four selected metals is like the one for Cu-Zn, although +the magnitude of hardness depends on the material. The hardness values at different distances from +the disc center excluding the center of discs are summarized in Fig. 1b as the steady-state hardness +versus strain rate. The strain rate at each radial distance was estimated as von Mises shear strain (ε += γ/√3, where γ = 2πrN/h, r: radial distance, N: the number of turns and h: thickness [2]) divided +by processing time (t = N/ω, ω: rotation speed). The steady-state hardness appears to be +independent of strain rate within the range of 0.004 to 20 s-1. The steady-state hardness is the +lowest for Al and is higher for Cu, Cu-Zn, Ti and Fe, respectively, in good agreement with earlier +publications [9,10]. + + + + +Figure 2. Effect of strain rate on steady-state hardness. Variations of hardness versus (a) distance +from disc center and (b) strain rate for (a) Cu-Zn and (b) Al, Cu, Cu-Zn, Ti and Fe processed by +HPT for 15 rotations with various rotation speeds. + + +(a) +300 +(Hv) +250 +Microhardness( +200 +Cu - 30 wt% Zn +HPT:P=6GPa.T=300K.N=15 +150 +@ = 0.06 rpm +100 +@ = 1 rpm +50 +@ = 8 rpm +@ = 60 rpm +0 +0 +1 +2 +3 +4 +5 +DistanceFrom Disc Center(mm) +(b) +(Hv) +HPT:P=6GPa. T=300K, N=15 +400 +Fe +Microhardness +300 +Cu-Zn +200 +100 +A +0 +0.001 +0.01 +0.1 +1 +10 +100 +Strain Rate (s-1)6 + + +Fig. 3 shows the variations of shear strength versus HPT rotations examined in situ by +torque measurements for Fe processed by HPT with different rotation speeds. Despite some +deviations in the stress-strain plots, the steady-state stresses at large stains appear to be independent +of the rotation speed. Therefore, both in-situ torque measurements and ex-situ hardness +measurements suggest that the flow stress is reasonably independent of strain rate, although an +earlier study suggested a rate-dependent torque behavior for severely deformed Fe [14]. + + + +Figure 3. Effect of strain rate on steady-state shear stress. Shear stress versus the number of +rotations, evaluated in-situ by torque measurement, for Fe processed by HPT by up to 2 rotations +with various rotation speeds. Each curve represents the average of 4-6 measurements with an +average standard deviation of 14 MPa (+/- 3 %) for shear stress. + + +Fig. 4a shows the typical XRD profiles for Cu-Zn processed by HPT with different rotation +speeds. The material has an FCC structure and does not show any phase transformations by HPT +processing. The XRD profiles for the four model metals also show the presence of single phases +(FCC for Al and Cu, HCP for Ti and BCC for Fe) without the occurrence of any phase +transformations after HPT processing. The XRD profiles for all materials were evaluated by the +Rietveld refinement to determine the crystallite size and dislocation density, as shown in Fig. 4b +and 4c, respectively. Both crystallite size and dislocation density are reasonably independent of +strain rate, although the dislocation density slightly decreases, and the crystallite size slightly +increases in some materials for a rotation speed of 60 rpm. These changes at 60 rpm can be due to +the localized temperature rise during high-speed HPT [29,30]. + + + +1000 +(MPa) +800 +-0.1 rpm +0.6rpm +600 +-1rpm +- 6 rpm +400 +Fe +200 +HPT: P=4GPa, T=300K, N=2 +0 +0 +0.4 +0.8 +1.2 +1.6 +2.0 +Rotations7 + + + +Figure 4. Effect of strain rate on steady-state microstructure. (a) XRD profiles and variations of +(b) crystallite size and (c) dislocation density versus average strain rate for (a) Cu-Zn and (b, c) Al, +Cu, Cu-Zn, Ti and Fe processed by HPT for 15 rotations with various rotation speeds. + + +(a) +Normalized Intensity +Cu - 30 wt% Zn +HPT:P=6GPa,T=300K,N=15 +(200) +(220) +D(311) +111 +△ +@=60 rpm +@ = 8 rpm +@ = 1 rpm +@=0.06rpm +40 +50 +60 +70 +80 +90 +Diffraction Angle, 20 (deg.) +(b) +Rotation Speed, @ (rpm) +0.01 +0.1 +1 +10 +100 +1000 +(nm) +HPT: P=6GPa, T=300K, N=15 +4 +Al. +Size +口 +Cuo +口 +口 +Fev +Crystallite +100 +Ti. +4fcu-Zna +2 +10 +0.01 +0.1 +1 +10 +Average Strain Rate (s-1) +(c) +Rotation Speed, @ (rpm) +0.01 +0.1 +1 +10 +100 +X1013 +HPT: P=6GPa, T=300K, N=15 +1000 +Densi +Ti- +100 +Cu-Zna +(z-w) +10 +Fev +cu +口 +1 +Al. +0.1 +0.01 +0.1 +1 +10 +Average Strain Rate (s-1)8 + + +Fig. 5a-d illustrates the TEM bright-field images, dark-field images and SAED patterns for +Cu-Zn samples, which were expected to show the most significant changes by strain rate changes +due to the solid-solution effect. Fig. 5e summarizes the mean grain sizes versus the average strain +rate, in which the average grain sizes were determined from the orthogonal sizes of bright regions +in the dark-field images for 20-80 grains. Both bright-field and dark-field images confirm the +presence of UFG microstructures in all regions, although the grain boundaries are not well-defined +in these micrographs due to significant lattice distortions within the grains. The ring shape of the +SAED pattern also suggests the presence of nanograins with the FCC structure. Fig. 5e indicates +that the average grain size is independent of the strain rate, although the grain size slightly +increases with increasing the rotation speed to 60 rpm in good agreement with the crystallite size +measurements in Fig. 4b. The significant reduction of grain size to the nanometer level in Cu-Zn +should be due to the interaction of solute atoms and dislocations with increased multiplication rate +[10], although some studies attributed this feature to the low stacking fault energy of the alloy [7,8]. +Fig. 6 shows the typical steady-state microstructures of (a) Al, (b) Cu, (c) Ti and (d) Fe and +Table 1 compares the average crystallite and grain sizes determined in this study with some earlier +publications on Al 1050 [17,31], Cu [7-9,32], Cu-Zn [7,8,10,33], Ti [17,34,35] and Fe [9,36,37]. +In Table 1, the crystallite sizes are the average of the values at four rotation speeds selected in this +study, and grain sizes are the average sizes at a constant rotation speed. It is evident that the +measured sizes in this study are reasonably consistent with earlier publications, although crystallite +sizes examined by XRD are naturally smaller than the grain sizes measured by TEM. + + + +9 + + + +Figure 5. Effect of strain rate on steady-state grain size. (a-d) TEM bright-field images (left), dark- +field images (center) and SAED patterns (right) and (e) variation of mean grain size versus average +strain rate for Cu-Zn processed by HPT for 15 turns, with rotation speeds of (a) 0.06 rpm, (b) 1 +rpm, (c) 8 rpm and (d) 60 rpm. Dark-field images were taken with diffracted beams indicated by +arrows in SAED patterns. + + + +100 nm +100 nm +(b)@=1 rpm +100 nm +100 nm +222 +c=8pn +100nm +100nm +222 +d）@=60rpm +100 nm +100 nm +222 +Rotation Speed, o (rpm) +(e) +0.01 +0.1 +10 +100 +Grain Size (nm) +100 +80 +60 +40 +Cu - 30 wt% Zn +20 +HPT:P=6GPa,T=300K,N=15 +0 +0.01 +0.1 +1 +10 +Average Strain Rate (s-1)10 + + + +Figure 6. Presence of ultrafine grains in severely deformed pure metals. TEM bright-field images +(left), dark-field images (center), and SAED patterns (right) for (a) Al, (b) Cu, (c) Ti and (e) Fe at +steady state. Dark-field images were taken with diffracted beams indicated by arrows in SAED +patterns. + + +(a) Al +400 nm +400 nm +(b)cu +400 nm +400 nm +400 nm +400 nm +(d) Fe +B0 +400 nm +400 nm11 + + +Table 1. Comparison of crystallite size and grain size measured in this study with those reported +in the literature. +Material +Crystallite Size (nm) +Grain Size (nm) + +This Study +Literature +This Study +Literature +Al +384±40 +--- +504±226 +500 [17], 600 [31] +Cu +207±51 +84 [7], 59 [8] +273±130 +200 [9], 290 [32] +Cu-Zn +39±9 +17 [7], 30 [8] +64±17 +74 [33], 75 [10] +Ti +52±4 +43 [35] +149±184 +200 [34], 200 [19] +Fe +135±38 +87 [37] +226±119 +200 [9], 200 [36] + + +Discussion +Two questions arise from the current study. (i) What are the possible reasons for the +independence of steady-state microstructure and flow stress on the strain rate in SPD processing? +(ii) What are the reasons for the inconsistency between the conclusion of this study and those from +some earlier studies? +Regarding the first question (i), it is well known that a high strain rate and a low processing +temperature in metal forming increase the accumulation rate of lattice defects and enhance the +fragmentation of grains [3,4]. It was suggested that the effects of strain rate and temperature can +be suitably quantified by the Zener-Hollomon parameter (Z) not only in low strain levels [4] but +also for the steady state [14]. One can understand the steady state in terms of the defect generation +rate compared to that of annihilation: with increasing the strain level, the densities of lattice defects +reach critical values which launch effects of dynamic recovery, recrystallization and grain +boundary migration [3]; this way, a balance between generation and annihilation of lattice defects +is reached [3,4]. While the increase of lattice defect densities and/or grain fragmentation leads to +strain hardening, the onset of dynamic recovery, recrystallization and grain boundary migration, +however, implies strain softening so that in total, no change in overall microstructural features and +flow stress occurs [3-5]. An increase in homologous processing temperature reduces the rate of +grain fragmentation, enhances the rate of dynamic recrystallization and accordingly leads to an +increase in the steady-state crystallite/grain size, as reported in numerous publications [1-6]. An +increase in the strain rate enhances the rate of dislocation generation and grain fragmentation but +also increases the rate of dynamic recrystallization [21,22]. Unlike the absolute temperature which +appears in exponential form in the Zener-Hollomon parameter, the strain rate enters proportionally, +and thus the rate of grain fragmentation is less sensitive to the strain rate than to the temperature +[4,11]. However, the rate of dynamic recrystallization is also directly proportional to the strain rate +[21,22]. Therefore, one may expect that the effect of strain rate on the balance between grain +fragmentation and dynamic recrystallization is insignificant, a fact which has - at least within the +achievable measurement resolution - been experimentally observed in the current study. +The second question (ii) concerns the contradicting conclusions reported in other studies +on the significance of shear strain rate on microstructural evolution during SPD [12-20]. It should +be noted that the focus of the current study is on the steady-state microstructure. With our +experiments, in order to make sure that the microstructure is really at a steady state, the HPT +process was conducted for 15 rotations which correspond to a maximum shear strain of 590. The +authors wonder whether the rate dependence reported in some studies may arise from the fact that +the microstructural features were not still at a steady state for given deformation modes. Another +source of discrepancies between other studies [12-20] and ours may be the fact that the ex-situ and + +12 + +in-situ measurements of strength do not combine because of static recrystallization effects that +may take place after the ex-situ experiments, i.e. during the unloading event before the ex-situ +strength measurements are done [38,39]. Concerning the in-situ experiments such as those reported +in [14], the way of torque measurement may be different for different facilities used, i.e. imply +different contributions of friction which may distort the results to make them strain rate-dependent. +Some other factors may also affect the current experimental observations. One of those is +the resolution limit of hardness test, torque measurement, XRD and TEM. Concerning the +mechanical measurements, their error is usually within +/- 3%; XRD and TEM are expected to be +sensitive enough to reveal visible changes by variation of strain rate from 0.004 to 20 s-1, but none +of those changes have been observed within the current investigation. The same is true for the +earlier deviating studies [12-20] as they did not report significant changes in microstructure at the +steady state or close to the steady-state conditions. +Another factor is the temperature rise at high strain rates which can influence the evolution +of microstructure, particularly when adiabatic shear bands are formed. However, it was shown +using both experimental measurements and finite element modeling that the temperature rise +during HPT processing is not so significant because massive anvils connected to large metallic +plates act as heat sinks for a small disc sample [29,30]. In this study, the magnitude of temperature +rise, which was measured during the process using a thermocouple located 10 mm away from the +disc in the upper anvil or by an optical thermometer, was quite small (< 323 K) at least for the +rotation speeds of 0.06, 1 and 8 rpm, i.e. strain rates up to 3 s-1. Only at the highest speed of 60 +rpm applied, some noticeable localized temperature rise may have occurred leading to effects of +dynamic and static recrystallization (Fig. 2a, 2b, 4b, 4c). For the samples processed with 60 rpm, +the crystallite size slightly increases, and dislocation density slightly decreases (Fig. 4), leading to +a slight softening (Fig. 2). However, for the other rotation speeds selected, significant change +neither in crystallite size nor in dislocation density occurs and accordingly the hardness/strength +is also constant. Taken altogether, the main reason for the independence of the steady state on +strain rate appears to be the fairly parallel change in the rates of hardening (crystallite/grain +fragmentation) and those of the softening (dynamic recrystallization) phenomena. + +Conclusions +In summary, the current study on the processing of four metals with different melting points +(Al, Cu, Ti and Fe) and a Cu-Zn alloy using HPT with strain rates of 0.004 to 20 s-1 confirms that +the steady-state microstructure, hardness and shear stress are independent of strain rate, at least +within the resolution limits of XRD, TEM, torque measurement and microhardness tests. These +findings suggest that although a high strain rate is effective in the enhancement of crystallite/grain +fragmentation and defect accumulation at the early stages of straining, the final microstructure at +the steady state, which is achieved by a balance between crystallite/grain fragmentation and +dynamic recrystallization, is reasonably independent of strain rate. + +Acknowledgments +This work has been supported in part by the Light Metals Educational Foundation of Japan, +in part by Grants-in-Aid for Scientific Research on Innovative Areas (JP19H05176 & +JP21H00150) and Challenging Research (Exploratory) (JP22K18737) from the MEXT, Japan, and +in part by the “Metals and Alloys under Extreme Impacts” Laboratory of Eurasian Center of +Excellence, USATU (assignment #075-03-2021-014/4). + + +13 + +References +[1] K. Edalati, A. Bachmaier, V.A. Beloshenko, Y. Beygelzimer, V.D. Blank, W.J. Botta, K. +Bryła, J. Čížek, S. Divinski, N.A. Enikeev, Y. Estrin, G. Faraji, R.B. 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Mater. 23 (2021) 2100159. + diff --git a/vNE0T4oBgHgl3EQf9wJp/content/tmp_files/load_file.txt b/vNE0T4oBgHgl3EQf9wJp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9c32e521016b178524cec34b0c48d203a5111539 --- /dev/null +++ b/vNE0T4oBgHgl3EQf9wJp/content/tmp_files/load_file.txt @@ -0,0 +1,632 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf,len=631 +page_content='1 Materials Science and Engineering A, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 859, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 144231, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' https:// https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='msea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='144231 Significance of Strain Rate in Severe Plastic Deformation on Steady-State Microstructure and Strength Kaveh Edalati1,*, Qing Wang1, Nariman A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Enikeev2,3, Laura-Jean Peters4, Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Zehetbauer4 and Erhard Schafler4 1 WPI, International Institute for Carbon-Neutral Energy Research (WPI-I2CNER), Kyushu University, Fukuoka 819-0395, Japan 2 Ufa State Aviation Technical University (USATU), K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Marx 12, 450008 Ufa, Russia 3 Center for Design of Functional Materials, Bashkir State University, 450076 Ufa, Russia 4 Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria The microstructure and mechanical properties of materials saturate to steady states after severe plastic deformation (SPD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Despite the well-known effect of temperature on the steady-state microstructure, there is no general agreement on the significance of strain rate and the applicability of the Zener-Hollomon parameter in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' In this study, several pure metals (aluminum, copper, titanium, and iron) and a Cu-30Zn (wt%) brass alloy have been processed by a high-speed high-pressure torsion (HPT) equipment with controllable rotation speeds in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='06 to 60 rpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' It is found that crystallite/grain size, dislocation density, microhardness and shear stress at the steady state are reasonably rate-independent for the von Mises strain rates in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='004 to 20 s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Because both rates of grain refinement and of dynamic recrystallization are proportional to the strain rate, it is suggested that their balance, which determines the steady state, is rate- independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Keywords: ultrafine-grained (UFG) materials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' nanostructured materials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' strain-rate hardening;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Zener-Hollomon parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' high-pressure torsion (HPT) Corresponding author (E mail: kaveh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='edalati@kyudai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='jp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Tel: +81 92 802 6744) 2 Introduction Severe plastic deformation (SPD) methods are efficient in achieving ultrafine-grained (UFG) microstructures in a wide range of metallic and non-metallic materials [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The microstructure and resultant mechanical and functional properties significantly change at the early stages of straining, but they finally saturate to the steady state at large strains particularly in single- phase materials [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The occurrence of a steady state, which was first recognized by Bridgman in the 1930s [2], has been attributed to a balance between the rate of grain refinement and defect generation on the one hand, and the rate of dynamic recovery, recrystallization and grain boundary migration on the other hand [3-5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Investigation of the parameters influencing the steady-state grain size has been of significant interest for the past two decades because one target in SPD processing has been introducing new strategies for further reduction of the final grain size [4-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' These investigations sometimes reported contradicting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' For example, while several studies suggested that stacking fault energy is the most important factor in determining the steady-state grain size [7,8], some others suggested that the final grain size is reasonably independent of stacking fault energy [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Most of the studies, however, suggested that homologous temperature and atomic diffusion are important factors in determining the steady-state grain size [3-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Despite a general agreement on the effect of temperature on the steady-state grain size, the significance of strain rate has not been well clarified so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' In conventional metal forming, it is generally accepted that strain rate and temperature affect the microstructural and strength evolution through the Zener-Hollomon parameter, 𝑍 = 𝜀̇ exp−𝑄/𝑅𝑇 (𝜀̇: strain rate, Q: activation energy for the operative thermally activated process, R: gas constant, T: absolute temperature) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' In SPD processing, some researchers suggested that higher strain rates can reduce grain size, particularly when the homologous temperature is low [12- 16], while others did not find any rate-dependent changes at the steady state [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Several studies showed that the strain rate can influence the steady-state microstructure through a change in the deformation mechanism (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=', dislocation activity to twinning) [18] or phase transformation [19], while other researchers reported only a small change in the microstructure even at extremely large strain rates [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Since a high strain rate can enhance the rate of both grain fragmentation [4,11] and dynamic recrystallization [21,22] phenomena, its influence on the steady-state grain size, where there is a balance between these two phenomena, still needs clarification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' In this study, high-pressure torsion (HPT), as an SPD method [1,2], is employed and the effect of von Mises strain rate in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='004 s-1 to 20 s-1 on the steady-state microstructure, hardness and shear strength is studied for five model materials (aluminum, copper, titanium, iron and brass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The results confirm that the steady state is independent of strain rate within the detection limits of employed characterization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Experimental Procedures Discs of Al 1050 (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='5%), Cu (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='99%), Ti (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='9%), Fe (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='96%) and Cu - 30 wt% Zn with 10 mm diameter and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='8 mm thickness were annealed in an argon atmosphere for 1 h at 773, 873, 1073, 1273 and 793 K, respectively, and processed by a high-speed HPT machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='. 1a and 1b, the machine was equipped with a hydraulic press, an electric motor and a control unit, and it had two anvils made of a composite of tungsten carbide and 11 wt% cobalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' There was a 10 mm diameter flat-bottomed hole with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='25 mm depth and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='1 mm roughness on the center of each anvil to account for full torsion deformation of disc samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The HPT process was conducted for 15 turns under a pressure of 6 GPa for Al, Cu, Fe and Cu-Zn and under 2 GPa for Ti to avoid 3 the formation of the high-pressure ω-Ti phase [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The rotation speed of HPT anvils was either 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='06, 1, 8 or 60 rotations per minute (rpm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The maximum temperature of anvils, which was measured during the process using a thermocouple located at 10 mm away from the disc in the upper anvil or by an optical thermometer, was below 323 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The Fe discs processed for 60 rpm were excluded from the experiments because they stack to the anvils, which made the reliability of their processing unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The surfaces of disc samples before and after HPT processing with 1 rpm are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The roughness of the sample surfaces was similar to the roughness of flat-bottomed holes on the anvils, which was introduced for avoiding slippage of samples during the HPT process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' After the HPT process, the discs were first ground using sandpapers to remove the roughened surface layers and then metallographically polished for examination by different methods of microstructural and mechanical property characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' First, the discs were polished to mirror-like surfaces and the Vickers hardness on the upper surface of the discs was measured in four different radial directions at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='15, 1, 2, 3 and 4 mm away from the disc center using the indentation loads of 2 N for Al, 3 N for Cu and 5 N for Ti, Fe and Cu-Zn and the indentation time of 15 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Second, in addition to the ex-situ evaluations of strength by hardness measurement, the strength of Fe was examined in situ by torque measurements during HPT under a pressure of 4 GPa for 2 turns with rotation speeds of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='6, 1 and 6 rpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The shear stress was estimated as 𝜏 = 3𝑞 2𝜋𝑅3 where q is the measured torque, and R is the radius of the disc (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Third, the polished discs were examined by X-ray diffraction (XRD) with the Rigaku SmartLab diffractometer using Cu Kα radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The XRD profiles were evaluated by the Rietveld refinement by the MAUD software to determine the microstrain and crystallite size [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The crystallite size is the average dimension of defect-free crystalline areas (domains) that scatter X- rays coherently [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' It was shown that crystallite size in severely deformed metals is equivalent to subgrain or dislocation cell size [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Since the strength of deformed and severely deformed metallic materials is mainly determined by dislocation cell sizes [25,26], the crystallite size is an important parameter to clarify the significance of strain rate dependence in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' For the Rietveld refinement, instrumental broadening was considered by measuring the silicon standard sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Moreover, the dislocation density was determined using the Williamson-Smallman method [27] for the materials with cubic symmetry and using the technique proposed in [28] for -type dislocations in Ti with hexagonal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Fourth, the microstructure of samples was additionally examined by transmission electron microscopy (TEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Discs with a 3 mm diameter were cut from the edge of HPT-processed samples using electric discharge machining and mechanically thinned to a thickness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='10-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='15 mm using sandpapers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The 3 mm discs were further thinned to foils with electron transparency using a twin- jet electro-polisher for Al (solution: 15 vol% HClO4 + 15 vol% C3H5(OH)3 + 70 vol% CH3OH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' voltage: 13 V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' temperature: 263 K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Cu (solution: 15 vol% HNO3 + 15 vol% C3H5(OH)3 + 70 vol% CH3OH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' voltage: 10 V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' temperature: 263 K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Ti (5 vol% HClO4 + 25 vol% C3H3(CH2)2CH2OH + 70 vol% CH3OH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' voltage: 13 V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' temperature: 263 K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Fe (90 vol% CH3COOH + 10 vol% HClO4 voltage: 14 V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' temperature: 300 K) and Cu-Zn (solution: 20 vol% HNO3 + 80 vol% CH3OH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' voltage: 20 V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' temperature: 263 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The thin foils were examined by TEM under an acceleration voltage of 200 keV or 300 keV using bright-field images, dark-field images and selected area electron diffraction (SAED) patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 4 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' (a) Photograph of high-speed HPT machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' (b) Photograph of processing unit of HPT machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' (c) Surfaces of disc samples before and after HPT processing with 1 rpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' (a) Processing Unit ControlUnit [201Y ALWAYS Electric Motor Hydraulic Press (b) Anvils Sample Pressure (c) Anneal HPT 10 mm Al Cu Cu Zn Ti Fe5 Results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 2a shows the typical variation of hardness against distance from the disc center for Cu- Zn discs processed by HPT with different rotation speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The hardness values are at the steady state, except for the center of discs which show slightly lower hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The variation of hardness against the distance from the center for the four selected metals is like the one for Cu-Zn, although the magnitude of hardness depends on the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The hardness values at different distances from the disc center excluding the center of discs are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 1b as the steady-state hardness versus strain rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The strain rate at each radial distance was estimated as von Mises shear strain (ε = γ/√3, where γ = 2πrN/h, r: radial distance, N: the number of turns and h: thickness [2]) divided by processing time (t = N/ω, ω: rotation speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The steady-state hardness appears to be independent of strain rate within the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='004 to 20 s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The steady-state hardness is the lowest for Al and is higher for Cu, Cu-Zn, Ti and Fe, respectively, in good agreement with earlier publications [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Effect of strain rate on steady-state hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Variations of hardness versus (a) distance from disc center and (b) strain rate for (a) Cu-Zn and (b) Al, Cu, Cu-Zn, Ti and Fe processed by HPT for 15 rotations with various rotation speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' (a) 300 (Hv) 250 Microhardness( 200 Cu 30 wt% Zn HPT:P=6GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='T=300K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='N=15 150 @ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='06 rpm 100 @ = 1 rpm 50 @ = 8 rpm @ = 60 rpm 0 0 1 2 3 4 5 DistanceFrom Disc Center(mm) (b) (Hv) HPT:P=6GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' T=300K, N=15 400 Fe Microhardness 300 Cu Zn 200 100 A 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='1 1 10 100 Strain Rate (s 1)6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 3 shows the variations of shear strength versus HPT rotations examined in situ by torque measurements for Fe processed by HPT with different rotation speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Despite some deviations in the stress-strain plots, the steady-state stresses at large stains appear to be independent of the rotation speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Therefore, both in-situ torque measurements and ex-situ hardness measurements suggest that the flow stress is reasonably independent of strain rate, although an earlier study suggested a rate-dependent torque behavior for severely deformed Fe [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Effect of strain rate on steady-state shear stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Shear stress versus the number of rotations, evaluated in-situ by torque measurement, for Fe processed by HPT by up to 2 rotations with various rotation speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Each curve represents the average of 4-6 measurements with an average standard deviation of 14 MPa (+/- 3 %) for shear stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 4a shows the typical XRD profiles for Cu-Zn processed by HPT with different rotation speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The material has an FCC structure and does not show any phase transformations by HPT processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The XRD profiles for the four model metals also show the presence of single phases (FCC for Al and Cu, HCP for Ti and BCC for Fe) without the occurrence of any phase transformations after HPT processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The XRD profiles for all materials were evaluated by the Rietveld refinement to determine the crystallite size and dislocation density, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 4b and 4c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Both crystallite size and dislocation density are reasonably independent of strain rate, although the dislocation density slightly decreases, and the crystallite size slightly increases in some materials for a rotation speed of 60 rpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' These changes at 60 rpm can be due to the localized temperature rise during high-speed HPT [29,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 1000 (MPa) 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='1 rpm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='6rpm 600 1rpm 6 rpm 400 Fe 200 HPT: P=4GPa, T=300K, N=2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='0 Rotations7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Effect of strain rate on steady-state microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' (a) XRD profiles and variations of (b) crystallite size and (c) dislocation density versus average strain rate for (a) Cu-Zn and (b, c) Al, Cu, Cu-Zn, Ti and Fe processed by HPT for 15 rotations with various rotation speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' (a) Normalized Intensity Cu 30 wt% Zn HPT:P=6GPa,T=300K,N=15 (200) (220) D(311) 111 △ @=60 rpm @ = 8 rpm @ = 1 rpm @=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='06rpm 40 50 60 70 80 90 Diffraction Angle, 20 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=') (b) Rotation Speed, @ (rpm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='1 1 10 100 1000 (nm) HPT: P=6GPa, T=300K, N=15 4 Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Size 口 Cuo 口 口 Fev Crystallite 100 Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 4fcu Zna 2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='1 1 10 Average Strain Rate (s 1) (c) Rotation Speed, @ (rpm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='1 1 10 100 X1013 HPT: P=6GPa, T=300K, N=15 1000 Densi Ti 100 Cu Zna (z w) 10 Fev cu 口 1 Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='1 1 10 Average Strain Rate (s 1)8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 5a-d illustrates the TEM bright-field images, dark-field images and SAED patterns for Cu-Zn samples, which were expected to show the most significant changes by strain rate changes due to the solid-solution effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 5e summarizes the mean grain sizes versus the average strain rate, in which the average grain sizes were determined from the orthogonal sizes of bright regions in the dark-field images for 20-80 grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Both bright-field and dark-field images confirm the presence of UFG microstructures in all regions, although the grain boundaries are not well-defined in these micrographs due to significant lattice distortions within the grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The ring shape of the SAED pattern also suggests the presence of nanograins with the FCC structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 5e indicates that the average grain size is independent of the strain rate, although the grain size slightly increases with increasing the rotation speed to 60 rpm in good agreement with the crystallite size measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The significant reduction of grain size to the nanometer level in Cu-Zn should be due to the interaction of solute atoms and dislocations with increased multiplication rate [10], although some studies attributed this feature to the low stacking fault energy of the alloy [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 6 shows the typical steady-state microstructures of (a) Al, (b) Cu, (c) Ti and (d) Fe and Table 1 compares the average crystallite and grain sizes determined in this study with some earlier publications on Al 1050 [17,31], Cu [7-9,32], Cu-Zn [7,8,10,33], Ti [17,34,35] and Fe [9,36,37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' In Table 1, the crystallite sizes are the average of the values at four rotation speeds selected in this study, and grain sizes are the average sizes at a constant rotation speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' It is evident that the measured sizes in this study are reasonably consistent with earlier publications, although crystallite sizes examined by XRD are naturally smaller than the grain sizes measured by TEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 9 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Effect of strain rate on steady-state grain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' (a-d) TEM bright-field images (left), dark- field images (center) and SAED patterns (right) and (e) variation of mean grain size versus average strain rate for Cu-Zn processed by HPT for 15 turns, with rotation speeds of (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='06 rpm, (b) 1 rpm, (c) 8 rpm and (d) 60 rpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Dark-field images were taken with diffracted beams indicated by arrows in SAED patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 100 nm 100 nm (b)@=1 rpm 100 nm 100 nm 222 c=8pn 100nm 100nm 222 d）@=60rpm 100 nm 100 nm 222 Rotation Speed, o (rpm) (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='1 10 100 Grain Size (nm) 100 80 60 40 Cu 30 wt% Zn 20 HPT:P=6GPa,T=300K,N=15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='1 1 10 Average Strain Rate (s 1)10 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Presence of ultrafine grains in severely deformed pure metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' TEM bright-field images (left), dark-field images (center), and SAED patterns (right) for (a) Al, (b) Cu, (c) Ti and (e) Fe at steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Dark-field images were taken with diffracted beams indicated by arrows in SAED patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' (a) Al 400 nm 400 nm (b)cu 400 nm 400 nm 400 nm 400 nm (d) Fe B0 400 nm 400 nm11 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Comparison of crystallite size and grain size measured in this study with those reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Material Crystallite Size (nm) Grain Size (nm) This Study Literature This Study Literature Al 384±40 -- 504±226 500 [17], 600 [31] Cu 207±51 84 [7], 59 [8] 273±130 200 [9], 290 [32] Cu Zn 39±9 17 [7], 30 [8] 64±17 74 [33], 75 [10] Ti 52±4 43 [35] 149±184 200 [34], 200 [19] Fe 135±38 87 [37] 226±119 200 [9], 200 [36] Discussion Two questions arise from the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' (i) What are the possible reasons for the independence of steady-state microstructure and flow stress on the strain rate in SPD processing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' (ii) What are the reasons for the inconsistency between the conclusion of this study and those from some earlier studies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Regarding the first question (i), it is well known that a high strain rate and a low processing temperature in metal forming increase the accumulation rate of lattice defects and enhance the fragmentation of grains [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' It was suggested that the effects of strain rate and temperature can be suitably quantified by the Zener-Hollomon parameter (Z) not only in low strain levels [4] but also for the steady state [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' One can understand the steady state in terms of the defect generation rate compared to that of annihilation: with increasing the strain level, the densities of lattice defects reach critical values which launch effects of dynamic recovery, recrystallization and grain boundary migration [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' this way, a balance between generation and annihilation of lattice defects is reached [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' While the increase of lattice defect densities and/or grain fragmentation leads to strain hardening, the onset of dynamic recovery, recrystallization and grain boundary migration, however, implies strain softening so that in total, no change in overall microstructural features and flow stress occurs [3-5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' An increase in homologous processing temperature reduces the rate of grain fragmentation, enhances the rate of dynamic recrystallization and accordingly leads to an increase in the steady-state crystallite/grain size, as reported in numerous publications [1-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' An increase in the strain rate enhances the rate of dislocation generation and grain fragmentation but also increases the rate of dynamic recrystallization [21,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Unlike the absolute temperature which appears in exponential form in the Zener-Hollomon parameter, the strain rate enters proportionally, and thus the rate of grain fragmentation is less sensitive to the strain rate than to the temperature [4,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' However, the rate of dynamic recrystallization is also directly proportional to the strain rate [21,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Therefore, one may expect that the effect of strain rate on the balance between grain fragmentation and dynamic recrystallization is insignificant, a fact which has - at least within the achievable measurement resolution - been experimentally observed in the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The second question (ii) concerns the contradicting conclusions reported in other studies on the significance of shear strain rate on microstructural evolution during SPD [12-20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' It should be noted that the focus of the current study is on the steady-state microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' With our experiments, in order to make sure that the microstructure is really at a steady state, the HPT process was conducted for 15 rotations which correspond to a maximum shear strain of 590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The authors wonder whether the rate dependence reported in some studies may arise from the fact that the microstructural features were not still at a steady state for given deformation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Another source of discrepancies between other studies [12-20] and ours may be the fact that the ex-situ and 12 in-situ measurements of strength do not combine because of static recrystallization effects that may take place after the ex-situ experiments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' during the unloading event before the ex-situ strength measurements are done [38,39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Concerning the in-situ experiments such as those reported in [14], the way of torque measurement may be different for different facilities used, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' imply different contributions of friction which may distort the results to make them strain rate-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Some other factors may also affect the current experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' One of those is the resolution limit of hardness test, torque measurement, XRD and TEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Concerning the mechanical measurements, their error is usually within +/- 3%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' XRD and TEM are expected to be sensitive enough to reveal visible changes by variation of strain rate from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='004 to 20 s-1, but none of those changes have been observed within the current investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' The same is true for the earlier deviating studies [12-20] as they did not report significant changes in microstructure at the steady state or close to the steady-state conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Another factor is the temperature rise at high strain rates which can influence the evolution of microstructure, particularly when adiabatic shear bands are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' However, it was shown using both experimental measurements and finite element modeling that the temperature rise during HPT processing is not so significant because massive anvils connected to large metallic plates act as heat sinks for a small disc sample [29,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' In this study, the magnitude of temperature rise, which was measured during the process using a thermocouple located 10 mm away from the disc in the upper anvil or by an optical thermometer, was quite small (< 323 K) at least for the rotation speeds of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='06, 1 and 8 rpm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' strain rates up to 3 s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Only at the highest speed of 60 rpm applied, some noticeable localized temperature rise may have occurred leading to effects of dynamic and static recrystallization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 2a, 2b, 4b, 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' For the samples processed with 60 rpm, the crystallite size slightly increases, and dislocation density slightly decreases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 4), leading to a slight softening (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' However, for the other rotation speeds selected, significant change neither in crystallite size nor in dislocation density occurs and accordingly the hardness/strength is also constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Taken altogether, the main reason for the independence of the steady state on strain rate appears to be the fairly parallel change in the rates of hardening (crystallite/grain fragmentation) and those of the softening (dynamic recrystallization) phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Conclusions In summary, the current study on the processing of four metals with different melting points (Al, Cu, Ti and Fe) and a Cu-Zn alloy using HPT with strain rates of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='004 to 20 s-1 confirms that the steady-state microstructure, hardness and shear stress are independent of strain rate, at least within the resolution limits of XRD, TEM, torque measurement and microhardness tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' These findings suggest that although a high strain rate is effective in the enhancement of crystallite/grain fragmentation and defect accumulation at the early stages of straining, the final microstructure at the steady state, which is achieved by a balance between crystallite/grain fragmentation and dynamic recrystallization, is reasonably independent of strain rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Acknowledgments This work has been supported in part by the Light Metals Educational Foundation of Japan, in part by Grants-in-Aid for Scientific Research on Innovative Areas (JP19H05176 & JP21H00150) and Challenging Research (Exploratory) (JP22K18737) from the MEXT, Japan, and in part by the “Metals and Alloys under Extreme Impacts” Laboratory of Eurasian Center of Excellence, USATU (assignment #075-03-2021-014/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' 13 References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Edalati, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Bachmaier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Beloshenko, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE0T4oBgHgl3EQf9wJp/content/2301.02805v1.pdf'} +page_content=' Beygelzimer, V.' metadata={'source': 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0000000000000000000000000000000000000000..ea19979f8376f9f6a288f020064338227a33d807 --- /dev/null +++ b/vNE1T4oBgHgl3EQfkQSg/content/tmp_files/2301.03272v1.pdf.txt @@ -0,0 +1,2257 @@ +A polytopal method for the Brinkman problem robust +in all regimes +Daniele A. Di Pietro1 and Jérôme Droniou2 +1IMAG, Univ Montpellier, CNRS, Montpellier, France, daniele.di-pietro@umontpellier.fr +2School of Mathematics, Monash University, Melbourne, Australia, jerome.droniou@monash.edu +January 10, 2023 +Abstract +In this work we develop a discretisation method for the Brinkman problem that is +uniformly well-behaved in all regimes (as identified by a local dimensionless number with +the meaning of a friction coefficient) and supports general meshes as well as arbitrary +approximation orders. The method is obtained combining ideas from the Hybrid High- +Order and Discrete de Rham methods, and its robustness rests on a potential reconstruction +and stabilisation terms that change in nature according to the value of the local friction +coefficient. We derive error estimates that, thanks to the presence of cut-off factors, are +valid across the all regimes and provide extensive numerical validation. +MSC: 65N30, 65N08, 76S05, 76D07 +Key words: Brinkman, Darcy, Stokes, Hybrid High-Order methods, Discrete de Rham +methods +1 +Introduction +The Brinkman problem governs the flow of a viscous fluid in an inhomogeneous material where +fractures, bubbles, or channels are present within a porous matrix. Mathematically, this problem +translates into a system of partial differential equations with saddle-point structure which can +be regarded as a superposition of the Stokes and Darcy systems. As pointed out in [30], the +construction of finite element approximations that are uniformly well-behaved across the entire +range of (Stokes- or Darcy-dominated) regimes is not straightforward; a representative, but by +far not exhaustive, list of references is [2, 4, 12–14, 26, 28, 29, 34]. In [10], we introduced a +numerical method for the Brinkman problem on matching simplicial meshes and derived what +appears to be the first error estimate accounting for the local regime through a dimensionless +number which can be interpreted as a friction coefficient. Thanks to the presence of cutoff +factors, this error estimate holds in all situations, including the Stokes problem as well as the +singular limit corresponding to the pure Darcy problem. +In this work, we provide a positive answer to an open question left in the above reference, +namely whether similar robustness features and error estimates can be obtained on general +polytopal meshes. As for the original method of [10], the discretisation of the Stokes term is +1 +arXiv:2301.03272v1 [math.NA] 9 Jan 2023 + +inspired by Hybrid High-Order (HHO) methods [19, 21, 23] while, for the Darcy and forcing +terms, a novel construction inspired by discrete de Rham methods [17, 20] (see also [22] for an +antecedent) replaces the one based on the Raviart–Thomas–Nédélec space [31, 32]. The first +central element in this construction is a discrete vector potential that changes in nature depending +on the value of the local friction coefficient. The other key ingredient are regime-dependent +stabilisation terms. Thanks to these novel tools, we are able to derive a robust estimate of +the adjoint error for the discrete divergence, which is the pivot result for the extension of the +techniques of [10] to polytopal meshes. The resulting error estimate, stated in Theorem 7 below, +is valid on the entire range of values for the local friction coefficient, from 0 (pure Stokes) to ++∞ (pure Darcy). +The rest of the work is organised as follows. In Section 2 we briefly recall the continuous and +discrete settings. In Section 3 we formulate the numerical scheme and state the main stability +and convergence results. Extensive numerical validation of these results on a variety of meshes +and regimes for analytical solutions is provided in Section 4, where a more physical three- +dimensional test case is also considered. Finally, the proofs of the main results are collected in +Section 5. +2 +Setting +2.1 +Continuous problem +Let Ω ⊂ R𝑑, 𝑑 ∈ {2, 3}, denote a bounded connected open polytopal (i.e., polygonal if 𝑑 = 2 +and polyhedral if 𝑑 = 3) domain with boundary 𝜕Ω. For the sake of simplicity, and without loss +of generality, we assume that Ω has unit diameter. Let two functions 𝜇 : Ω → R and 𝜈 : Ω → R +be given. In what follows, we assume that there exist real numbers 𝜇, 𝜇, and 𝜈 such that, almost +everywhere in Ω, +0 < 𝜇 ≤ 𝜇 ≤ 𝜇, +0 ≤ 𝜈 ≤ 𝜈. +(1) +Let 𝒇 : Ω → R𝑑 and 𝑔 : Ω → R denote volumetric source terms. The Brinkman problem reads: +Find the velocity 𝒖 : Ω → R𝑑 and the pressure 𝑝 : Ω → R such that +−∇·(𝜇∇𝒖) + 𝜈𝒖 + ∇𝑝 = 𝒇 +in Ω, +(2a) +∇·𝒖 = 𝑔 +in Ω, +(2b) +𝒖 = 0 +on 𝜕Ω, +(2c) +∫ +Ω +𝑝 = 0. +(2d) +A few simplifications are made to make the exposition more compact while retaining all the +difficulties related to the robustness across the entire range of values for 𝜇 and 𝜈. First of all, +in (2a) we have considered a viscous term expressed in terms of the full gradient instead of +its symmetric part ∇s. The changes to replace ∇ with ∇s are standard in the HHO literature; +see, e.g., [11, 21] and [19, Chapter 7]. Second, we assume henceforth that both 𝜇 and 𝜈 are +piecewise constant on a polytopal partition 𝑃Ω of the domain. The extension to coefficients +that vary smoothly inside each element, and are possibly full tensors, is also standard; see, in +particular, [19, Section 4.2]. +2 + +2.2 +Discrete setting +2.2.1 +Mesh and notation for inequalities up to a constant +We consider polytopal meshes Mℎ ≔ Tℎ ∪ Fℎ matching the geometrical requirements detailed +in [19, Definition 1.4], with Tℎ set of elements and Fℎ set of faces. To avoid dealing with jumps +of the problem coefficients 𝜇 and 𝜈 inside mesh elements, we additionally assume that Tℎ is +compatible with 𝑃Ω, meaning that, for each 𝑇 ∈ Tℎ, there exists 𝜔 ∈ 𝑃Ω such that 𝑇 ⊂ 𝜔. +We then set 𝜇𝑇 ≔ 𝜇|𝑇 and 𝜈𝑇 ≔ 𝜈|𝑇 for all 𝑇 ∈ Tℎ, noticing that these constant values are +uniquely defined in each element. For any 𝑌 ∈ Mℎ, we denote by ℎ𝑌 its diameter, so that +ℎ = max𝑇∈Tℎ ℎ𝑇 > 0. For every mesh element 𝑇 ∈ Tℎ, we denote by F𝑇 the subset of Fℎ +containing the faces that lie on the boundary 𝜕𝑇 of 𝑇. For any mesh face 𝐹 ∈ Fℎ, we fix once +and for all a unit normal vector 𝒏𝐹 and, for any mesh element 𝑇 ∈ Tℎ such that 𝐹 ∈ F𝑇, we let +𝜔𝑇𝐹 ∈ {−1, +1} denote the orientation of 𝐹 relative to 𝑇, selected so that 𝜔𝑇𝐹𝒏𝐹 points out of +𝑇. Boundary faces lying on 𝜕Ω are collected in the set F b +ℎ . +Our focus being on the ℎ-convergence analysis, we assume that Mℎ belongs to a sequence of +refined polygonal or polyhedral meshes that is regular in the sense of [19, Definition 1.9]. This +implies, in particular, that the number of faces of each mesh element is bounded from above by +an integer independent of ℎ; see [19, Lemma 1.12]. +From this point on, 𝑎 ≲ 𝑏 (resp. 𝑎 ≳ 𝑏) means 𝑎 ≤ 𝐶𝑏 (resp. 𝑎 ≥ 𝐶𝑏) with 𝐶 only +depending on Ω, the mesh regularity parameter, and the polynomial degree 𝑘 of the scheme +defined in Section 3. We stress that this means, in particular, that 𝐶 is independent of the +problem parameters 𝜇 and 𝜈. We also write 𝑎 ≃ 𝑏 as a shorthand for “𝑎 ≲ 𝑏 and 𝑏 ≲ 𝑎”. +2.2.2 +Polynomial spaces +Given 𝑌 ∈ Tℎ ∪ Fℎ and an integer 𝑚 ≥ 0, we denote by P𝑚(𝑌) the space spanned by the +restriction to 𝑌 of 𝑑-variate polynomials of total degree ≤ 𝑚. The symbols P𝑚(𝑌; R𝑑) and +P𝑚(𝑌; R𝑑×𝑑) respectively denote the sets of vector- and tensor-valued functions over 𝑌 whose +components are in P𝑚(𝑌). For 𝑇 ∈ Tℎ, we will need the following direct decomposition of +P𝑚(𝑇; R𝑑) (see, e.g., [5, Corollary 7.4]): +P𝑚(𝑇; R𝑑) = G𝑚(𝑇) ⊕ Gc,𝑚(𝑇), +with +G𝑚(𝑇) ≔ ∇P𝑚+1(𝑇) +and +Gc,𝑚(𝑇) ≔ +� +(𝒙 − 𝒙𝑇)⊥P𝑚−1(𝑇) +if 𝑑 = 2, +(𝒙 − 𝒙𝑇) × P𝑚−1(𝑇; R3) +if 𝑑 = 3, +(3) +where 𝒙𝑇 is a point such that 𝑇 is star-shaped with respect to a ball of radius ≳ ℎ𝑇 and, in the +case 𝑑 = 2, for any 𝒗 ∈ R2 we denote by 𝒗⊥ the vector obtained rotating 𝒗 by − 𝜋 +2 radians. Given +a polynomial (sub)space X𝑚(𝑌) on 𝑌 ∈ Tℎ ∪ Fℎ, the corresponding 𝐿2-orthogonal projector is +denoted by 𝜋𝑚 +X,𝑌. Boldface fonts will be used when the elements of X𝑚(𝑌) are vector-valued. +The set of broken polynomials of total degree ≤ 𝑚 on the mesh is denoted by P𝑚(Tℎ), and the +corresponding 𝐿2-orthogonal projector by 𝜋𝑚 +P,ℎ. +3 + +2.2.3 +Local friction coefficient +The regime inside each mesh element 𝑇 ∈ Tℎ is identified by the following dimensionless +number, which can be interpreted as a friction coefficient: +𝐶f,𝑇 ≔ 𝜈𝑇ℎ2 +𝑇 +𝜇𝑇 +. +(4) +Elements for which 𝐶f,𝑇 < 1 are in the Stokes-dominated regime, while elements for which +𝐶f,𝑇 ≥ 1 are in the Darcy-dominated regime. The values 𝐶f,𝑇 = 0 and 𝐶f,𝑇 = +∞ correspond +to pure Stokes and pure Darcy, respectively. Notice that 𝐶f,𝑇 = +∞ is a singular limit which, +despite requiring to modify the continuous formulation (2), can be handled seamlessly by the +method developed in the next section; see Remark 8 below. +3 +A robust numerical scheme for the Brinkman problem +3.1 +Spaces +Let an integer 𝑘 ≥ 0 be fixed. We define the following HHO space: +𝑼𝑘 +ℎ ≔ +� +𝒗ℎ = �(𝒗𝑇)𝑇∈Tℎ, (𝒗𝐹)𝐹∈Fℎ +� : +𝒗𝑇 ∈ P𝑘 (𝑇; R𝑑) for all 𝑇 ∈ Tℎ and 𝒗𝐹 ∈ P𝑘 (𝐹; R𝑑) for all 𝐹 ∈ Fℎ +� +. +The meaningofthepolynomialcomponentsin𝑼𝑘 +ℎ isprovidedbytheinterpolator 𝑰𝑘 +ℎ : 𝑯1(Ω; R𝑑) → +𝑼𝑘 +ℎ such that, for all 𝒗 ∈ 𝑯1(Ω; R𝑑), +𝑰𝑘 +ℎ𝒗 ≔ �(𝝅𝑘 +P,𝑇𝒗)𝑇∈Tℎ, (𝝅𝑘 +P,𝐹𝒗)𝐹∈Tℎ, � ∈ 𝑼𝑘 +ℎ, +where it is understood that 𝐿2-orthogonal projectors are applied to restrictions or traces as +needed. The restrictions of 𝑼𝑘 +ℎ, 𝒗ℎ ∈ 𝑼𝑘 +ℎ, and 𝑰𝑘 +ℎ to a mesh element 𝑇, respectively denoted by +𝑼𝑘 +𝑇, 𝒗𝑇 ∈ 𝑼𝑘 +𝑇, and 𝑰𝑘 +𝑇, are obtained collecting the components attached to 𝑇 and its faces. +In what follows, given a logical proposition 𝑃, we denote by ⟨𝑃⟩ its truth value such that +⟨𝑃⟩ ≔ +� +0 +if 𝑃 is false, +1 +if 𝑃 is true. +(5) +We define the following 𝐿2-like product in 𝑼𝑘 +ℎ: For all (𝒘ℎ, 𝒗ℎ) ∈ 𝑼𝑘 +ℎ × 𝑼𝑘 +ℎ, +(𝒘ℎ, 𝒗ℎ)𝑼,ℎ ≔ +∑︁ +𝑇∈Tℎ +(𝒘𝑇, 𝒗𝑇)𝑼,𝑇 +with +(𝒘𝑇, 𝒗𝑇)𝑼,𝑇 ≔ 𝜆𝑇 +∫ +𝑇 +𝒘𝑇 · 𝒗𝑇 + ℎ𝑇 +∑︁ +𝐹∈F𝑇 +⟨𝐶f,𝑇 < 1 or 𝐹 ∉ F b +ℎ ⟩ +∫ +𝐹 +𝒘𝐹 · 𝒗𝐹, +(6) +where 𝜆𝑇 ≃ 1 is a factor, based on the regularity of the element 𝑇, chosen to balance out the +element and face contributions to (·, ·)𝑼,𝑇 (see Section 4). The corresponding local and global +seminorms are obtained setting, for • ∈ Tℎ ∪ {ℎ}, +∥𝒗•∥𝑼,• ≔ (𝒗•, 𝒗•) +1/2 +𝑼,•. +(7) +4 + +The following boundedness property of the interpolator in the ∥·∥𝑼,ℎ-norm follows from the +definition of this norm along with the uniform boundedness of the 𝐿2-orthogonal projectors +𝝅𝑘 +P,𝑌, 𝑌 ∈ Mℎ, and continuous trace inequalities (cf. [19, Lemma 1.31]): For all 𝑇 ∈ Tℎ and all +𝒗 ∈ 𝑯1(𝑇; R𝑑), +∥𝑰𝑘 +𝑇𝒗∥𝑼,𝑇 ≲ ∥𝒗∥𝑳2(𝑇;R𝑑) + ℎ𝑇 |𝒗|𝑯1(𝑇;R𝑑). +(8) +The velocity and pressure spaces, respectively incorporating the boundary and zero-average +conditions, are +𝑼𝑘 +ℎ,0 ≔ +� +𝒗ℎ ∈ 𝑼𝑘 +ℎ : 𝒗𝐹 = 0 for all 𝐹 ∈ F b +ℎ +� +, +𝑃𝑘 +ℎ ≔ P𝑘 (Tℎ) ∩ 𝐿2 +0(Ω), +where, as usual, 𝐿2 +0(Ω) = +� +𝑞 ∈ 𝐿2(Ω) : +∫ +Ω 𝑞 = 0 +� +. +Remark 1 (Boundary degrees of freedom). Note that the degrees of freedom on the boundary +faces of a vector in 𝑼𝑘 +ℎ may not be controlled by the seminorms ∥·∥𝑼,•. This is, however, not an +issue as the final problem will be set on 𝑼𝑘 +ℎ,0 (see also Remark 8 for the handling of boundary +values in the limiting case of the pure Darcy problem). +3.2 +Viscous term +Let 𝑇 ∈ Tℎ be fixed. For the discretisation of the viscous term, we define the discrete gradient +𝑮𝑘 +𝑇 : 𝑼𝑘 +𝑇 → P𝑘 (𝑇; R𝑑×𝑑) and the Stokes potential 𝑷𝑘+1 +S,𝑇 : 𝑼𝑘 +𝑇 → P𝑘 (𝑇; R𝑑) such that, for all +𝒗𝑇 ∈ 𝑼𝑘 +𝑇, +∫ +𝑇 +𝑮𝑘 +𝑇𝒗𝑇 : 𝝉 = − +∫ +𝑇 +𝒗𝑇 · ∇·𝝉 + +∑︁ +𝐹∈F𝑇 +𝜔𝑇𝐹 +∫ +𝐹 +𝒗𝐹 · 𝝉𝒏𝐹 +∀𝝉 ∈ P𝑘 (𝑇; R𝑑×𝑑), +(9) +and +∇𝑷𝑘+1 +S,𝑇 𝒗𝑇 = 𝝅𝑘 +G,𝑇𝑮𝑘 +𝑇𝒗𝑇, +∫ +𝑇 +𝑷𝑘+1 +S,𝑇 𝒗𝑇 = +∫ +𝑇 +𝒗𝑇, +(10) +with 𝝅𝑘 +G,𝑇 applied to tensor-valued fields also acting row-wise. Likewise, in the formulas above, +∇· and ∇ are understood to act row-wise. +The Stokes term in (2a) is discretised through the bilinear form 𝑎𝜇,ℎ : 𝑼𝑘 +ℎ × 𝑼𝑘 +ℎ → R such +that, for all (𝒘ℎ, 𝒗ℎ) ∈ 𝑼𝑘 +ℎ × 𝑼𝑘 +ℎ, +𝑎𝜇,ℎ(𝒘ℎ, 𝒗ℎ) ≔ +∑︁ +𝑇∈Tℎ +𝜇𝑇𝑎S,𝑇 (𝒘𝑇, 𝒗𝑇), +(11) +where, for all 𝑇 ∈ Tℎ, +𝑎S,𝑇 (𝒘𝑇, 𝒗𝑇) ≔ +∫ +𝑇 +𝑮𝑘 +𝑇𝒘𝑇 : 𝑮𝑘 +𝑇𝒗𝑇 + +min(1, 𝐶−1 +f,𝑇) +ℎ2 +𝑇 +(𝒘𝑇 − 𝑰𝑘 +𝑇 𝑷𝑘+1 +S,𝑇 𝒘𝑇, 𝒗𝑇 − 𝑰𝑘 +𝑇 𝑷𝑘+1 +S,𝑇 𝒗𝑇)𝑼,𝑇. (12) +We define the following induced seminorms: For all 𝒗ℎ ∈ 𝑼𝑘 +ℎ, +∥𝒗ℎ∥𝜇,ℎ ≔ 𝑎𝜇,ℎ(𝒗ℎ, 𝒗ℎ) +1/2 +and +∥𝒗𝑇 ∥S,𝑇 ≔ 𝑎S,𝑇 (𝒗𝑇, 𝒗𝑇) +1/2 for all 𝑇 ∈ Tℎ. +(13) +5 + +Lemma 2 (Norm equivalence). Let 𝑇 ∈ Tℎ and 𝒗𝑇 ∈ 𝑼𝑘 +𝑇. Then, it holds +∥∇𝒗𝑇 ∥2 +𝑳2(𝑇;R𝑑×𝑑) + 1 +ℎ𝑇 +∑︁ +𝐹∈F𝑇 +∥𝒗𝑇 − 𝒗𝐹∥2 +𝑳2(𝐹;R𝑑) ≳ ∥𝒗𝑇 ∥2 +S,𝑇. +(14) +Assuming, moreover, 𝐶f,𝑇 < 1, we also have +∥∇𝒗𝑇 ∥2 +𝑳2(𝑇;R𝑑×𝑑) + 1 +ℎ𝑇 +∑︁ +𝐹∈F𝑇 +∥𝒗𝑇 − 𝒗𝐹∥2 +𝑳2(𝐹;R𝑑) ≲ ∥𝒗𝑇 ∥2 +S,𝑇. +(15) +Proof. For the sake of brevity, we only prove (15). +The proof of (14) hinges on similar +arguments, together with the fact that min(1, 𝐶−1 +f,𝑇) ≤ 1, and is left to the reader. Taking 𝝉 = ∇𝒗𝑇 +in (9), integrating by parts the first term in the right-hand side, and using Cauchy–Schwarz and +discrete trace inequalities (see [19, Lemma 1.32]) as in the proof of [19, Eq. (2.25)], we get, +after simplifying and raising to the square, +∥∇𝒗𝑇 ∥2 +𝑳2(𝑇;R𝑑×𝑑) ≲ ∥𝑮𝑘 +𝑇𝒗𝑇 ∥2 +𝑳2(𝑇;R𝑑×𝑑) + ℎ−1 +𝑇 +∑︁ +𝐹∈F𝑇 +∥𝒗𝑇 − 𝒗𝐹∥2 +𝑳2(𝐹;R𝑑). +(16) +To estimate the second term, for any 𝐹 ∈ F𝑇, we insert ±(𝝅𝑘 +P,𝑇 𝑷𝑘+1 +S,𝑇 𝒗𝑇 − 𝝅𝑘 +P,𝐹𝑷𝑘+1 +S,𝑇 𝒗𝑇) and use +triangle inequalities to get +ℎ−1 +𝑇 ∥𝒗𝑇 − 𝒗𝐹∥2 +𝑳2(𝐹;R𝑑) ≲ ℎ−1 +𝑇 ∥𝒗𝑇 − 𝝅𝑘 +P,𝑇 𝑷𝑘+1 +S,𝑇 𝒗𝑇 ∥2 +𝑳2(𝐹;R𝑑) + ℎ−1 +𝑇 ∥𝒗𝐹 − 𝝅𝑘 +P,𝐹𝑷𝑘+1 +S,𝑇 𝒗𝑇 ∥2 +𝑳2(𝐹;R𝑑) ++ ℎ−1 +𝑇 ∥𝝅𝑘 +P,𝐹(𝑷𝑘+1 +S,𝑇 𝒗𝑇 − 𝝅𝑘 +P,𝑇 𝑷𝑘+1 +S,𝑇 𝒗𝑇)∥2 +𝑳2(𝐹;R𝑑) +≲ ℎ−2 +𝑇 ∥𝒗𝑇 − 𝝅𝑘 +P,𝑇 𝑷𝑘+1 +S,𝑇 𝒗𝑇 ∥2 +𝑳2(𝑇;R𝑑) + ℎ−1 +𝑇 ∥𝒗𝐹 − 𝝅𝑘 +P,𝐹𝑷𝑘+1 +S,𝑇 𝒗𝑇 ∥2 +𝑳2(𝐹;R𝑑) ++ ℎ−2 +𝑇 ∥𝑷𝑘+1 +S,𝑇 𝒗𝑇 − 𝝅𝑘 +P,𝑇 𝑷𝑘+1 +S,𝑇 𝒗𝑇 ∥2 +𝑳2(𝑇;R𝑑) +≲ ℎ−2 +𝑇 ∥𝒗𝑇 − 𝑰𝑘 +𝑇 𝑷𝑘+1 +S,𝑇 𝒗𝑇 ∥2 +𝑼,𝑇 + ∥∇𝑷𝑘+1 +S,𝑇 𝒗𝑇 ∥2 +𝑳2(𝑇;R𝑑×𝑑) +≲ +min(1, 𝐶−1 +f,𝑇) +ℎ2 +𝑇 +∥𝒗𝑇 − 𝑰𝑘 +𝑇 𝑷𝑘+1 +S,𝑇 𝒗𝑇 ∥2 +𝑼,𝑇 + ∥𝑮𝑘 +𝑇𝒗𝑇 ∥2 +𝑳2(𝑇;R𝑑×𝑑) = ∥𝒗𝑇 ∥2 +S,𝑇, +(17) +where we have used the 𝐿2-boundedness of 𝝅𝑘 +P,𝐹 along with discrete trace inequalities in the +second passage, the definition (7) of ∥·∥𝑼,𝑇 along with 𝐶f,𝑇 < 1 for the first two terms and the +approximation properties of 𝝅𝑘 +P,𝑇 for the last term in the third passage, and concluded noticing +that 1 = min(1, 𝐶−1 +f,𝑇) and that ∇𝑷𝑘+1 +S,𝑇 𝒗𝑇 is by definition the 𝐿2-orthogonal projection of 𝑮𝑘 +𝑇𝒗𝑇 on +G𝑘 (𝑇)𝑑 (see (10)), so that ∥∇𝑷𝑘+1 +S,𝑇 𝒗𝑇 ∥𝑳2(𝑇;R𝑑×𝑑) ≤ ∥𝑮𝑘 +𝑇𝒗𝑇 ∥𝑳2(𝑇;R𝑑×𝑑). Plugging (17) into (16) +and using the fact that card(F𝑇) ≲ 1 by mesh regularity, we get ∥∇𝒗𝑇 ∥2 +𝑳2(𝑇;R𝑑×𝑑) ≲ ∥𝒗𝑇 ∥2 +S,𝑇, +which is the sought estimate for the first term in the left-hand side of (15). The fact second term +is ≲ ∥𝒗𝑇 ∥2 +S,𝑇 is an immediate consequence of (17) along with card(F𝑇) ≲ 1. +□ +Remark 3 (HHO stabilisation). It is not difficult to check that the bilinear form 𝑼𝑘 +𝑇 × 𝑼𝑘 +𝑇 ∋ +(𝒘𝑇, 𝒗𝑇) ↦→ (𝒘𝑇 − 𝑰𝑘 +𝑇 𝑷𝑘+1 +S,𝑇 𝒘𝑇, 𝒗𝑇 − 𝑰𝑘 +𝑇 𝑷𝑘+1 +S,𝑇 𝒗𝑇)𝑼,𝑇 matches [19, Assumption 8.10]. As a matter +of fact, this bilinear form is clearly positive-semidefinite, it satisfies the requested seminorm +equivalence by (14) and (15), and is polynomially consistent since it only depends on its +arguments through the difference operators defined by [19, Eq. (8.30)]. +6 + +3.3 +Darcy term +Let again 𝑇 ∈ Tℎ. The discretisation of the Darcy and coupling terms hinges on the discrete +divergence 𝐷𝑘 +𝑇 : 𝑼𝑘 +𝑇 → P𝑘 (𝑇) such that +𝐷𝑘 +𝑇𝒗𝑇 ≔ tr(𝑮𝑘 +𝑇𝒗𝑇) +∀𝒗𝑇 ∈ 𝑼𝑘 +𝑇. +(18) +Based on this operator, we define the Darcy potential 𝑷𝑘 +D,𝑇 : 𝑼𝑘 +𝑇 → P𝑘 (𝑇; R𝑑) such that, for all +𝒗𝑇 ∈ 𝑼𝑘 +𝑇 and all (𝑞, 𝒘) ∈ P𝑘+1(𝑇) × Gc,𝑘 (𝑇), +∫ +𝑇 +𝑷𝑘 +D,𝑇𝒗𝑇 · (∇𝑞 + 𝒘) = − +∫ +𝑇 +𝐷𝑘 +𝑇𝒗𝑇 𝑞 + +∑︁ +𝐹∈F𝑇 +𝜔𝑇𝐹 +∫ +𝐹 +(𝒗𝐹 · 𝒏𝐹) 𝑞 + +∫ +𝑇 +𝒗𝑇 · 𝒘. +(19) +This Darcy potential will play a key role in the discretisation of the source term, to ensure that +the scheme is fully robust in the whole range of friction coefficients; see Remark 13. +Remark 4 (Link with DDR). Recall the Discrete De Rham 𝑯(div; Ω)-like space +𝑿𝑘 +div,ℎ ≔ +� +𝒗ℎ = �(𝒗G,𝑇, 𝒗c +G,𝑇)𝑇∈Tℎ, (𝑣𝐹)𝐹∈Fℎ +� : +𝒗G,𝑇 ∈ G𝑘−1(𝑇) and 𝒗c +G,𝑇 ∈ Gc,𝑘 (𝑇) for all 𝑇 ∈ Tℎ, +𝑣𝐹 ∈ P𝑘 (𝐹) for all 𝐹 ∈ Fℎ +� +. +Noticing that G𝑘−1(𝑇) ⊂ G𝑘 (𝑇) (cf. (3)), this space injects into 𝑼𝑘 +ℎ through the mapping +𝑿𝑘 +div,ℎ ∋ 𝒗ℎ ↦→ �(ℜ𝑘 +G,𝑇 (𝒗G,𝑇, 𝒗c +G,𝑇))𝑇∈Tℎ, (𝑣𝐹𝒏𝐹)𝐹∈Fℎ +� ∈ 𝑼𝑘 +ℎ, +where ℜ𝑘 +G,𝑇 : G𝑘 (𝑇) × Gc,𝑘 (𝑇) → P𝑘 (𝑇; R𝑑) denotes the recovery operator [17, Eq. (2.17)], +which satisfies 𝝅𝑘 +G,𝑇ℜ𝑘 +G,𝑇 (𝒗G,𝑇, 𝒗c +G,𝑇) = 𝒗G,𝑇 and 𝝅c,𝑘 +G,𝑇ℜ𝑘 +G,𝑇 (𝒗G,𝑇, 𝒗c +G,𝑇) = 𝒗c +G,𝑇 (where 𝝅c,𝑘 +G,𝑇 is +the 𝐿2-orthogonal projector on Gc,𝑘 (𝑇). It can be checked that the discrete divergence (18) +and the Darcy potential (19) only depend on the polynomial components shared by 𝑼𝑘 +𝑇 and +𝑿𝑘 +div,𝑇, and that they coincide with the corresponding DDR operators respectively defined by +[17, Eqs. (3.32) and (4.9)–(4.10)]. +Accounting for the previous remark and recalling [17, Eq. (4.12) and (4.13)], it holds +𝝅𝑘−1 +P,𝑇 𝑷𝑘 +D,𝑇𝒗𝑇 = 𝝅𝑘−1 +P,𝑇𝒗𝑇 +∀𝒗𝑇 ∈ 𝑼𝑘 +𝑇, +(20) +𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒗 = 𝒗 +∀𝒗 ∈ P𝑘 (𝑇; R𝑑). +(21) +The approximation properties of 𝑷𝑘 +D,𝑇 in the 𝐿2-norm have been studied in [17, Theorem 6]. +The following proposition extends the above results to general Hilbert seminorms. +Proposition 5 (Approximation properties of the Darcy potential). Let an integer 𝑟 ∈ {0, . . . , 𝑘} +be given. Then, for all 𝑇 ∈ Tℎ, all 𝒗 ∈ 𝑯𝑟+1(𝑇; R𝑑), and all 𝑚 ∈ {0, . . . , 𝑟 + 1}, +|𝒗 − 𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒗|𝑯𝑚(𝑇;R𝑑) ≲ ℎ𝑟+1−𝑚 +𝑇 +|𝒗|𝑯𝑟+1(𝑇;R𝑑). +(22) +7 + +Proof. By [19, Proposition 1.35], 𝑷𝑘 +D,𝑇 ◦ 𝑰𝑘 +𝑇 : 𝑯1(𝑇; R𝑑) → P𝑘 (𝑇; R𝑑) is a projector owing to +(21). By [19, Lemma 1.43], it then suffices to prove that, for all 𝒗 ∈ 𝑯1(𝑇; R𝑑), +∥𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒗∥𝑳2(𝑇;R𝑑) ≲ ∥𝒗∥𝑳2(𝑇;R𝑑) + ℎ𝑇 |𝒗|𝑯1(𝑇;R𝑑) +if 𝑚 = 0, +(23) +|𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒗|𝑯1(𝑇;R𝑑) ≲ |𝒗|𝑯1(𝑇;R𝑑) +if 𝑚 ≥ 1. +(24) +To prove (23), it suffices to recall Remark 4 and use [18, Eqs. (4.24) and (4.28)]. To prove (24), +we write +|𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒗|𝑯1(𝑇;R𝑑) = |𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇 (𝒗 − 𝝅0 +P,𝑇𝒗)|𝑯1(𝑇;R𝑑) +≲ ℎ−1 +𝑇 ∥𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇 (𝒗 − 𝝅0 +P,𝑇𝒗)∥𝑳2(𝑇;R𝑑) +[19, Eq. (1.46)] +≲ ℎ−1 +𝑇 ∥𝒗 − 𝝅0 +P,𝑇𝒗∥𝑳2(𝑇;R𝑑) + |𝒗 − 𝝅0 +P,𝑇𝒗|𝑯1(𝑇;R𝑑) +Eq. (23) +≲ |𝒗|𝑯1(𝑇;R𝑑), +where the first line follows using the polynomial consistency (21) of 𝑷𝑘 +D,𝑇 to write 0 = +|𝝅0 +P,𝑇𝒗|𝑯1(𝑇;R𝑑) = |𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇𝝅0 +P,𝑇𝒗|𝑯1(𝑇;R𝑑), while the conclusion follows from a Poincaré– +Wirtinger inequality on the zero-average function 𝒗 − 𝝅0 +P,𝑇𝒗. +□ +Let �𝑷 +𝑘 +D,𝑇 : 𝑼𝑘 +𝑇 → P𝑘 (𝑇; R𝑑) be such that +�𝑷 +𝑘 +D,𝑇𝒗𝑇 ≔ ⟨𝐶f,𝑇 < 1⟩𝒗𝑇 + ⟨𝐶f,𝑇 ≥ 1⟩𝑷𝑘 +D,𝑇𝒗𝑇 +∀𝒗𝑇 ∈ 𝑼𝑘 +𝑇. +(25) +The Darcy term in (2a) is discretised by means of the bilinear form 𝑎𝜈,ℎ : 𝑼𝑘 +ℎ × 𝑼𝑘 +ℎ → R such +that, for all (𝒘ℎ, 𝒗ℎ) ∈ 𝑼𝑘 +ℎ × 𝑼𝑘 +ℎ, +𝑎𝜈,ℎ(𝒘ℎ, 𝒗ℎ) ≔ +∑︁ +𝑇∈Tℎ +𝜈𝑇𝑎D,𝑇 (𝒘𝑇, 𝒗𝑇) +(26) +with, for all 𝑇 ∈ Tℎ, +𝑎D,𝑇 (𝒘𝑇, 𝒗𝑇) ≔ +∫ +𝑇 +�𝑷 +𝑘 +D,𝑇𝒘𝑇 ·�𝑷 +𝑘 +D,𝑇𝒗𝑇 +min(1, 𝐶f,𝑇)(𝒘𝑇 −𝑰𝑘 +𝑇 𝑷𝑘 +D,𝑇𝒘𝑇, 𝒗𝑇 −𝑰𝑘 +𝑇 𝑷𝑘 +D,𝑇𝒗𝑇, )𝑼,𝑇. (27) +We define the following induced norms: For all 𝒗ℎ ∈ 𝑼𝑘 +ℎ, +∥𝒗ℎ∥𝜈,ℎ ≔ 𝑎𝜈,ℎ(𝒗ℎ, 𝒗ℎ) +1/2 +and +∥𝒗𝑇 ∥D,𝑇 ≔ 𝑎D,𝑇 (𝒗𝑇, 𝒗𝑇) +1/2 for all 𝑇 ∈ Tℎ. +(28) +3.4 +Coupling +The coupling terms in (2a) and (2b) are discretised by the bilinear form 𝑏ℎ : 𝑼𝑘 +ℎ × P𝑘 (Tℎ) → R +such that, for all (𝒗ℎ, 𝑞ℎ) ∈ 𝑼𝑘 +ℎ × P𝑘 (Tℎ), +𝑏ℎ(𝒗ℎ, 𝑞ℎ) ≔ − +∑︁ +𝑇∈Tℎ +∫ +𝑇 +𝐷𝑘 +𝑇𝒗𝑇 𝑞𝑇, +(29) +where 𝑞𝑇 denotes the restriction of 𝑞ℎ to 𝑇. +Recalling [19, Eq. (8.36)], it holds: For all +𝒗 ∈ 𝑯1(Ω; R𝑑), +𝑏ℎ(𝑰𝑘 +ℎ𝒗, 𝑞ℎ) = − +∫ +Ω +∇·𝒗 𝑞ℎ +∀𝑞ℎ ∈ P𝑘 (Tℎ). +(30) +8 + +3.5 +Discrete problem and main results +The discrete problem reads: Find (𝒖ℎ, 𝑝ℎ) ∈ 𝑼𝑘 +ℎ,0 × 𝑃𝑘 +ℎ such that +𝑎𝜇,ℎ(𝒖ℎ, 𝒗ℎ) + 𝑎𝜈,ℎ(𝒖ℎ, 𝒗ℎ) + 𝑏ℎ(𝒗ℎ, 𝑝ℎ) = +∑︁ +𝑇∈Tℎ +∫ +𝑇 +𝒇 · �𝑷 +𝑘 +D,𝑇𝒗𝑇 +∀𝒗ℎ ∈ 𝑼𝑘 +ℎ,0, +−𝑏ℎ(𝒖ℎ, 𝑞ℎ) = +∫ +Ω +𝑔𝑞ℎ +∀𝑞ℎ ∈ 𝑃𝑘 +ℎ. +(31) +The equivalent variational formulation is: Find (𝒖ℎ, 𝑝ℎ) ∈ 𝑼𝑘 +ℎ,0 × 𝑃𝑘 +ℎ such that +Aℎ((𝒖ℎ, 𝑝ℎ), (𝒗ℎ, 𝑞ℎ)) = +∑︁ +𝑇∈Tℎ +∫ +𝑇 +𝒇 · �𝑷 +𝑘 +D,𝑇𝒗𝑇 + +∫ +Ω +𝑔𝑞ℎ, +(32) +with Aℎ : �𝑼𝑘 +ℎ × 𝑃𝑘 +ℎ +�2 → R such that, for all (𝒘ℎ, 𝑟ℎ) and all (𝒗ℎ, 𝑞ℎ) in 𝑼𝑘 +ℎ × 𝑃𝑘 +ℎ, +Aℎ((𝒘ℎ, 𝑟ℎ), (𝒗ℎ, 𝑞ℎ)) ≔ 𝑎𝜇,ℎ(𝒘ℎ, 𝒗ℎ) + 𝑎𝜈,ℎ(𝒘ℎ, 𝒗ℎ) + 𝑏ℎ(𝒗ℎ, 𝑟ℎ) − 𝑏ℎ(𝒘ℎ, 𝑞ℎ). +(33) +Recalling (13) and (28), we equip the space 𝑼𝑘 +ℎ,0 with the following natural energy norm: +For all 𝒗ℎ ∈ 𝑼𝑘 +ℎ,0, +∥𝒗ℎ∥𝜇,𝜈,ℎ ≔ +� +∥𝒗ℎ∥2 +𝜇,ℎ + ∥𝒗ℎ∥2 +𝜈,ℎ +�1/2 +(34) +and, given a linear form ℓℎ : 𝑼𝑘 +ℎ,0 → R, we denote its dual norm by +∥ℓℎ∥𝜇,𝜈,ℎ,∗ ≔ +sup +𝒗ℎ∈𝑼𝑘 +ℎ,0\{0} +ℓℎ(𝒗ℎ) +∥𝒗ℎ∥𝜇,𝜈,ℎ +. +The bilinear form 𝑎𝜇,ℎ + 𝑎𝜈,ℎ is ∥·∥𝜇,𝜈,ℎ-coercive with unit coercivity constant. +The well- +posedness of (31) then classically follows from the theory of mixed methods (see, e.g., [19, +Lemma A.11]) thanks to the inf-sup condition on 𝑏ℎ stated in the following lemma. +Lemma 6 (Inf-sup condition on 𝑏ℎ). Letting 𝛽 ≔ (𝜇 + 𝜈)−1/2, it holds, for all 𝑞ℎ ∈ 𝑃𝑘 +ℎ, +𝛽∥𝑞ℎ∥𝐿2(Ω) ≲ +sup +𝒗ℎ∈𝑼𝑘 +ℎ,0\{0} +𝑏ℎ(𝒗ℎ, 𝑞ℎ) +∥𝒗ℎ∥𝜇,𝜈,ℎ +. +Proof. See Section 5.1. +□ +Thanks to the presence of cut-off factors, the following error estimate is robust across the +entire range of (local) regimes. +Theorem 7 (Error estimate). Denote by (𝒖, 𝑝) ∈ 𝑯1 +0(Ω; R𝑑) × 𝐿2 +0(Ω) the unique solution to +the standard weak formulation of (2) and by (𝒖ℎ, 𝑝ℎ) ∈ 𝑼𝑘 +ℎ,0 × 𝑃𝑘 +ℎ the unique solution of the +numerical scheme (31) (or, equivalently, (32)). Then, recalling the notation (5) for the truth value +9 + +of a logical proposition and assuming, for some 𝑟 ∈ {0, . . . , 𝑘}, 𝒖 ∈ 𝑯𝑟+2(Tℎ; R𝑑), 𝑝 ∈ 𝐻1(Ω), +and, for all 𝑇 ∈ Tℎ, 𝑝 ∈ 𝐻𝑟+1+⟨𝐶f,𝑇 ≥1⟩(𝑇), it holds, +∥𝒖ℎ − 𝑰𝑘 +ℎ𝒖∥2 +𝜇,𝜈,ℎ + ∥𝑝ℎ − 𝜋𝑘 +P,ℎ𝑝∥2 +𝐿2(Ω) +≲ 1 +𝛾2 +� ∑︁ +𝑇∈Tℎ +𝜇𝑇 min(1, 𝐶−1 +f,𝑇)ℎ2(𝑟+1) +𝑇 +|𝒖|2 +𝑯𝑟+2(𝑇;R𝑑) + +∑︁ +𝑇∈Tℎ +𝜈𝑇 min(1, 𝐶f,𝑇)ℎ2(𝑟+1) +𝑇 +|𝒖|2 +𝑯𝑟+1(𝑇;R𝑑) ++ +∑︁ +𝑇∈Tℎ +� +𝜇−1 +𝑇 ⟨𝐶f,𝑇 < 1⟩ℎ2(𝑟+1) +𝑇 +|𝑝|2 +𝐻𝑟+1(𝑇) + 𝜈−1 +𝑇 ⟨𝐶f,𝑇 ≥ 1⟩ℎ2(𝑟+1) +𝑇 +|𝑝|2 +𝐻𝑟+2(𝑇) +� � +, +(35) +where 𝛾−2 ≔ 4𝛽−4 + 8𝛽−2 + 1 with 𝛽 as in Lemma 6, while, for all 𝑇 ∈ Tℎ, 𝜈−1 +𝑇 ⟨𝐶f,𝑇 ≥ 1⟩ ≔ 0 if +𝜈𝑇 = 0. +Proof. See Section 5.2. +□ +Remark 8 (Robustness of the error estimate and application to the Darcy problem). In the spirit +of [10, Remark 13], the presence of the cutoff factors min(1, 𝐶−1 +f,𝑇), min(1, 𝐶f,𝑇), 𝜇−1 +𝑇 ⟨𝐶f,𝑇 < 1⟩, +and 𝜈−1 +𝑇 ⟨𝐶f,𝑇 ≥ 1⟩ makes the above estimate robust across the entire range 𝐶f,𝑇 ∈ [0, +∞). +The case 𝐶f,𝑇 = +∞ corresponds to the pure Darcy problem, which is the singular limit +obtained assuming minΩ 𝜈 > 0 and 𝐶f,𝑇 = +∞ for all 𝑇 ∈ Tℎ. In this case, a more in-depth +discussion is in order. Denoting by 𝛾𝒏 the normal trace operator on 𝜕Ω, the space for the velocity +becomes 𝑯0(div; Ω) ≔ {𝒗 ∈ 𝑯(div; Ω) : 𝛾𝒏(𝒗) = 0 on 𝜕Ω}, and the weak formulation of (2) +yields the Darcy problem in mixed form. +The error estimate (35) remains valid under the +regularity assumption 𝒖 ∈ 𝑯𝑟+1(Tℎ; R𝑑), and provided the following conventions are adopted: +𝜇−1 +𝑇 ⟨𝐶f,𝑇 < 1⟩ ≔ 0 and, for any 𝒗 ∈ 𝑯0(div; Ω) ∩ 𝑯1(Tℎ; R𝑑), all the components of the +boundary values of 𝑰𝑘 +ℎ𝒗 are forced to zero, i.e., (𝑰𝑘 +ℎ𝒗)𝐹 ≔ 0 for all 𝐹 ∈ F b +ℎ . Notice that the +tangential components of the velocity on boundary faces do not appear in the formulation of the +method when 𝜇 = 0. To check this fact: +• Concerning the Darcy contribution 𝑎D,𝑇 (cf. (27)), recall Remark 4 for the consistent term +while, for the stabilisation term, notice that, by (6), boundary faces are not present in +(·, ·)𝑼,𝑇 since 𝐶f,𝑇 ≥ 1 for all 𝑇 ∈ Tℎ ; +• Concerning the coupling term 𝑏ℎ (cf. (29)), notice that the following equivalent formula- +tion results applying the definition (9) of 𝑮𝑘 +𝑇 with 𝝉 = 𝑞𝑇 𝑰𝑑 ≔ (𝑞ℎ)|𝑇 𝑰𝑑 for all 𝑇 ∈ Tℎ: +𝑏ℎ(𝒗ℎ, 𝑞ℎ) = +∑︁ +𝑇∈Tℎ +�∫ +𝑇 +𝒗𝑇 · ∇𝑞𝑇 − +∑︁ +𝐹∈F𝑇 +𝜔𝑇𝐹 +∫ +𝐹 +(𝒗𝐹 · 𝒏𝐹) 𝑞𝑇 +� +, +clearly showing that 𝑏ℎ is independent of the tangential component of 𝒗𝐹 for all 𝐹 ∈ Fℎ. +The method obtained for the pure Darcy problem has more unknowns than, say, the mixed +method of [22] or a similar one that could be obtained starting from the space 𝑿𝑘 +div,ℎ of [17]. In +particular, the tangential components of interface unknowns are not present in the consistency +term of 𝑎D,𝑇 (see again Remark 4), but are controlled by the stabilisation term. Despite this +difference in the discrete space for the flux, the estimate for the error on 𝒖 resulting from (35) +in the pure Darcy case is analogous to the one given in [22, Theorem 6] (where the highest +regularity case corresponding to 𝑟 = 𝑘 is considered). +10 + +4 +Numerical tests +In this section we numerically assess the convergence properties of the scheme (31) for different +values of the friction coefficient (including the limit cases) and on both standard and genuinely +polyhedral meshes. +The code used for the numerical tests is part of the open source C++ HArDCore3D library; +see https://github.com/jdroniou/HArDCore. In order to reduce the size of the global +linear systems, static condensation was applied the scheme (31) in accordance with the principles +outlined in [19, Appendix B]; see [24, Section 6] for a discussion specific to the Stokes equations +and [9] for a study of the effect of static condensation on 𝑝-multilevel preconditioners for the +Stokes problem. We have chosen to locally eliminate all element degrees of freedom except for +the average value of the pressure inside each element. The linear systems were solved using the +Intel MKL PARDISO library (see https://software.intel.com/en-us/mkl). +The parameter 𝜆𝑇 in (6) was chosen as +ℎ3 +𝑇 +|𝑇| card(F𝑇), to give a larger weight to the element +contribution in (7) when 𝑇 is elongated or has many faces: this compensates the relatively larger +contribution, in these circumstances, of the boundary terms in this local norm. We have also +applied scalings to the stabilisation terms in (12) and (27): 3 for the Stokes stabilisation, 0.3 for +the Darcy stabilisation. Introducing scalings in the stabilisation terms is not strictly necessary to +observe the convergence of the scheme at the expected rates, but we noticed that they improve the +magnitudes of the relative errors. Understanding the optimal scaling of stabilisations involved +in polytopal methods is an ongoing subject of investigation; here, these numbers were found by +quick trial and error on unexpensive tests (low degree 𝑘, coarse meshes), before being used in +all the tests below. +4.1 +Convergence in various regimes +Following [10], we consider a constant viscosity 𝜇 and inverse permeability 𝜈, and we evaluate +the relative velocity–pressure error +𝐸𝒖,𝑝 = +� +∥𝒖ℎ − 𝑰𝑘 +ℎ𝒖∥2 +𝜇,𝜈,ℎ + ∥𝑝ℎ − 𝜋𝑘 +P,ℎ𝑝∥2 +𝐿2(Ω) +�1/2 +� +∥𝑰𝑘 +ℎ𝒖∥2 +𝜇,𝜈,ℎ + ∥𝜋𝑘 +P,ℎ𝑝∥2 +𝐿2(Ω) +�1/2 +, +when the nature of the exact solution (𝒖, 𝑝) is determined by the global friction coefficient +𝐶f,Ω = 𝜈/𝜇, with the convention 𝐶f,Ω = +∞ if 𝜇 = 0. Specifically, we consider the domain +Ω = (0, 1)3 and, setting 𝜒S(𝐶f,Ω) ≔ exp(−𝐶f,Ω), the pressure and velocity are chosen as +𝑝(𝑥, 𝑦, 𝑧) = sin(2𝜋𝑥) sin(2𝜋𝑦) sin(2𝜋𝑧) +∀(𝑥, 𝑦, 𝑧) ∈ Ω, +𝒖 = 𝜒S(𝐶f,Ω)𝒖S + (1 − 𝜒S(𝐶f,Ω))𝒖D, +where 𝒖S and 𝒖D are the velocity obtained in the Stokes (𝐶f,Ω = 0) and Darcy (𝐶f,Ω = +∞) +limits, and are given by +𝒖S(𝑥, 𝑦, 𝑧) = 1 +2 +������ +sin(2𝜋𝑥) cos(2𝜋𝑦) cos(2𝜋𝑧) +cos(2𝜋𝑥) sin(2𝜋𝑦) cos(2𝜋𝑧) +−2 cos(2𝜋𝑥) cos(2𝜋𝑦) sin(2𝜋𝑧) +������ +∀(𝑥, 𝑦, 𝑧) ∈ Ω, +𝒖D = +� −𝜈−1∇𝑝 +if 𝜈 > 0, +0 +otherwise. +11 + +We notice that ∇·𝒖S = 0 and that 𝜈𝒖D + ∇𝑝 = 0; these are expected relations, respectively, +for a solution of an incompressible Stokes equation, and for a solution of a Darcy equation in +mixed form (when gravity is neglected). The meshes used for the test correspond to the families +of Voronoi meshes “Voro-small-0”, of tetrahedral meshes “Tetgen-Cube-0” and of random +hexahedral meshes “Random-Hexahedra” available on the HArDCore3D repository. The errors +as a function of ℎ are presented in Figures 1, 2 and 3, showing that the predicted convergence is +observed in practice for all the considered mesh families and polynomial degrees, and that both +orders of convergence and magnitudes of errors are robust in all regimes. +𝑘 = 0; +𝑘 = 1; +𝑘 = 2 +𝑘 = 3 +10−1 +100 +10−3 +10−2 +10−1 +100 +1 +1 +1 +2 +1 +3 +1 +4 +(a) 𝜇 = 𝜈 = 1 +10−1 +100 +10−3 +10−2 +10−1 +100 +1 +1 +1 +2 +1 +3 +1 +4 +(b) 𝜇 = 1, 𝜈 = 0 +10−1 +100 +10−3 +10−2 +10−1 +100 +1 +1 +1 +2 +1 +3 +1 +4 +(c) 𝜇 = 0, 𝜈 = 1 +Figure 1: Voronoi meshes: errors 𝐸𝑢,𝑝 with respect to ℎ +4.2 +Lid-driven cavity in porous medium +The tests in this section are inspired by situations described in [1, 6]. In these references, a +V-crack is realised at the top of a homogeneous porous medium, and plays the role of a lid-driven +cavity (with a Stokes-dominated model in this cavity, while the rest of the medium is modelled +using pure Darcy flow), and low-order mixed finite elements on triangles/tetrahedra are used to +simulate the flow. +12 + +𝑘 = 0; +𝑘 = 1; +𝑘 = 2 +𝑘 = 3 +10−0.6 +10−0.4 +10−0.2 +100 +10−3 +10−2 +10−1 +100 +1 +1 +1 +2 +1 +3 +1 +4 +(a) 𝜇 = 𝜈 = 1 +10−0.6 +10−0.4 +10−0.2 +100 +10−3 +10−2 +10−1 +100 +101 +1 +1 +1 +2 +1 +3 +1 +4 +(b) 𝜇 = 1, 𝜈 = 0 +10−0.6 +10−0.4 +10−0.2 +100 +10−2 +10−1 +100 +1 +1 +1 +2 +1 +3 +1 +4 +(c) 𝜇 = 0, 𝜈 = 1 +Figure 2: Tetrahedral meshes: errors 𝐸𝑢,𝑝 with respect to ℎ +We consider here a cavity, where a pure Stokes flow occurs, sitting in a porous medium, +with pure Darcy flow; the porous medium is heterogeneous, with permeability equal to 10−7 +in the surrounding “box” and 10−2 in a “wedge” at the outset of the cavity; see Figure 4, left. +The domain is Ω = (−1, 2) × (−1, 2) × (−2, 0), with the cavity being (0, 1)3 and the wedge +{(𝑥, 𝑦, 𝑧) ∈ R3 : 1 < 𝑥 < 2 , 0 < 𝑦 < 1 , −0.75(𝑥 − 1) + 0.25 < 𝑧 < 0}. The domain has been +meshed using gmsh (https://gmsh.info/), with cubic elements in the cavity, and mostly tetrahedral +elements in the porous medium (together with a few pyramidal elements at the junctions cavity– +porous medium); see Figure 4, right, for an example of mesh, and Table 1 for the characteristic +of all meshes. The files describing the geometry are available in the HArDCore repository. +The forcing term 𝒇 = (0, 0, −0.98) represents the gravity, while we fix 𝑔 = 0. The boundary +conditions on the velocity are 𝒖(𝑥, 𝑦, 𝑧) = (𝑥(1 − 𝑥), 0, 0) on top of the cavity, and 𝒖 = 0 +elsewhere. Figure 5 presents the streamlines obtained on the third mesh in the family with +𝑘 = 2. These streamlines show the usual form of circulation inside the cavity for a pure Stokes +lid-driven cavity, which drives some (slower) motion inside the wedge section of the porous +medium; given the very low permeability of the rest of the medium, little material is transferred +13 + +𝑘 = 0; +𝑘 = 1; +𝑘 = 2 +𝑘 = 3 +10−1 +10−0.8 +10−0.6 +10−0.4 +10−0.2 +10−4 +10−3 +10−2 +10−1 +100 +1 +1 +1 +2 +1 +3 +1 +4 +(a) 𝜇 = 𝜈 = 1 +10−1 +10−0.8 +10−0.6 +10−0.4 +10−0.2 +10−4 +10−3 +10−2 +10−1 +100 +1 +1 +1 +2 +1 +3 +1 +4 +(b) 𝜇 = 1, 𝜈 = 0 +10−1 +10−0.8 +10−0.6 +10−0.4 +10−0.2 +10−3 +10−2 +10−1 +100 +1 +1 +1 +2 +1 +3 +1 +4 +(c) 𝜇 = 0, 𝜈 = 1 +Figure 3: Random hexahedral meshes: errors 𝐸𝑢,𝑝 with respect to ℎ +into this medium, in which the velocity remains almost zero; in the region 𝑧 < −1 below the +cavity, for example, the maximum of the vertex values (obtained by averaging the potential +reconstructions in each element surrounding the vertices) of the velocity is below 6 × 10−5. +To qualitatively assess the impact of increasing the degree of approximation 𝑘 of the method, +we evaluate for various meshes and degrees the flux across the interface Γ = {0} × (0, 1) × +(−0.75, 1) between the cavity and the wedge. All the meshes Mℎ we consider are compatible +with this interface, that is, setting Γℎ = {𝐹 ∈ Fℎ : 𝐹 ⊂ Γ} we have Γ = ∪𝐹∈Γℎ𝐹. We then +consider the numerical convergence of the numerical flux defined by +∑︁ +𝐹∈Γℎ +∫ +𝐹 +𝒖𝐹 · 𝒏Γ, +where 𝒏Γ = (1, 0, 0) is the unit normal to Γ pointing inside the wedge. The values of this +flux for different degrees of approximations 𝑘 are provided in Figure 6 (left: w.r.t. the mesh +size; right: w.r.t. the total wall time, including assembly and solution time – notice that the +HArDCore library uses multi-threading processes). These results show that the lowest order of +14 + +Figure 4: Left: geometry of the cavity (green) inside the porous medium, comprising a wedge +(green) and the surrounding box (shadow). Right: example of mesh used in the simulations. +Mesh index +1 +2 +3 +4 +5 +Mesh size +0.95 +0.61 +0.54 +0.22 +0.17 +Num. of elements +1,326 +5,935 +7,963 +99,748 +201,653 +Table 1: Characteristics of the mesh family for the tests in Section 4.2. +approximation struggles to provide what seems to be a correct value of the flux, and that the +mesh must be extremely fine to get close to this value; on the contrary, for 𝑘 ≥ 1, all results, even +on coarse meshes and with a low computational cost, seem to be very close to a given value, +indicating that convergence has already occurred in these cases. These results corroborate a +conclusion already highlighted in [3]: even on a problem where the solution is not expected to +be very regular, slightly increasing the order of approximation of the scheme (here, going from +𝑘 = 0 to 𝑘 = 1) can lead to a vastly improved accuracy of the numerical outputs at a very low +computational cost. +5 +Analysis +5.1 +Stability +Proposition 9 (∥·∥𝜇,𝜈,ℎ-boundedness of the interpolator). With 𝛽 as in Lemma 6, it holds, for all +𝒗 ∈ 𝑯1(Ω; R𝑑), +𝛽∥𝑰𝑘 +ℎ𝒗∥𝜇,𝜈,ℎ ≲ ∥𝒗∥𝑯1(Ω;R𝑑). +(36) +Proof. It holds, by definition, ∥𝑰𝑘 +ℎ𝒗∥2 +𝜇,𝜈,ℎ = � +𝑇∈Tℎ [𝜇𝑇𝔗1(𝑇) + 𝜈𝑇𝔗2(𝑇) + 𝜈𝑇𝔗3(𝑇)] with +𝔗1(𝑇) ≔ ∥𝑮𝑘 +𝑇 𝑰𝑘 +𝑇𝒗∥2 +𝑳2(𝑇;R𝑑×𝑑) + +min(1, 𝐶−1 +f,𝑇) +ℎ2 +𝑇 +∥𝑰𝑘 +𝑇 (𝒗 − 𝑷𝑘+1 +S,𝑇 𝑰𝑘 +𝑇𝒗)∥2 +𝑼,𝑇, +𝔗2(𝑇) ≔ ∥�𝑷 +𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒗∥2 +𝑳2(𝑇;R𝑑), +𝔗3(𝑇) ≔ min(1, 𝐶f,𝑇)∥𝑰𝑘 +𝑇 (𝒗 − 𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒗)∥2 +𝑼,𝑇. +For the first term, combining (14) and [19, Eqs. (8.25)], we obtain 𝔗1(𝑇) ≲ |𝒗|2 +𝑯1(𝑇;R𝑑). For +15 + +X Axis +2 +1 +0 +0 +-1 +0 +Z Axis -1 + +-2 +Y Axis +0 +Y Axis +-1 +2 +01 +1 +X Axis +0 +-1Figure 5: Streamlines for the test case of Section 4.2 (cavity and wedge displayed in shadow). +16 + +2.5e-01 +0.2 +velocity Magnitude +0.15 +0.1 +0.05 +0.0e+00𝑘 = 0; +𝑘 = 1; +𝑘 = 2 +𝑘 = 3 +0.2 +0.4 +0.6 +0.8 +1 +0 +1 +2 +3 +4 +·10−8 +(a) w.r.t. mesh size +100 +101 +102 +103 +10−8.5 +10−8 +10−7.5 +(b) w.r.t. wall time (seconds) +Figure 6: Convergence of flux values from the cavity to the wedge. +the second term, if 𝐶f,𝑇 < 1, we can write 𝔗2(𝑇) = ∥𝝅𝑘 +P,𝑇𝒗∥2 +𝑳2(𝑇;R𝑑) ≤ ∥𝒗∥2 +𝑳2(𝑇;R𝑑) using the +boundedness of 𝝅𝑘 +P,𝑇, while, if 𝐶f,𝑇 ≥ 1, (23) gives 𝔗2(𝑇) ≲ ∥𝒗∥2 +𝑳2(𝑇;R𝑑) + ℎ2 +𝑇 |𝒗|2 +𝑯1(𝑇;R𝑑) ≤ +∥𝒗∥2 +𝑯1(𝑇;R𝑑), where the conclusion follows observing that ℎ𝑇 ≤ 1 since Ω has unit diameter by +assumption. Finally, for the third term, the boundedness (8) of the interpolator in the ∥·∥𝑼,𝑇- +norm followed by the approximation properties (22) of 𝑷𝑘 +D,𝑇 ◦ 𝑰𝑘 +𝑇 with (𝑟, 𝑚) = (0, 0) and +(𝑟, 𝑚) = (0, 1) yield +𝔗3(𝑇) ≲ ∥𝒗 − 𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒗∥2 +𝑳2(𝑇;R𝑑) + ℎ2 +𝑇 |𝒗 − 𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒗|2 +𝑯1(𝑇;R𝑑) ≲ ℎ2 +𝑇 |𝒗|2 +𝑯1(𝑇;R𝑑). +Gathering the above estimates and recalling the bounds (1) on 𝜇 and 𝜈, the result follows. +□ +Proof of Lemma 6. Classical consequence of the continuous inf-sup condition for the divergence +∇· : 𝑯1 +0(Ω; R𝑑) → 𝐿2 +0(Ω) (see, e.g., [8, 25, 27, 33]) along with the Fortin properties for the +interpolator corresponding to (30) and (36); see, e.g., [7, Section 5.4.3] for further details. +□ +5.2 +Convergence +The purpose of this section is to prove Theorem 7. The proof rests on consistency results for the +Stokes, Darcy, and coupling bilinear forms as well as the forcing term linear form which make +the object of the following subsections. +5.2.1 +Consistency of the Stokes bilinear form +Lemma 10 (Consistency of the Stokes bilinear form). Given 𝒘 ∈ 𝑯1 +0(Ω; R𝑑) such that +∇·(𝜇∇𝒘) ∈ 𝑳2(Ω; R𝑑), let the Stokes consistency error linear form E𝑘 +S,ℎ(𝒘; ·) : 𝑼𝑘 +ℎ,0 → R +be such that, for all 𝒗ℎ ∈ 𝑼𝑘 +ℎ,0, +E𝑘 +S,ℎ(𝒘; 𝒗ℎ) ≔ − +∑︁ +𝑇∈Tℎ +∫ +𝑇 +∇·(𝜇𝑇∇𝒘) · 𝒗𝑇 − 𝑎𝜇,ℎ(𝑰𝑘 +ℎ𝒘, 𝒗ℎ). +(37) +17 + +Then, further assuming 𝒘 ∈ 𝑯𝑟+2(Tℎ; R𝑑) for some 𝑟 ∈ {0, . . . , 𝑘}, it holds +∥E𝑘 +S,ℎ(𝒘; ·)∥𝜇,𝜈,ℎ,∗ ≲ +� ∑︁ +𝑇∈Tℎ +𝜇𝑇 min(1, 𝐶−1 +f,𝑇)ℎ2(𝑟+1) +𝑇 +|𝒘|2 +𝑯𝑟+2(𝑇;R𝑑) +�1/2 +. +(38) +Proof. Let 𝒗ℎ ∈ 𝑼𝑘 +ℎ,0 \ {0}. Proceeding as in [19, Point (ii) in Lemma 2.18] using an integration +by parts for the first term in the definition of E𝑘 +S,ℎ along with the definitions (11) of 𝑎𝜇,ℎ and (9) +of 𝑮𝑘 +𝑇 for the second term, we get the following reformulation of the error: +E𝑘 +S,ℎ(𝒘; 𝒗ℎ) = +∑︁ +𝑇∈Tℎ +∑︁ +𝐹∈F𝑇 +𝜔𝑇𝐹 +∫ +𝐹 +𝜇𝑇 (∇𝒘 − 𝑮𝑘 +𝑇 𝑰𝑘 +𝑇𝒘)𝒏𝐹 · (𝒗𝐹 − 𝒗𝑇) +− +∑︁ +𝑇∈Tℎ +𝜇𝑇 min(1, 𝐶−1 +f,𝑇) +ℎ2 +𝑇 +(𝑰𝑘 +𝑇 (𝒘 − 𝑷𝑘+1 +S,𝑇 𝑰𝑘 +𝑇𝒘), 𝒗𝑇 − 𝑰𝑘 +𝑇 𝑷𝑘+1 +S,𝑇 𝒗𝑇)𝑼,𝑇. +Using Cauchy–Schwarz and Hölder inequalities along with ∥𝒏𝐹∥𝑳∞(𝐹;R𝑑) ≤ 1 for all 𝐹 ∈ Fℎ, +we can write +E𝑘 +S,ℎ(𝒘; 𝒗ℎ) ≲ +∑︁ +𝑇∈Tℎ +[𝔗1(𝑇) + 𝔗2(𝑇)] +(39) +with +𝔗1(𝑇) ≔ 𝜇 +1/2 +𝑇 ℎ +1/2 +𝑇 ∥∇𝒘 − 𝑮𝑘 +𝑇 𝑰𝑘 +𝑇𝒘∥𝑳2(𝜕𝑇;R𝑑×𝑑) +� +𝜇𝑇 +ℎ𝑇 +∑︁ +𝐹∈F𝑇 +∥𝒗𝐹 − 𝒗𝑇 ∥2 +𝑳2(𝐹;R𝑑) +�1/2 +, +𝔗2(𝑇) ≔ +𝜇𝑇 min(1, 𝐶−1 +f,𝑇) +ℎ2 +𝑇 +∥𝑰𝑘 +𝑇 (𝒘 − 𝑷𝑘+1 +S,𝑇 𝑰𝑘 +𝑇𝒘)∥𝑼,𝑇 ∥𝒗𝑇 − 𝑰𝑘 +𝑇 𝑷𝑘+1 +S,𝑇 𝒗𝑇 ∥𝑼,𝑇. +Let us estimate 𝔗1(𝑇). Recalling that 𝑮𝑘 +𝑇 ◦𝑰𝑘 +𝑇 = 𝝅𝑘 +P,𝑇 and using the approximation properties +of this projector (cf. [15] and [19, Chapter 1] concerning the extension to non-star-shaped +elements), it is readily inferred for the first factor +𝜇 +1/2 +𝑇 ℎ +1/2 +𝑇 ∥∇𝒘 − 𝑮𝑘 +𝑇 𝑰𝑘 +𝑇𝒘∥𝑳2(𝜕𝑇;R𝑑×𝑑) ≲ 𝜇 +1/2 +𝑇 ℎ𝑟+1 +𝑇 +|𝒘|𝑯𝑟+2(𝑇;R𝑑). +The estimate of the second factor depends on the regime. If 𝐶f,𝑇 < 1, using (15) we write +𝜇𝑇 +ℎ𝑇 +∑︁ +𝐹∈F𝑇 +∥𝒗𝐹 − 𝒗𝑇 ∥2 +𝑳2(𝐹;R𝑑) ≲ 𝜇𝑇 ∥𝒗𝑇 ∥2 +S,𝑇 = 𝜇𝑇 min(1, 𝐶−1 +f,𝑇)∥𝒗𝑇 ∥2 +S,𝑇, +(40) +where the conclusion follows observing that 1 = min(1, 𝐶−1 +f,𝑇). If, on the other hand, 𝐶f,𝑇 ≥ 1 +(which implies, in particular, 𝜈𝑇 > 0), we insert ±𝑷𝑘 +D,𝑇𝒗𝑇 into the norm and use triangle and +discrete trace inequalities to write +𝜇𝑇 +ℎ𝑇 +∑︁ +𝐹∈F𝑇 +∥𝒗𝐹 − 𝒗𝑇 ∥2 +𝑳2(𝐹;R𝑑) +≲ 𝜈𝑇𝐶−1 +f,𝑇 +� +ℎ𝑇 +∑︁ +𝐹∈F𝑇 +∥𝒗𝐹 − 𝑷𝑘 +D,𝑇𝒗𝑇 ∥2 +𝑳2(𝐹;R𝑑) + ∥𝒗𝑇 − 𝑷𝑘 +D,𝑇𝒗𝑇 ∥2 +𝑳2(𝑇;R𝑑) +� +18 + +≲ 𝜈𝑇𝐶−1 +f,𝑇 +� +ℎ𝑇 +∑︁ +𝐹∈F𝑇 ∩F b +ℎ +∥𝒗𝐹 − 𝑷𝑘 +D,𝑇𝒗𝑇 ∥2 +𝑳2(𝐹;R𝑑) + ∥𝒗𝑇 − 𝑰𝑘 +𝑇 𝑷𝑘 +D,𝑇𝒗𝑇 ∥2 +𝑼,𝑇 +� +where we have additionally used the definition (4) of 𝐶f,𝑇 in the first inequality, and continued +invoking the definition of ∥·∥𝑼,𝑇 (see (6)–(7)) to bound the element term and the non-boundary +face terms in the second line by ∥𝒗𝑇 − 𝑰𝑘 +𝑇 𝑷𝑘 +D,𝑇𝒗𝑇 ∥2 +𝑼,𝑇. For all 𝐹 ∈ F𝑇 ∩ F b +ℎ , we have 𝒗𝐹 = 0 +by definition of 𝑼𝑘 +ℎ,0 and, using discrete trace inequalities and the mesh regularity to write +card(F𝑇) ≲ 1, we infer that +𝜇𝑇 +ℎ𝑇 +∑︁ +𝐹∈F𝑇 +∥𝒗𝐹 − 𝒗𝑇 ∥2 +𝑳2(𝐹;R𝑑) ≲ 𝜈𝑇𝐶−1 +f,𝑇 +� +∥𝑷𝑘 +D,𝑇𝒗𝑇 ∥2 +𝑳2(𝑇;R𝑑) + ∥𝒗𝑇 − 𝑰𝑘 +𝑇 𝑷𝑘 +D,𝑇𝒗𝑇 ∥2 +𝑼,𝑇 +� +≲ 𝜈𝑇𝐶−1 +f,𝑇 ∥𝒗𝑇 ∥2 +D,𝑇, +where the last passage follows recalling the definitions (28) of the ∥·∥D,𝑇-norm, (25) of �𝑷 +𝑘 +D,𝑇 +(which is equal to 𝑷𝑘 +D,𝑇 since 𝐶f,𝑇 ≥ 1), and observing that 1 = min(1, 𝐶f,𝑇). Hence, further +observing that 𝐶−1 +f,𝑇 = min(1, 𝐶−1 +f,𝑇), we can go on writing +𝜇𝑇 +ℎ𝑇 +∑︁ +𝐹∈F𝑇 +∥𝒗𝐹 − 𝒗𝑇 ∥2 +𝑳2(𝐹;R𝑑) ≲ 𝜈𝑇 min(1, 𝐶−1 +f,𝑇)∥𝒗𝑇 ∥2 +D,𝑇. +(41) +Gathering (40) and (41), we arrive at +𝔗1(𝑇) ≲ 𝜇 +1/2 +𝑇 min(1, 𝐶−1 +f,𝑇) +1/2ℎ𝑟+1 +𝑇 +|𝒘|𝑯𝑟+2(𝑇;R𝑑) +� +𝜇𝑇 ∥𝒗𝑇 ∥2 +S,𝑇 + 𝜈𝑇 ∥𝒗𝑇 ∥2 +D,𝑇 +�1/2 +. +(42) +Moving to 𝔗2(𝑇), using the ∥·∥𝑼,𝑇-boundedness (8) of 𝑰𝑘 +𝑇 followed by the approximation +properties of 𝑷𝑘+1 +S,𝑇 ◦ 𝑰𝑘 +𝑇 (consequence, for each of its components, of [19, Eq. (2.14) and +Theorem 1.48]), we have +∥𝑰𝑘 +𝑇 (𝒘−𝑷𝑘+1 +S,𝑇 𝑰𝑘 +𝑇𝒘)∥𝑼,𝑇 ≲ ∥𝒘−𝑷𝑘+1 +S,𝑇 𝑰𝑘 +𝑇𝒘∥𝑳2(𝑇;R𝑑)+ℎ𝑇 |𝒘−𝑷𝑘+1 +S,𝑇 𝑰𝑘 +𝑇𝒘|𝑯1(𝑇;R𝑑) ≲ ℎ𝑟+2 +𝑇 +|𝒘|𝑯𝑟+2(𝑇;R𝑑). +Plugging this estimate into the definition of 𝔗2(𝑇) and recalling the definition (13) of ∥·∥S,𝑇, we +get +𝔗2(𝑇) ≲ 𝜇 +1/2 +𝑇 min(1, 𝐶−1 +f,𝑇) +1/2ℎ𝑟+1 +𝑇 +|𝒘|𝑯𝑟+2(𝑇;R𝑑) 𝜇 +1/2 +𝑇 ∥𝒗𝑇 ∥S,𝑇. +(43) +Using (42) and (43) to estimate the right-hand side of (39), we obtain +E𝑘 +S,ℎ(𝒘; 𝒗ℎ) ≲ +∑︁ +𝑇∈Tℎ +𝜇 +1/2 +𝑇 min(1, 𝐶−1 +f,𝑇) +1/2ℎ𝑟+1 +𝑇 +|𝒘|𝑯𝑟+2(𝑇;R𝑑) +� +𝜇𝑇 ∥𝒗𝑇 ∥2 +S,𝑇 + 𝜈𝑇 ∥𝒗𝑇 ∥2 +D,𝑇 +�1/2 +≤ +� ∑︁ +𝑇∈Tℎ +𝜇𝑇 min(1, 𝐶−1 +f,𝑇)ℎ2(𝑟+1) +𝑇 +|𝒘|2 +𝑯𝑟+2(𝑇;R𝑑) +�1/2 +∥𝒗ℎ∥𝜇,𝜈,ℎ, +where the conclusion follows using a discrete Cauchy–Schwarz inequality on the sum over +𝑇 ∈ Tℎ along with the definition (34) of ∥·∥𝜇,𝜈,ℎ. Dividing by ∥𝒗ℎ∥𝜇,𝜈,ℎ and passing to the +supremum concludes the proof of (38). +□ +19 + +5.2.2 +Consistency of the Darcy bilinear form +Lemma 11 (Consistency of the Darcy bilinear form). Given 𝒘 ∈ 𝑯1(Ω; R𝑑), let the Darcy +consistency error linear form E𝑘 +D,ℎ(𝒘; ·) : 𝑼𝑘 +ℎ → R be such that, for all 𝒗ℎ ∈ 𝑼𝑘 +ℎ, +E𝑘 +D,ℎ(𝒘; 𝒗ℎ) ≔ +∑︁ +𝑇∈Tℎ +∫ +𝑇 +𝜈𝑇𝒘 · �𝑷 +𝑘 +D,𝑇𝒗𝑇 − 𝑎𝜈,ℎ(𝑰𝑘 +ℎ𝒘, 𝒗ℎ). +(44) +Then, further assuming 𝒘 ∈ 𝑯𝑟+1(Tℎ; R𝑑) for some 𝑟 ∈ {0, . . . , 𝑘}, it holds +∥E𝑘 +D,ℎ(𝒘; ·)∥𝜇,𝜈,ℎ,∗ ≲ +� ∑︁ +𝑇∈Tℎ +𝜈𝑇 min(1, 𝐶f,𝑇)ℎ2(𝑟+1) +𝑇 +|𝒘|2 +𝑯𝑟+1(𝑇;R𝑑) +�1/2 +. +(45) +Proof. Let 𝒗ℎ ∈ 𝑼𝑘 +ℎ,0 \ {0}. Expanding 𝑎𝜈,ℎ according to its definition (26), we get +E𝑘 +D,ℎ(𝒘; 𝒗ℎ) = +∑︁ +𝑇∈Tℎ +[𝔗1(𝑇) + 𝔗2(𝑇)] , +(46) +with +𝔗1(𝑇) ≔ +∫ +𝑇 +𝜈𝑇 (𝒘 − �𝑷 +𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒘) · �𝑷 +𝑘 +D,𝑇𝒗𝑇, +𝔗2(𝑇) ≔ −𝜈𝑇 min(1, 𝐶f,𝑇)(𝑰𝑘 +𝑇 (𝒘 − 𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒘), 𝒗𝑇 − 𝑰𝑘 +𝑇 𝑷𝑘 +D,𝑇𝒗𝑇)𝑼,𝑇. +The estimate of 𝔗1(𝑇) depends on the regime. Let us start with the case 𝐶f,𝑇 ≥ 1. Recalling +(25) to replace �𝑷 +𝑘 +D,𝑇 with 𝑷𝑘 +D,𝑇 and applying a Cauchy–Schwarz inequality, we get +|𝔗1(𝑇)| ≲ 𝜈𝑇 ∥𝒘 − 𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒘∥𝑳2(𝑇;R𝑑)∥𝑷𝑘 +D,𝑇𝒗𝑇 ∥𝑳2(𝑇;R𝑑) +≲ 𝜈 +1/2 +𝑇 min(1, 𝐶f,𝑇) +1/2ℎ𝑟+1 +𝑇 +|𝒘|𝑯𝑟+1(𝑇;R𝑑) 𝜈 +1/2 +𝑇 ∥𝒗𝑇 ∥D,𝑇, +where, to pass to the second line, we have used the approximation properties (22) of 𝑷𝑘 +D,𝑇 ◦ 𝑰𝑘 +𝑇 +with 𝑚 = 0, the definition (28) of the ∥·∥D,𝑇-norm, and observed that 1 = min(1, 𝐶f,𝑇). +Let us now consider the case 𝐶f,𝑇 < 1. Recalling that �𝑷 +𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒘 = 𝝅𝑘 +P,𝑇𝒘 in this case, we +can write 𝔗1(𝑇) = +∫ +𝑇 𝜈𝑇 (𝒘 − 𝝅𝑘 +P,𝑇𝒘) · (𝒗𝑇 − 𝝅0 +P,𝑇𝒗𝑇) and, using Cauchy–Schwarz inequalities, +continue with +|𝔗1(𝑇)| ≤ 𝜈𝑇 ∥𝒘 − 𝝅𝑘 +P,𝑇𝒘∥𝑳2(𝑇;R𝑑)∥𝒗𝑇 − 𝝅0 +P,𝑇𝒗𝑇 ∥𝑳2(𝑇;R𝑑). +(47) +Using the approximation properties of 𝝅𝑘 +P,𝑇, it is readily inferred that the first factor is ≲ +ℎ𝑟+1 +𝑇 +|𝒘|𝑯𝑟+1(𝑇;R𝑑). To estimate the last factor, we use a Poincaré–Wirtinger inequality to write +∥𝒗𝑇 − 𝝅0 +P,𝑇𝒗𝑇 ∥𝑳2(𝑇;R𝑑) ≲ ℎ𝑇 ∥∇𝒗𝑇 ∥𝑳2(𝑇;R𝑑×𝑑) ≲ ℎ𝑇 ∥𝒗𝑇 ∥S,𝑇, where the conclusion follows from +(15). Plugging the above estimates into (47), we can go on writing +|𝔗1(𝑇)| ≲ 𝜈 +1/2 +𝑇 ℎ𝑟+1 +𝑇 +|𝒘|𝑯𝑟+1(𝑇;R𝑑) 𝜈 +1/2 +𝑇 ℎ𝑇 ∥𝒗𝑇 ∥S,𝑇 += 𝜈 +1/2 +𝑇 ℎ𝑟+1 +𝑇 +|𝒘|𝑯𝑟+1(𝑇;R𝑑) 𝜇 +1/2 +𝑇 𝐶 +1/2 +f,𝑇 ∥𝒗𝑇 ∥S,𝑇, += 𝜈 +1/2 +𝑇 min(1, 𝐶f,𝑇) +1/2ℎ𝑟+1 +𝑇 +|𝒘|𝑯𝑟+1(𝑇;R𝑑) 𝜇 +1/2 +𝑇 ∥𝒗𝑇 ∥S,𝑇, +20 + +where we have used the definition (4) of 𝐶f,𝑇 to pass to the second line and, after rearranging the +factors, the fact that 𝐶f,𝑇 = min(1, 𝐶f,𝑇) to conclude. Gathering the above estimates, we thus +have +|𝔗1(𝑇)| ≲ 𝜈 +1/2 +𝑇 min(1, 𝐶f,𝑇) +1/2ℎ𝑟+1 +𝑇 +|𝒘|𝑯𝑟+1(𝑇;R𝑑) +� +𝜇𝑇 ∥𝒗𝑇 ∥2 +S,𝑇 + 𝜈𝑇 ∥𝒗𝑇 ∥2 +D,𝑇 +�1/2 +. +(48) +To estimate 𝔗2(𝑇), we use a Cauchy–Schwarz inequality to write +|𝔗2(𝑇)| +≤ 𝜈 +1/2 +𝑇 min(1, 𝐶f,𝑇) +1/2∥𝑰𝑘 +𝑇 (𝒘 − 𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒘)∥𝑼,𝑇 𝜈 +1/2 +𝑇 min(1, 𝐶f,𝑇) +1/2∥𝒗𝑇 − 𝑰𝑘 +𝑇 𝑷𝑘 +D,𝑇𝒗𝑇 ∥𝑼,𝑇 +≲ 𝜈 +1/2 +𝑇 min(1, 𝐶f,𝑇) +1/2 � +∥𝒘 − 𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒘∥𝑳2(𝑇;R𝑑) + ℎ𝑇 |𝒘 − 𝑷𝑘 +D,𝑇 𝑰𝑘 +𝑇𝒘|𝑯1(𝑇;R𝑑) +� +𝜈 +1/2 +𝑇 ∥𝒗𝑇 ∥D,𝑇 +≲ 𝜈 +1/2 +𝑇 min(1, 𝐶f,𝑇) +1/2ℎ𝑟+1 +𝑇 +|𝒘|𝑯𝑟+1(𝑇;R𝑑) 𝜈 +1/2 +𝑇 ∥𝒗𝑇 ∥D,𝑇, +(49) +where we have used the ∥·∥𝑼,𝑇-boundedness (8) of 𝑰𝑘 +𝑇 along with the definition (28) of the +∥·∥D,𝑇-norm in the second inequality and the approximation properties (22) of 𝑷𝑘 +D,𝑇 ◦ 𝑰𝑘 +𝑇 with +𝑚 = 0 and 𝑚 = 1 to conclude. Plugging (48) and (49) into (46), using discrete Cauchy–Schwarz +inequalities, dividing by ∥𝒗ℎ∥𝜇,𝜈,ℎ, and passing to the supremum, the conclusion follows. +□ +5.2.3 +Consistency of the coupling bilinear form +The quantity estimated in the following lemma can be interpreted as an adjoint consistency error +for the discrete divergence. +Lemma 12 (Consistency of the coupling bilinear form). Given 𝑞 ∈ 𝐻1(Ω), let the coupling +consistency error linear form E𝑘 +c,ℎ(𝑞; ·) : 𝑼𝑘 +ℎ,0 → R be such that, for all 𝒗ℎ ∈ 𝑼𝑘 +ℎ,0, +E𝑘 +c,ℎ(𝑞; 𝒗ℎ) ≔ +∑︁ +𝑇∈Tℎ +∫ +𝑇 +∇𝑞 · �𝑷 +𝑘 +D,𝑇𝒗𝑇 − 𝑏ℎ(𝒗ℎ, 𝜋𝑘 +P,ℎ𝑞). +(50) +Then, further assuming, for some 𝑟 ∈ {0, . . . , 𝑘}, 𝑞 ∈ 𝐻𝑟+1+⟨𝐶f,𝑇 ≥1⟩(𝑇) for all 𝑇 ∈ Tℎ, it holds +∥E𝑘 +c,ℎ(𝑞; ·)∥𝜇,𝜈,ℎ,∗ +≲ +� ∑︁ +𝑇∈Tℎ +� +𝜇−1 +𝑇 ⟨𝐶f,𝑇 < 1⟩ℎ2(𝑟+1) +𝑇 +|𝑞|2 +𝐻𝑟+1(𝑇) + 𝜈−1 +𝑇 ⟨𝐶f,𝑇 ≥ 1⟩ℎ2(𝑟+1) +𝑇 +|𝑞|2 +𝐻𝑟+2(𝑇) +�� 1/2 +, +(51) +where 𝜈−1 +𝑇 ⟨𝐶f,𝑇 ≥ 1⟩ ≔ 0 if 𝜈𝑇 = 0 as in Theorem 7. +Proof. Let 𝒗ℎ ∈ 𝑼𝑘 +ℎ,0 \ {0}. We start by noticing that, expanding the bilinear form 𝑏ℎ according +to its definition (29), +E𝑘 +c,ℎ(𝑞; 𝒗ℎ) = +∑︁ +𝑇∈Tℎ +�∫ +𝑇 +∇𝑞 · �𝑷 +𝑘 +D,𝑇𝒗𝑇 + +∫ +𝑇 +𝜋𝑘 +P,𝑇𝑞 𝐷𝑘 +𝑇𝒗𝑇 − +∑︁ +𝐹∈F𝑇 +𝜔𝑇𝐹 +∫ +𝐹 +𝑞 (𝒗𝐹 · 𝒏𝐹) +� +, +(52) +where the insertion of the last term in parenthesis is made possible by the single-valuedness of +𝑞 (𝒗𝐹 · 𝒏𝐹) at interfaces along with the fact that 𝒗𝐹 · 𝒏𝐹 = 0 for all 𝐹 ∈ F b +ℎ . Denote by 𝔗(𝑇) the +21 + +argument of the summation in (52). To estimate this quantity, we distinguish two cases based +on the value of 𝐶f,𝑇. +If 𝐶f,𝑇 < 1, �𝑷 +𝑘 +D,𝑇𝒗𝑇 = 𝒗𝑇 by (25), so that +𝔗(𝑇) = +∫ +𝑇 +∇𝑞 · 𝒗𝑇 + +∫ +𝑇 +𝜋𝑘 +P,𝑇𝑞 𝐷𝑘 +𝑇𝒗𝑇 − +∑︁ +𝐹∈F𝑇 +𝜔𝑇𝐹 +∫ +𝐹 +𝑞 (𝒗𝐹 · 𝒏𝐹). +Thus, proceeding as in the derivation of [19, Eq. (8.41)], we get +𝔗(𝑇) ≲ ℎ𝑟+1 +𝑇 +|𝑞|𝐻𝑟+1(𝑇) +� +1 +ℎ𝑇 +∑︁ +𝐹∈F𝑇 +∥𝒗𝑇 − 𝒗𝐹∥2 +𝑳2(𝐹;R𝑑) +�1/2 +≲ 𝜇−1/2 +𝑇 +⟨𝐶f,𝑇 < 1⟩ +1/2ℎ𝑟+1 +𝑇 +|𝑞|𝐻𝑟+1(𝑇) 𝜇 +1/2 +𝑇 ∥𝒗𝑇 ∥S,𝑇, +(53) +where we have used (15) to conclude. +If 𝐶f,𝑇 ≥ 1, on the other hand, we have �𝑷 +𝑘 +D,𝑇 = 𝑷𝑘 +D,𝑇 (cf. (25)), so that +𝔗(𝑇) = +�∫ +𝑇 +∇𝑞 · 𝑷𝑘 +D,𝑇𝒗𝑇 + +∫ +𝑇 +𝜋𝑘 +P,𝑇𝑞 𝐷𝑘 +𝑇𝒗𝑇 − +∑︁ +𝐹∈F𝑇 +𝜔𝑇𝐹 +∫ +𝐹 +𝑞 (𝒗𝐹 · 𝒏𝐹) +� +. +Using the definition (19) of 𝑷𝑘 +D,𝑇 to proceed as in [17, Theorem 11], we get +𝔗(𝑇) ≲ ℎ𝑟+1 +𝑇 +|𝑞|𝐻𝑟+2(𝑇) ∥𝒗𝑇 ∥D,𝑇 ≤ 𝜈−1/2 +𝑇 +⟨𝐶f,𝑇 ≥ 1⟩ +1/2ℎ𝑟+1 +𝑇 +|𝑞|𝐻𝑟+2(𝑇) 𝜈 +1/2 +𝑇 ∥𝒗𝑇 ∥D,𝑇, +(54) +where we have additionally noticed that 𝐶f,𝑇 ≥ 1 implies 𝜈𝑇 > 0. +To conclude, we plug (53) and (54) into (52), use a Cauchy–Schwarz inequality on the sum +over 𝑇 ∈ Tℎ, recall the definition (34) of the ∥·∥𝜇,𝜈,ℎ-norm, and pass to the supremum after +dividing by ∥𝒗ℎ∥𝜇,𝜈,ℎ. +□ +Remark 13 (Discretisation of the source term). The use of 𝑷𝑘 +D,𝑇 in the discretisation of the source +term when 𝐶f,𝑇 ≥ 1 (see (32) and (25)) is crucial to ensure that, in this case, the consistency +error of the coupling bilinear form can be bounded above using the Darcy norm instead of the +Stokes norm; compare (54) and (53). This bound is key to establishing an error estimate in ℎ𝑟+1 +that remains robust in the Darcy limit. +5.2.4 +Consistency of the forcing term linear form +The following lemma estimates the difference between the standard HHO right-hand side linear +form and the one obtained, as in (2a), using �𝑷 +𝑘 +D,𝑇𝒗𝑇 instead of 𝒗𝑇 as a test function. +Lemma 14 (Consistency of the forcing term). For any 𝝋 ∈ 𝑳2(Ω; R𝑑), define the right-hand +side consistency error linear form E𝑘 +rhs,ℎ(𝝋; ·) : 𝑼𝑘 +ℎ → R such that, for all 𝒗ℎ ∈ 𝑼𝑘 +ℎ, +E𝑘 +rhs,ℎ(𝝋; 𝒗ℎ) ≔ +∑︁ +𝑇∈Tℎ +∫ +𝑇 +𝝋 · (𝒗𝑇 − �𝑷 +𝑘 +D,𝑇𝒗𝑇). +(55) +Further assuming 𝝋 ∈ 𝑯𝑟(Tℎ; R𝑑) for some 𝑟 ∈ {0, . . . , 𝑘}, it holds +∥E𝑘 +rhs,ℎ(𝝋; ·)∥𝜇,𝜈,ℎ,∗ ≲ +� ∑︁ +𝑇∈Tℎ +𝜇−1 +𝑇 min(1, 𝐶−1 +f,𝑇)ℎ2(𝑟+1) +𝑇 +|𝝋|2 +𝑯𝑟 (𝑇;R𝑑) +�1/2 +. +(56) +22 + +Proof. Denote by 𝔗(𝑇) the argument of the summation in (55). If 𝐶f,𝑇 < 1, the definition (25) +of �𝑷 +𝑘 +D,𝑇 yields 𝔗(𝑇) = 0. Consider now the case 𝐶f,𝑇 ≥ 1 (which implies, in particular, 𝜈𝑇 > 0). +We first notice that, letting 𝝅𝑘−1 +P,𝑇 𝝋 ≔ 0 if 𝑘 = 0, +∥𝝋 − 𝝅𝑘−1 +P,𝑇 𝝋∥𝑳2(𝑇;R𝑑) ≲ ℎ𝑟 +𝑇 |𝝋|𝑯𝑟 (𝑇;R𝑑), +(57) +where the result is trivial if 𝑘 = 0 (which imposes 𝑟 = 0) and otherwise follows from the +approximation properties of 𝝅𝑘−1 +P,𝑇, see [19, Theorem 1.45]. Recalling that, for 𝐶f,𝑇 ≥ 1, we have +�𝑷 +𝑘 +D,𝑇 = 𝑷𝑘 +D,𝑇 by (25) and invoking (20) (which trivially holds also for 𝑘 = 0), we then write +𝔗(𝑇) = +∫ +𝑇 +(𝝋 − 𝝅𝑘−1 +P,𝑇 𝝋) · (𝒗𝑇 − 𝑷𝑘 +D,𝑇𝒗𝑇) +≤ ∥𝝋 − 𝝅𝑘−1 +P,𝑇 𝝋∥𝑳2(𝑇;R𝑑)∥𝒗𝑇 − 𝑷𝑘 +D,𝑇𝒗𝑇 ∥𝑳2(𝑇;R𝑑) +≲ 𝜇−1/2 +𝑇 +ℎ𝑟 +𝑇 |𝝋|𝑯𝑟 (𝑇;R𝑑) ℎ𝑇 𝜇 +1/2 +𝑇 ℎ−1 +𝑇 ∥𝒗𝑇 ∥D,𝑇 += 𝜇−1/2 +𝑇 +ℎ𝑟+1 +𝑇 +|𝝋|𝑯𝑟 (𝑇;R𝑑) 𝜈 +1/2 +𝑇 min(1, 𝐶−1 +f,𝑇) +1/2∥𝒗𝑇 ∥D,𝑇, +where we have used Cauchy–Schwarz inequalities in the first passage, the approximation prop- +erties (57) of the 𝐿2-orthogonal projector for the first factor together with the definitions (7) and +(28) of ∥·∥𝑼,𝑇 and ∥·∥D,𝑇 to write ∥𝒗𝑇 − 𝑷𝑘 +D,𝑇𝒗𝑇 ∥𝑳2(𝑇;R𝑑) ≤ ∥𝒗𝑇 − 𝑰𝑘 +𝑇 𝑷𝑘 +D,𝑇𝒗𝑇 ∥𝑼,𝑇 ≤ ∥𝒗𝑇 ∥D,𝑇 +in the second passage, while the conclusion follows from the definition (4) of 𝐶f,𝑇 along with +𝐶−1 +f,𝑇 = min(1, 𝐶−1 +f,𝑇). Using the above estimate in (55), applying a Cauchy–Schwarz inequality +on the sum over 𝑇 ∈ Tℎ, and recalling the definition (34) of ∥·∥𝜇,𝜈,ℎ, (56) follows. +□ +5.2.5 +Proof of Theorem 7 +Proof of Theorem 7. Since 𝑎𝜇,ℎ + 𝑎𝜈,ℎ is 1-coercive and has norm 1 for the ∥·∥𝜇,𝜈,ℎ norm, +Lemma 6 and [19, Lemma A.11] show that Aℎ is 𝛾-inf-sup stable for the norm in the left-hand +side of (35). Hence, in the spirit of the third Strang lemma [16], this error estimate follows if +we bound the consistency error by the bracketed term in the right-hand side. The consistency +error for the scheme (32) is +E𝑘 +ℎ (𝒖, 𝑝; 𝒗ℎ) ≔ +∑︁ +𝑇∈Tℎ +∫ +𝑇 +𝒇 · �𝑷 +𝑘 +D,𝑇𝒗𝑇 + +∫ +Ω +𝑔𝑞ℎ − Aℎ((𝑰𝑘 +ℎ𝒖, 𝜋ℎ +P,𝑘 𝑝), (𝒗ℎ, 𝑞ℎ)) += +∑︁ +𝑇∈Tℎ +∫ +𝑇 +∇·(𝜇𝑇∇𝒖) · (𝒗𝑇 − �𝑷 +𝑘 +D,𝑇𝒗𝑇) − +∑︁ +𝑇∈Tℎ +∫ +𝑇 +∇·(𝜇𝑇∇𝒖) · 𝒗𝑇 − 𝑎𝜇,ℎ(𝑰𝑘 +ℎ𝒖, 𝒗ℎ) ++ +∑︁ +𝑇∈Tℎ +∫ +𝑇 +𝜈𝑇𝒖 · �𝑷 +𝑘 +D,𝑇𝒗𝑇 − 𝑎𝜈,ℎ(𝑰𝑘 +ℎ𝒖, 𝒗ℎ) + +∑︁ +𝑇∈Tℎ +∫ +𝑇 +∇𝑝 · �𝑷 +𝑘 +D,𝑇𝒗𝑇 − 𝑏ℎ(𝒗ℎ, 𝜋𝑘 +P,ℎ𝑝) ++ + +∫ +Ω +𝑔𝑞ℎ + 𝑏ℎ(𝑰𝑘 +ℎ𝒖, 𝑞ℎ) += E𝑘 +rhs,ℎ(∇·(𝜇∇𝒖); 𝒗ℎ) + E𝑘 +S,ℎ(𝒖; 𝒗ℎ) + E𝑘 +D,ℎ(𝒖; 𝒗ℎ) + E𝑘 +c,ℎ(𝑝; 𝒗ℎ), +(58) +where we have we have replaced 𝒇 with the left-hand side of (2a), expanded Aℎ according to +its definition (33), and used (30) along with (2b) to cancel the last term in the first passage, and +used the definitions of the consistency errors (55) with 𝝋 = ∇·(𝜇∇𝒖), (37) and (44) with 𝒘 = 𝒖, +and (50) with 𝑞 = 𝑝 to conclude. +23 + +Using, respectively, (56) (further noticing that |∇·(𝜇𝑇∇𝒖)|𝑯𝑟 (𝑇;R𝑑) ≲ 𝜇𝑇 |𝒖|𝑯𝑟+2(𝑇;R𝑑) for all +𝑇 ∈ Tℎ), (38), (45), and (51) to estimate the terms in the right-hand side of (58), the result +follows. +□ +Acknowledgements +This research received support from the ANR “NEMESIS” (ANR-20-MRS2-0004) and the +Australian Research Council’s Discovery Projects funding scheme (DP210103092). 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ModelsMethodsAppl.Sci. 28.1(2018),pp. 159–194. doi: 10.1142/S0218202518500057. +26 + diff --git a/vNE1T4oBgHgl3EQfkQSg/content/tmp_files/load_file.txt b/vNE1T4oBgHgl3EQfkQSg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7ec2ad90cc3954dd41295a3c673aba1ab55cd4ed --- /dev/null +++ b/vNE1T4oBgHgl3EQfkQSg/content/tmp_files/load_file.txt @@ -0,0 +1,1113 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf,len=1112 +page_content='A polytopal method for the Brinkman problem robust in all regimes Daniele A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Di Pietro1 and Jérôme Droniou2 1IMAG, Univ Montpellier, CNRS, Montpellier, France, daniele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='di-pietro@umontpellier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='fr 2School of Mathematics, Monash University, Melbourne, Australia, jerome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='droniou@monash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='edu January 10, 2023 Abstract In this work we develop a discretisation method for the Brinkman problem that is uniformly well-behaved in all regimes (as identified by a local dimensionless number with the meaning of a friction coefficient) and supports general meshes as well as arbitrary approximation orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The method is obtained combining ideas from the Hybrid High- Order and Discrete de Rham methods, and its robustness rests on a potential reconstruction and stabilisation terms that change in nature according to the value of the local friction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' We derive error estimates that, thanks to the presence of cut-off factors, are valid across the all regimes and provide extensive numerical validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' MSC: 65N30, 65N08, 76S05, 76D07 Key words: Brinkman, Darcy, Stokes, Hybrid High-Order methods, Discrete de Rham methods 1 Introduction The Brinkman problem governs the flow of a viscous fluid in an inhomogeneous material where fractures, bubbles, or channels are present within a porous matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Mathematically, this problem translates into a system of partial differential equations with saddle-point structure which can be regarded as a superposition of the Stokes and Darcy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' As pointed out in [30], the construction of finite element approximations that are uniformly well-behaved across the entire range of (Stokes- or Darcy-dominated) regimes is not straightforward;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' a representative, but by far not exhaustive, list of references is [2, 4, 12–14, 26, 28, 29, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In [10], we introduced a numerical method for the Brinkman problem on matching simplicial meshes and derived what appears to be the first error estimate accounting for the local regime through a dimensionless number which can be interpreted as a friction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Thanks to the presence of cutoff factors, this error estimate holds in all situations, including the Stokes problem as well as the singular limit corresponding to the pure Darcy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In this work, we provide a positive answer to an open question left in the above reference, namely whether similar robustness features and error estimates can be obtained on general polytopal meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' As for the original method of [10], the discretisation of the Stokes term is 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='03272v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='NA] 9 Jan 2023 inspired by Hybrid High-Order (HHO) methods [19, 21, 23] while, for the Darcy and forcing terms, a novel construction inspired by discrete de Rham methods [17, 20] (see also [22] for an antecedent) replaces the one based on the Raviart–Thomas–Nédélec space [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The first central element in this construction is a discrete vector potential that changes in nature depending on the value of the local friction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The other key ingredient are regime-dependent stabilisation terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Thanks to these novel tools, we are able to derive a robust estimate of the adjoint error for the discrete divergence, which is the pivot result for the extension of the techniques of [10] to polytopal meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The resulting error estimate, stated in Theorem 7 below, is valid on the entire range of values for the local friction coefficient, from 0 (pure Stokes) to +∞ (pure Darcy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The rest of the work is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In Section 2 we briefly recall the continuous and discrete settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In Section 3 we formulate the numerical scheme and state the main stability and convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Extensive numerical validation of these results on a variety of meshes and regimes for analytical solutions is provided in Section 4, where a more physical three- dimensional test case is also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Finally, the proofs of the main results are collected in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 2 Setting 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1 Continuous problem Let Ω ⊂ R𝑑, 𝑑 ∈ {2, 3}, denote a bounded connected open polytopal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=', polygonal if 𝑑 = 2 and polyhedral if 𝑑 = 3) domain with boundary 𝜕Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' For the sake of simplicity, and without loss of generality, we assume that Ω has unit diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Let two functions 𝜇 : Ω → R and 𝜈 : Ω → R be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In what follows, we assume that there exist real numbers 𝜇, 𝜇, and 𝜈 such that, almost everywhere in Ω, 0 < 𝜇 ≤ 𝜇 ≤ 𝜇, 0 ≤ 𝜈 ≤ 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (1) Let 𝒇 : Ω → R𝑑 and 𝑔 : Ω → R denote volumetric source terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The Brinkman problem reads: Find the velocity 𝒖 : Ω → R𝑑 and the pressure 𝑝 : Ω → R such that −∇·(𝜇∇𝒖) + 𝜈𝒖 + ∇𝑝 = 𝒇 in Ω, (2a) ∇·𝒖 = 𝑔 in Ω, (2b) 𝒖 = 0 on 𝜕Ω, (2c) ∫ Ω 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (2d) A few simplifications are made to make the exposition more compact while retaining all the difficulties related to the robustness across the entire range of values for 𝜇 and 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' First of all, in (2a) we have considered a viscous term expressed in terms of the full gradient instead of its symmetric part ∇s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The changes to replace ∇ with ∇s are standard in the HHO literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=', [11, 21] and [19, Chapter 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Second, we assume henceforth that both 𝜇 and 𝜈 are piecewise constant on a polytopal partition 𝑃Ω of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The extension to coefficients that vary smoothly inside each element, and are possibly full tensors, is also standard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' see, in particular, [19, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 Discrete setting 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1 Mesh and notation for inequalities up to a constant We consider polytopal meshes Mℎ ≔ Tℎ ∪ Fℎ matching the geometrical requirements detailed in [19, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='4], with Tℎ set of elements and Fℎ set of faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' To avoid dealing with jumps of the problem coefficients 𝜇 and 𝜈 inside mesh elements, we additionally assume that Tℎ is compatible with 𝑃Ω, meaning that, for each 𝑇 ∈ Tℎ, there exists 𝜔 ∈ 𝑃Ω such that 𝑇 ⊂ 𝜔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' We then set 𝜇𝑇 ≔ 𝜇|𝑇 and 𝜈𝑇 ≔ 𝜈|𝑇 for all 𝑇 ∈ Tℎ, noticing that these constant values are uniquely defined in each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' For any 𝑌 ∈ Mℎ, we denote by ℎ𝑌 its diameter, so that ℎ = max𝑇∈Tℎ ℎ𝑇 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' For every mesh element 𝑇 ∈ Tℎ, we denote by F𝑇 the subset of Fℎ containing the faces that lie on the boundary 𝜕𝑇 of 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' For any mesh face 𝐹 ∈ Fℎ, we fix once and for all a unit normal vector 𝒏𝐹 and, for any mesh element 𝑇 ∈ Tℎ such that 𝐹 ∈ F𝑇, we let 𝜔𝑇𝐹 ∈ {−1, +1} denote the orientation of 𝐹 relative to 𝑇, selected so that 𝜔𝑇𝐹𝒏𝐹 points out of 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Boundary faces lying on 𝜕Ω are collected in the set F b ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Our focus being on the ℎ-convergence analysis, we assume that Mℎ belongs to a sequence of refined polygonal or polyhedral meshes that is regular in the sense of [19, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' This implies, in particular, that the number of faces of each mesh element is bounded from above by an integer independent of ℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' see [19, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' From this point on, 𝑎 ≲ 𝑏 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑎 ≳ 𝑏) means 𝑎 ≤ 𝐶𝑏 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑎 ≥ 𝐶𝑏) with 𝐶 only depending on Ω, the mesh regularity parameter, and the polynomial degree 𝑘 of the scheme defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' We stress that this means, in particular, that 𝐶 is independent of the problem parameters 𝜇 and 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' We also write 𝑎 ≃ 𝑏 as a shorthand for “𝑎 ≲ 𝑏 and 𝑏 ≲ 𝑎”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 Polynomial spaces Given 𝑌 ∈ Tℎ ∪ Fℎ and an integer 𝑚 ≥ 0, we denote by P𝑚(𝑌) the space spanned by the restriction to 𝑌 of 𝑑-variate polynomials of total degree ≤ 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The symbols P𝑚(𝑌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) and P𝑚(𝑌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑×𝑑) respectively denote the sets of vector- and tensor-valued functions over 𝑌 whose components are in P𝑚(𝑌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' For 𝑇 ∈ Tℎ, we will need the following direct decomposition of P𝑚(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=', [5, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='4]): P𝑚(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) = G𝑚(𝑇) ⊕ Gc,𝑚(𝑇), with G𝑚(𝑇) ≔ ∇P𝑚+1(𝑇) and Gc,𝑚(𝑇) ≔ � (𝒙 − 𝒙𝑇)⊥P𝑚−1(𝑇) if 𝑑 = 2, (𝒙 − 𝒙𝑇) × P𝑚−1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R3) if 𝑑 = 3, (3) where 𝒙𝑇 is a point such that 𝑇 is star-shaped with respect to a ball of radius ≳ ℎ𝑇 and, in the case 𝑑 = 2, for any 𝒗 ∈ R2 we denote by 𝒗⊥ the vector obtained rotating 𝒗 by − 𝜋 2 radians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Given a polynomial (sub)space X𝑚(𝑌) on 𝑌 ∈ Tℎ ∪ Fℎ, the corresponding 𝐿2-orthogonal projector is denoted by 𝜋𝑚 X,𝑌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Boldface fonts will be used when the elements of X𝑚(𝑌) are vector-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The set of broken polynomials of total degree ≤ 𝑚 on the mesh is denoted by P𝑚(Tℎ), and the corresponding 𝐿2-orthogonal projector by 𝜋𝑚 P,ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='3 Local friction coefficient The regime inside each mesh element 𝑇 ∈ Tℎ is identified by the following dimensionless number, which can be interpreted as a friction coefficient: 𝐶f,𝑇 ≔ 𝜈𝑇ℎ2 𝑇 𝜇𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (4) Elements for which 𝐶f,𝑇 < 1 are in the Stokes-dominated regime, while elements for which 𝐶f,𝑇 ≥ 1 are in the Darcy-dominated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The values 𝐶f,𝑇 = 0 and 𝐶f,𝑇 = +∞ correspond to pure Stokes and pure Darcy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Notice that 𝐶f,𝑇 = +∞ is a singular limit which, despite requiring to modify the continuous formulation (2), can be handled seamlessly by the method developed in the next section;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' see Remark 8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 3 A robust numerical scheme for the Brinkman problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1 Spaces Let an integer 𝑘 ≥ 0 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' We define the following HHO space: 𝑼𝑘 ℎ ≔ � 𝒗ℎ = �(𝒗𝑇)𝑇∈Tℎ, (𝒗𝐹)𝐹∈Fℎ � : 𝒗𝑇 ∈ P𝑘 (𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) for all 𝑇 ∈ Tℎ and 𝒗𝐹 ∈ P𝑘 (𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) for all 𝐹 ∈ Fℎ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The meaningofthepolynomialcomponentsin𝑼𝑘 ℎ isprovidedbytheinterpolator 𝑰𝑘 ℎ : 𝑯1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) → 𝑼𝑘 ℎ such that, for all 𝒗 ∈ 𝑯1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑), 𝑰𝑘 ℎ𝒗 ≔ �(𝝅𝑘 P,𝑇𝒗)𝑇∈Tℎ, (𝝅𝑘 P,𝐹𝒗)𝐹∈Tℎ, � ∈ 𝑼𝑘 ℎ, where it is understood that 𝐿2-orthogonal projectors are applied to restrictions or traces as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The restrictions of 𝑼𝑘 ℎ, 𝒗ℎ ∈ 𝑼𝑘 ℎ, and 𝑰𝑘 ℎ to a mesh element 𝑇, respectively denoted by 𝑼𝑘 𝑇, 𝒗𝑇 ∈ 𝑼𝑘 𝑇, and 𝑰𝑘 𝑇, are obtained collecting the components attached to 𝑇 and its faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In what follows, given a logical proposition 𝑃, we denote by ⟨𝑃⟩ its truth value such that ⟨𝑃⟩ ≔ � 0 if 𝑃 is false, 1 if 𝑃 is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (5) We define the following 𝐿2-like product in 𝑼𝑘 ℎ: For all (𝒘ℎ, 𝒗ℎ) ∈ 𝑼𝑘 ℎ × 𝑼𝑘 ℎ, (𝒘ℎ, 𝒗ℎ)𝑼,ℎ ≔ ∑︁ 𝑇∈Tℎ (𝒘𝑇, 𝒗𝑇)𝑼,𝑇 with (𝒘𝑇, 𝒗𝑇)𝑼,𝑇 ≔ 𝜆𝑇 ∫ 𝑇 𝒘𝑇 · 𝒗𝑇 + ℎ𝑇 ∑︁ 𝐹∈F𝑇 ⟨𝐶f,𝑇 < 1 or 𝐹 ∉ F b ℎ ⟩ ∫ 𝐹 𝒘𝐹 · 𝒗𝐹, (6) where 𝜆𝑇 ≃ 1 is a factor, based on the regularity of the element 𝑇, chosen to balance out the element and face contributions to (·, ·)𝑼,𝑇 (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The corresponding local and global seminorms are obtained setting, for • ∈ Tℎ ∪ {ℎ}, ∥𝒗•∥𝑼,• ≔ (𝒗•, 𝒗•) 1/2 𝑼,•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (7) 4 The following boundedness property of the interpolator in the ∥·∥𝑼,ℎ-norm follows from the definition of this norm along with the uniform boundedness of the 𝐿2-orthogonal projectors 𝝅𝑘 P,𝑌, 𝑌 ∈ Mℎ, and continuous trace inequalities (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [19, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='31]): For all 𝑇 ∈ Tℎ and all 𝒗 ∈ 𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑), ∥𝑰𝑘 𝑇𝒗∥𝑼,𝑇 ≲ ∥𝒗∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + ℎ𝑇 |𝒗|𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (8) The velocity and pressure spaces, respectively incorporating the boundary and zero-average conditions, are 𝑼𝑘 ℎ,0 ≔ � 𝒗ℎ ∈ 𝑼𝑘 ℎ : 𝒗𝐹 = 0 for all 𝐹 ∈ F b ℎ � , 𝑃𝑘 ℎ ≔ P𝑘 (Tℎ) ∩ 𝐿2 0(Ω), where, as usual, 𝐿2 0(Ω) = � 𝑞 ∈ 𝐿2(Ω) : ∫ Ω 𝑞 = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Remark 1 (Boundary degrees of freedom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Note that the degrees of freedom on the boundary faces of a vector in 𝑼𝑘 ℎ may not be controlled by the seminorms ∥·∥𝑼,•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' This is, however, not an issue as the final problem will be set on 𝑼𝑘 ℎ,0 (see also Remark 8 for the handling of boundary values in the limiting case of the pure Darcy problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 Viscous term Let 𝑇 ∈ Tℎ be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' For the discretisation of the viscous term, we define the discrete gradient 𝑮𝑘 𝑇 : 𝑼𝑘 𝑇 → P𝑘 (𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑×𝑑) and the Stokes potential 𝑷𝑘+1 S,𝑇 : 𝑼𝑘 𝑇 → P𝑘 (𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) such that, for all 𝒗𝑇 ∈ 𝑼𝑘 𝑇, ∫ 𝑇 𝑮𝑘 𝑇𝒗𝑇 : 𝝉 = − ∫ 𝑇 𝒗𝑇 · ∇·𝝉 + ∑︁ 𝐹∈F𝑇 𝜔𝑇𝐹 ∫ 𝐹 𝒗𝐹 · 𝝉𝒏𝐹 ∀𝝉 ∈ P𝑘 (𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑×𝑑), (9) and ∇𝑷𝑘+1 S,𝑇 𝒗𝑇 = 𝝅𝑘 G,𝑇𝑮𝑘 𝑇𝒗𝑇, ∫ 𝑇 𝑷𝑘+1 S,𝑇 𝒗𝑇 = ∫ 𝑇 𝒗𝑇, (10) with 𝝅𝑘 G,𝑇 applied to tensor-valued fields also acting row-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Likewise, in the formulas above, ∇· and ∇ are understood to act row-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The Stokes term in (2a) is discretised through the bilinear form 𝑎𝜇,ℎ : 𝑼𝑘 ℎ × 𝑼𝑘 ℎ → R such that, for all (𝒘ℎ, 𝒗ℎ) ∈ 𝑼𝑘 ℎ × 𝑼𝑘 ℎ, 𝑎𝜇,ℎ(𝒘ℎ, 𝒗ℎ) ≔ ∑︁ 𝑇∈Tℎ 𝜇𝑇𝑎S,𝑇 (𝒘𝑇, 𝒗𝑇), (11) where, for all 𝑇 ∈ Tℎ, 𝑎S,𝑇 (𝒘𝑇, 𝒗𝑇) ≔ ∫ 𝑇 𝑮𝑘 𝑇𝒘𝑇 : 𝑮𝑘 𝑇𝒗𝑇 + min(1, 𝐶−1 f,𝑇) ℎ2 𝑇 (𝒘𝑇 − 𝑰𝑘 𝑇 𝑷𝑘+1 S,𝑇 𝒘𝑇, 𝒗𝑇 − 𝑰𝑘 𝑇 𝑷𝑘+1 S,𝑇 𝒗𝑇)𝑼,𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (12) We define the following induced seminorms: For all 𝒗ℎ ∈ 𝑼𝑘 ℎ, ∥𝒗ℎ∥𝜇,ℎ ≔ 𝑎𝜇,ℎ(𝒗ℎ, 𝒗ℎ) 1/2 and ∥𝒗𝑇 ∥S,𝑇 ≔ 𝑎S,𝑇 (𝒗𝑇, 𝒗𝑇) 1/2 for all 𝑇 ∈ Tℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (13) 5 Lemma 2 (Norm equivalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Let 𝑇 ∈ Tℎ and 𝒗𝑇 ∈ 𝑼𝑘 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Then, it holds ∥∇𝒗𝑇 ∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑×𝑑) + 1 ℎ𝑇 ∑︁ 𝐹∈F𝑇 ∥𝒗𝑇 − 𝒗𝐹∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≳ ∥𝒗𝑇 ∥2 S,𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (14) Assuming, moreover, 𝐶f,𝑇 < 1, we also have ∥∇𝒗𝑇 ∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑×𝑑) + 1 ℎ𝑇 ∑︁ 𝐹∈F𝑇 ∥𝒗𝑇 − 𝒗𝐹∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ ∥𝒗𝑇 ∥2 S,𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' For the sake of brevity, we only prove (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The proof of (14) hinges on similar arguments, together with the fact that min(1, 𝐶−1 f,𝑇) ≤ 1, and is left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Taking 𝝉 = ∇𝒗𝑇 in (9), integrating by parts the first term in the right-hand side, and using Cauchy–Schwarz and discrete trace inequalities (see [19, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='32]) as in the proof of [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='25)], we get, after simplifying and raising to the square, ∥∇𝒗𝑇 ∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑×𝑑) ≲ ∥𝑮𝑘 𝑇𝒗𝑇 ∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑×𝑑) + ℎ−1 𝑇 ∑︁ 𝐹∈F𝑇 ∥𝒗𝑇 − 𝒗𝐹∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (16) To estimate the second term, for any 𝐹 ∈ F𝑇, we insert ±(𝝅𝑘 P,𝑇 𝑷𝑘+1 S,𝑇 𝒗𝑇 − 𝝅𝑘 P,𝐹𝑷𝑘+1 S,𝑇 𝒗𝑇) and use triangle inequalities to get ℎ−1 𝑇 ∥𝒗𝑇 − 𝒗𝐹∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ ℎ−1 𝑇 ∥𝒗𝑇 − 𝝅𝑘 P,𝑇 𝑷𝑘+1 S,𝑇 𝒗𝑇 ∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + ℎ−1 𝑇 ∥𝒗𝐹 − 𝝅𝑘 P,𝐹𝑷𝑘+1 S,𝑇 𝒗𝑇 ∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + ℎ−1 𝑇 ∥𝝅𝑘 P,𝐹(𝑷𝑘+1 S,𝑇 𝒗𝑇 − 𝝅𝑘 P,𝑇 𝑷𝑘+1 S,𝑇 𝒗𝑇)∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ ℎ−2 𝑇 ∥𝒗𝑇 − 𝝅𝑘 P,𝑇 𝑷𝑘+1 S,𝑇 𝒗𝑇 ∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + ℎ−1 𝑇 ∥𝒗𝐹 − 𝝅𝑘 P,𝐹𝑷𝑘+1 S,𝑇 𝒗𝑇 ∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + ℎ−2 𝑇 ∥𝑷𝑘+1 S,𝑇 𝒗𝑇 − 𝝅𝑘 P,𝑇 𝑷𝑘+1 S,𝑇 𝒗𝑇 ∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ ℎ−2 𝑇 ∥𝒗𝑇 − 𝑰𝑘 𝑇 𝑷𝑘+1 S,𝑇 𝒗𝑇 ∥2 𝑼,𝑇 + ∥∇𝑷𝑘+1 S,𝑇 𝒗𝑇 ∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑×𝑑) ≲ min(1, 𝐶−1 f,𝑇) ℎ2 𝑇 ∥𝒗𝑇 − 𝑰𝑘 𝑇 𝑷𝑘+1 S,𝑇 𝒗𝑇 ∥2 𝑼,𝑇 + ∥𝑮𝑘 𝑇𝒗𝑇 ∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑×𝑑) = ∥𝒗𝑇 ∥2 S,𝑇, (17) where we have used the 𝐿2-boundedness of 𝝅𝑘 P,𝐹 along with discrete trace inequalities in the second passage, the definition (7) of ∥·∥𝑼,𝑇 along with 𝐶f,𝑇 < 1 for the first two terms and the approximation properties of 𝝅𝑘 P,𝑇 for the last term in the third passage, and concluded noticing that 1 = min(1, 𝐶−1 f,𝑇) and that ∇𝑷𝑘+1 S,𝑇 𝒗𝑇 is by definition the 𝐿2-orthogonal projection of 𝑮𝑘 𝑇𝒗𝑇 on G𝑘 (𝑇)𝑑 (see (10)), so that ∥∇𝑷𝑘+1 S,𝑇 𝒗𝑇 ∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑×𝑑) ≤ ∥𝑮𝑘 𝑇𝒗𝑇 ∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑×𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Plugging (17) into (16) and using the fact that card(F𝑇) ≲ 1 by mesh regularity, we get ∥∇𝒗𝑇 ∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑×𝑑) ≲ ∥𝒗𝑇 ∥2 S,𝑇, which is the sought estimate for the first term in the left-hand side of (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The fact second term is ≲ ∥𝒗𝑇 ∥2 S,𝑇 is an immediate consequence of (17) along with card(F𝑇) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' □ Remark 3 (HHO stabilisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' It is not difficult to check that the bilinear form 𝑼𝑘 𝑇 × 𝑼𝑘 𝑇 ∋ (𝒘𝑇, 𝒗𝑇) ↦→ (𝒘𝑇 − 𝑰𝑘 𝑇 𝑷𝑘+1 S,𝑇 𝒘𝑇, 𝒗𝑇 − 𝑰𝑘 𝑇 𝑷𝑘+1 S,𝑇 𝒗𝑇)𝑼,𝑇 matches [19, Assumption 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' As a matter of fact, this bilinear form is clearly positive-semidefinite, it satisfies the requested seminorm equivalence by (14) and (15), and is polynomially consistent since it only depends on its arguments through the difference operators defined by [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='30)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='3 Darcy term Let again 𝑇 ∈ Tℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The discretisation of the Darcy and coupling terms hinges on the discrete divergence 𝐷𝑘 𝑇 : 𝑼𝑘 𝑇 → P𝑘 (𝑇) such that 𝐷𝑘 𝑇𝒗𝑇 ≔ tr(𝑮𝑘 𝑇𝒗𝑇) ∀𝒗𝑇 ∈ 𝑼𝑘 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (18) Based on this operator, we define the Darcy potential 𝑷𝑘 D,𝑇 : 𝑼𝑘 𝑇 → P𝑘 (𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) such that, for all 𝒗𝑇 ∈ 𝑼𝑘 𝑇 and all (𝑞, 𝒘) ∈ P𝑘+1(𝑇) × Gc,𝑘 (𝑇), ∫ 𝑇 𝑷𝑘 D,𝑇𝒗𝑇 · (∇𝑞 + 𝒘) = − ∫ 𝑇 𝐷𝑘 𝑇𝒗𝑇 𝑞 + ∑︁ 𝐹∈F𝑇 𝜔𝑇𝐹 ∫ 𝐹 (𝒗𝐹 · 𝒏𝐹) 𝑞 + ∫ 𝑇 𝒗𝑇 · 𝒘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (19) This Darcy potential will play a key role in the discretisation of the source term, to ensure that the scheme is fully robust in the whole range of friction coefficients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' see Remark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Remark 4 (Link with DDR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Recall the Discrete De Rham 𝑯(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Ω)-like space 𝑿𝑘 div,ℎ ≔ � 𝒗ℎ = �(𝒗G,𝑇, 𝒗c G,𝑇)𝑇∈Tℎ, (𝑣𝐹)𝐹∈Fℎ � : 𝒗G,𝑇 ∈ G𝑘−1(𝑇) and 𝒗c G,𝑇 ∈ Gc,𝑘 (𝑇) for all 𝑇 ∈ Tℎ, 𝑣𝐹 ∈ P𝑘 (𝐹) for all 𝐹 ∈ Fℎ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Noticing that G𝑘−1(𝑇) ⊂ G𝑘 (𝑇) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (3)), this space injects into 𝑼𝑘 ℎ through the mapping 𝑿𝑘 div,ℎ ∋ 𝒗ℎ ↦→ �(ℜ𝑘 G,𝑇 (𝒗G,𝑇, 𝒗c G,𝑇))𝑇∈Tℎ, (𝑣𝐹𝒏𝐹)𝐹∈Fℎ � ∈ 𝑼𝑘 ℎ, where ℜ𝑘 G,𝑇 : G𝑘 (𝑇) × Gc,𝑘 (𝑇) → P𝑘 (𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) denotes the recovery operator [17, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='17)], which satisfies 𝝅𝑘 G,𝑇ℜ𝑘 G,𝑇 (𝒗G,𝑇, 𝒗c G,𝑇) = 𝒗G,𝑇 and 𝝅c,𝑘 G,𝑇ℜ𝑘 G,𝑇 (𝒗G,𝑇, 𝒗c G,𝑇) = 𝒗c G,𝑇 (where 𝝅c,𝑘 G,𝑇 is the 𝐿2-orthogonal projector on Gc,𝑘 (𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' It can be checked that the discrete divergence (18) and the Darcy potential (19) only depend on the polynomial components shared by 𝑼𝑘 𝑇 and 𝑿𝑘 div,𝑇, and that they coincide with the corresponding DDR operators respectively defined by [17, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='32) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='9)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='10)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Accounting for the previous remark and recalling [17, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='13)], it holds 𝝅𝑘−1 P,𝑇 𝑷𝑘 D,𝑇𝒗𝑇 = 𝝅𝑘−1 P,𝑇𝒗𝑇 ∀𝒗𝑇 ∈ 𝑼𝑘 𝑇, (20) 𝑷𝑘 D,𝑇 𝑰𝑘 𝑇𝒗 = 𝒗 ∀𝒗 ∈ P𝑘 (𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (21) The approximation properties of 𝑷𝑘 D,𝑇 in the 𝐿2-norm have been studied in [17, Theorem 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The following proposition extends the above results to general Hilbert seminorms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Proposition 5 (Approximation properties of the Darcy potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Let an integer 𝑟 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' , 𝑘} be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Then, for all 𝑇 ∈ Tℎ, all 𝒗 ∈ 𝑯𝑟+1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑), and all 𝑚 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' , 𝑟 + 1}, |𝒗 − 𝑷𝑘 D,𝑇 𝑰𝑘 𝑇𝒗|𝑯𝑚(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ ℎ𝑟+1−𝑚 𝑇 |𝒗|𝑯𝑟+1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (22) 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' By [19, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='35], 𝑷𝑘 D,𝑇 ◦ 𝑰𝑘 𝑇 : 𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) → P𝑘 (𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) is a projector owing to (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' By [19, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='43], it then suffices to prove that, for all 𝒗 ∈ 𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑), ∥𝑷𝑘 D,𝑇 𝑰𝑘 𝑇𝒗∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ ∥𝒗∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + ℎ𝑇 |𝒗|𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) if 𝑚 = 0, (23) |𝑷𝑘 D,𝑇 𝑰𝑘 𝑇𝒗|𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ |𝒗|𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) if 𝑚 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (24) To prove (23), it suffices to recall Remark 4 and use [18, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='24) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='28)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' To prove (24), we write |𝑷𝑘 D,𝑇 𝑰𝑘 𝑇𝒗|𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) = |𝑷𝑘 D,𝑇 𝑰𝑘 𝑇 (𝒗 − 𝝅0 P,𝑇𝒗)|𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ ℎ−1 𝑇 ∥𝑷𝑘 D,𝑇 𝑰𝑘 𝑇 (𝒗 − 𝝅0 P,𝑇𝒗)∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='46)] ≲ ℎ−1 𝑇 ∥𝒗 − 𝝅0 P,𝑇𝒗∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + |𝒗 − 𝝅0 P,𝑇𝒗|𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (23) ≲ |𝒗|𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑), where the first line follows using the polynomial consistency (21) of 𝑷𝑘 D,𝑇 to write 0 = |𝝅0 P,𝑇𝒗|𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) = |𝑷𝑘 D,𝑇 𝑰𝑘 𝑇𝝅0 P,𝑇𝒗|𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑), while the conclusion follows from a Poincaré– Wirtinger inequality on the zero-average function 𝒗 − 𝝅0 P,𝑇𝒗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' □ Let �𝑷 𝑘 D,𝑇 : 𝑼𝑘 𝑇 → P𝑘 (𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) be such that �𝑷 𝑘 D,𝑇𝒗𝑇 ≔ ⟨𝐶f,𝑇 < 1⟩𝒗𝑇 + ⟨𝐶f,𝑇 ≥ 1⟩𝑷𝑘 D,𝑇𝒗𝑇 ∀𝒗𝑇 ∈ 𝑼𝑘 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (25) The Darcy term in (2a) is discretised by means of the bilinear form 𝑎𝜈,ℎ : 𝑼𝑘 ℎ × 𝑼𝑘 ℎ → R such that, for all (𝒘ℎ, 𝒗ℎ) ∈ 𝑼𝑘 ℎ × 𝑼𝑘 ℎ, 𝑎𝜈,ℎ(𝒘ℎ, 𝒗ℎ) ≔ ∑︁ 𝑇∈Tℎ 𝜈𝑇𝑎D,𝑇 (𝒘𝑇, 𝒗𝑇) (26) with, for all 𝑇 ∈ Tℎ, 𝑎D,𝑇 (𝒘𝑇, 𝒗𝑇) ≔ ∫ 𝑇 �𝑷 𝑘 D,𝑇𝒘𝑇 ·�𝑷 𝑘 D,𝑇𝒗𝑇 +min(1, 𝐶f,𝑇)(𝒘𝑇 −𝑰𝑘 𝑇 𝑷𝑘 D,𝑇𝒘𝑇, 𝒗𝑇 −𝑰𝑘 𝑇 𝑷𝑘 D,𝑇𝒗𝑇, )𝑼,𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (27) We define the following induced norms: For all 𝒗ℎ ∈ 𝑼𝑘 ℎ, ∥𝒗ℎ∥𝜈,ℎ ≔ 𝑎𝜈,ℎ(𝒗ℎ, 𝒗ℎ) 1/2 and ∥𝒗𝑇 ∥D,𝑇 ≔ 𝑎D,𝑇 (𝒗𝑇, 𝒗𝑇) 1/2 for all 𝑇 ∈ Tℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (28) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='4 Coupling The coupling terms in (2a) and (2b) are discretised by the bilinear form 𝑏ℎ : 𝑼𝑘 ℎ × P𝑘 (Tℎ) → R such that, for all (𝒗ℎ, 𝑞ℎ) ∈ 𝑼𝑘 ℎ × P𝑘 (Tℎ), 𝑏ℎ(𝒗ℎ, 𝑞ℎ) ≔ − ∑︁ 𝑇∈Tℎ ∫ 𝑇 𝐷𝑘 𝑇𝒗𝑇 𝑞𝑇, (29) where 𝑞𝑇 denotes the restriction of 𝑞ℎ to 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Recalling [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='36)], it holds: For all 𝒗 ∈ 𝑯1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑), 𝑏ℎ(𝑰𝑘 ℎ𝒗, 𝑞ℎ) = − ∫ Ω ∇·𝒗 𝑞ℎ ∀𝑞ℎ ∈ P𝑘 (Tℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (30) 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='5 Discrete problem and main results The discrete problem reads: Find (𝒖ℎ, 𝑝ℎ) ∈ 𝑼𝑘 ℎ,0 × 𝑃𝑘 ℎ such that 𝑎𝜇,ℎ(𝒖ℎ, 𝒗ℎ) + 𝑎𝜈,ℎ(𝒖ℎ, 𝒗ℎ) + 𝑏ℎ(𝒗ℎ, 𝑝ℎ) = ∑︁ 𝑇∈Tℎ ∫ 𝑇 𝒇 · �𝑷 𝑘 D,𝑇𝒗𝑇 ∀𝒗ℎ ∈ 𝑼𝑘 ℎ,0, −𝑏ℎ(𝒖ℎ, 𝑞ℎ) = ∫ Ω 𝑔𝑞ℎ ∀𝑞ℎ ∈ 𝑃𝑘 ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (31) The equivalent variational formulation is: Find (𝒖ℎ, 𝑝ℎ) ∈ 𝑼𝑘 ℎ,0 × 𝑃𝑘 ℎ such that Aℎ((𝒖ℎ, 𝑝ℎ), (𝒗ℎ, 𝑞ℎ)) = ∑︁ 𝑇∈Tℎ ∫ 𝑇 𝒇 · �𝑷 𝑘 D,𝑇𝒗𝑇 + ∫ Ω 𝑔𝑞ℎ, (32) with Aℎ : �𝑼𝑘 ℎ × 𝑃𝑘 ℎ �2 → R such that, for all (𝒘ℎ, 𝑟ℎ) and all (𝒗ℎ, 𝑞ℎ) in 𝑼𝑘 ℎ × 𝑃𝑘 ℎ, Aℎ((𝒘ℎ, 𝑟ℎ), (𝒗ℎ, 𝑞ℎ)) ≔ 𝑎𝜇,ℎ(𝒘ℎ, 𝒗ℎ) + 𝑎𝜈,ℎ(𝒘ℎ, 𝒗ℎ) + 𝑏ℎ(𝒗ℎ, 𝑟ℎ) − 𝑏ℎ(𝒘ℎ, 𝑞ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (33) Recalling (13) and (28), we equip the space 𝑼𝑘 ℎ,0 with the following natural energy norm: For all 𝒗ℎ ∈ 𝑼𝑘 ℎ,0, ∥𝒗ℎ∥𝜇,𝜈,ℎ ≔ � ∥𝒗ℎ∥2 𝜇,ℎ + ∥𝒗ℎ∥2 𝜈,ℎ �1/2 (34) and, given a linear form ℓℎ : 𝑼𝑘 ℎ,0 → R, we denote its dual norm by ∥ℓℎ∥𝜇,𝜈,ℎ,∗ ≔ sup 𝒗ℎ∈𝑼𝑘 ℎ,0\\{0} ℓℎ(𝒗ℎ) ∥𝒗ℎ∥𝜇,𝜈,ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The bilinear form 𝑎𝜇,ℎ + 𝑎𝜈,ℎ is ∥·∥𝜇,𝜈,ℎ-coercive with unit coercivity constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The well- posedness of (31) then classically follows from the theory of mixed methods (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=', [19, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='11]) thanks to the inf-sup condition on 𝑏ℎ stated in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Lemma 6 (Inf-sup condition on 𝑏ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Letting 𝛽 ≔ (𝜇 + 𝜈)−1/2, it holds, for all 𝑞ℎ ∈ 𝑃𝑘 ℎ, 𝛽∥𝑞ℎ∥𝐿2(Ω) ≲ sup 𝒗ℎ∈𝑼𝑘 ℎ,0\\{0} 𝑏ℎ(𝒗ℎ, 𝑞ℎ) ∥𝒗ℎ∥𝜇,𝜈,ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' See Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' □ Thanks to the presence of cut-off factors, the following error estimate is robust across the entire range of (local) regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Theorem 7 (Error estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Denote by (𝒖, 𝑝) ∈ 𝑯1 0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) × 𝐿2 0(Ω) the unique solution to the standard weak formulation of (2) and by (𝒖ℎ, 𝑝ℎ) ∈ 𝑼𝑘 ℎ,0 × 𝑃𝑘 ℎ the unique solution of the numerical scheme (31) (or, equivalently, (32)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Then, recalling the notation (5) for the truth value 9 of a logical proposition and assuming, for some 𝑟 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' , 𝑘}, 𝒖 ∈ 𝑯𝑟+2(Tℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑), 𝑝 ∈ 𝐻1(Ω), and, for all 𝑇 ∈ Tℎ, 𝑝 ∈ 𝐻𝑟+1+⟨𝐶f,𝑇 ≥1⟩(𝑇), it holds, ∥𝒖ℎ − 𝑰𝑘 ℎ𝒖∥2 𝜇,𝜈,ℎ + ∥𝑝ℎ − 𝜋𝑘 P,ℎ𝑝∥2 𝐿2(Ω) ≲ 1 𝛾2 � ∑︁ 𝑇∈Tℎ 𝜇𝑇 min(1, 𝐶−1 f,𝑇)ℎ2(𝑟+1) 𝑇 |𝒖|2 𝑯𝑟+2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + ∑︁ 𝑇∈Tℎ 𝜈𝑇 min(1, 𝐶f,𝑇)ℎ2(𝑟+1) 𝑇 |𝒖|2 𝑯𝑟+1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + ∑︁ 𝑇∈Tℎ � 𝜇−1 𝑇 ⟨𝐶f,𝑇 < 1⟩ℎ2(𝑟+1) 𝑇 |𝑝|2 𝐻𝑟+1(𝑇) + 𝜈−1 𝑇 ⟨𝐶f,𝑇 ≥ 1⟩ℎ2(𝑟+1) 𝑇 |𝑝|2 𝐻𝑟+2(𝑇) � � , (35) where 𝛾−2 ≔ 4𝛽−4 + 8𝛽−2 + 1 with 𝛽 as in Lemma 6, while, for all 𝑇 ∈ Tℎ, 𝜈−1 𝑇 ⟨𝐶f,𝑇 ≥ 1⟩ ≔ 0 if 𝜈𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' See Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' □ Remark 8 (Robustness of the error estimate and application to the Darcy problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In the spirit of [10, Remark 13], the presence of the cutoff factors min(1, 𝐶−1 f,𝑇), min(1, 𝐶f,𝑇), 𝜇−1 𝑇 ⟨𝐶f,𝑇 < 1⟩, and 𝜈−1 𝑇 ⟨𝐶f,𝑇 ≥ 1⟩ makes the above estimate robust across the entire range 𝐶f,𝑇 ∈ [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The case 𝐶f,𝑇 = +∞ corresponds to the pure Darcy problem, which is the singular limit obtained assuming minΩ 𝜈 > 0 and 𝐶f,𝑇 = +∞ for all 𝑇 ∈ Tℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In this case, a more in-depth discussion is in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Denoting by 𝛾𝒏 the normal trace operator on 𝜕Ω, the space for the velocity becomes 𝑯0(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Ω) ≔ {𝒗 ∈ 𝑯(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Ω) : 𝛾𝒏(𝒗) = 0 on 𝜕Ω}, and the weak formulation of (2) yields the Darcy problem in mixed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The error estimate (35) remains valid under the regularity assumption 𝒖 ∈ 𝑯𝑟+1(Tℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑), and provided the following conventions are adopted: 𝜇−1 𝑇 ⟨𝐶f,𝑇 < 1⟩ ≔ 0 and, for any 𝒗 ∈ 𝑯0(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Ω) ∩ 𝑯1(Tℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑), all the components of the boundary values of 𝑰𝑘 ℎ𝒗 are forced to zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=', (𝑰𝑘 ℎ𝒗)𝐹 ≔ 0 for all 𝐹 ∈ F b ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Notice that the tangential components of the velocity on boundary faces do not appear in the formulation of the method when 𝜇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' To check this fact: Concerning the Darcy contribution 𝑎D,𝑇 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (27)), recall Remark 4 for the consistent term while, for the stabilisation term, notice that, by (6), boundary faces are not present in (·, ·)𝑼,𝑇 since 𝐶f,𝑇 ≥ 1 for all 𝑇 ∈ Tℎ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Concerning the coupling term 𝑏ℎ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (29)), notice that the following equivalent formula- tion results applying the definition (9) of 𝑮𝑘 𝑇 with 𝝉 = 𝑞𝑇 𝑰𝑑 ≔ (𝑞ℎ)|𝑇 𝑰𝑑 for all 𝑇 ∈ Tℎ: 𝑏ℎ(𝒗ℎ, 𝑞ℎ) = ∑︁ 𝑇∈Tℎ �∫ 𝑇 𝒗𝑇 · ∇𝑞𝑇 − ∑︁ 𝐹∈F𝑇 𝜔𝑇𝐹 ∫ 𝐹 (𝒗𝐹 · 𝒏𝐹) 𝑞𝑇 � , clearly showing that 𝑏ℎ is independent of the tangential component of 𝒗𝐹 for all 𝐹 ∈ Fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The method obtained for the pure Darcy problem has more unknowns than, say, the mixed method of [22] or a similar one that could be obtained starting from the space 𝑿𝑘 div,ℎ of [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In particular, the tangential components of interface unknowns are not present in the consistency term of 𝑎D,𝑇 (see again Remark 4), but are controlled by the stabilisation term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Despite this difference in the discrete space for the flux, the estimate for the error on 𝒖 resulting from (35) in the pure Darcy case is analogous to the one given in [22, Theorem 6] (where the highest regularity case corresponding to 𝑟 = 𝑘 is considered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 10 4 Numerical tests In this section we numerically assess the convergence properties of the scheme (31) for different values of the friction coefficient (including the limit cases) and on both standard and genuinely polyhedral meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The code used for the numerical tests is part of the open source C++ HArDCore3D library;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' see https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='com/jdroniou/HArDCore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In order to reduce the size of the global linear systems, static condensation was applied the scheme (31) in accordance with the principles outlined in [19, Appendix B];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' see [24, Section 6] for a discussion specific to the Stokes equations and [9] for a study of the effect of static condensation on 𝑝-multilevel preconditioners for the Stokes problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' We have chosen to locally eliminate all element degrees of freedom except for the average value of the pressure inside each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The linear systems were solved using the Intel MKL PARDISO library (see https://software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='intel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='com/en-us/mkl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The parameter 𝜆𝑇 in (6) was chosen as ℎ3 𝑇 |𝑇| card(F𝑇), to give a larger weight to the element contribution in (7) when 𝑇 is elongated or has many faces: this compensates the relatively larger contribution, in these circumstances, of the boundary terms in this local norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' We have also applied scalings to the stabilisation terms in (12) and (27): 3 for the Stokes stabilisation, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='3 for the Darcy stabilisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Introducing scalings in the stabilisation terms is not strictly necessary to observe the convergence of the scheme at the expected rates, but we noticed that they improve the magnitudes of the relative errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Understanding the optimal scaling of stabilisations involved in polytopal methods is an ongoing subject of investigation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' here, these numbers were found by quick trial and error on unexpensive tests (low degree 𝑘, coarse meshes), before being used in all the tests below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1 Convergence in various regimes Following [10], we consider a constant viscosity 𝜇 and inverse permeability 𝜈, and we evaluate the relative velocity–pressure error 𝐸𝒖,𝑝 = � ∥𝒖ℎ − 𝑰𝑘 ℎ𝒖∥2 𝜇,𝜈,ℎ + ∥𝑝ℎ − 𝜋𝑘 P,ℎ𝑝∥2 𝐿2(Ω) �1/2 � ∥𝑰𝑘 ℎ𝒖∥2 𝜇,𝜈,ℎ + ∥𝜋𝑘 P,ℎ𝑝∥2 𝐿2(Ω) �1/2 , when the nature of the exact solution (𝒖, 𝑝) is determined by the global friction coefficient 𝐶f,Ω = 𝜈/𝜇, with the convention 𝐶f,Ω = +∞ if 𝜇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' we consider the domain Ω = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 1)3 and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' setting 𝜒S(𝐶f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='Ω) ≔ exp(−𝐶f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' the pressure and velocity are chosen as 𝑝(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑧) = sin(2𝜋𝑥) sin(2𝜋𝑦) sin(2𝜋𝑧) ∀(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑧) ∈ Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒖 = 𝜒S(𝐶f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='Ω)𝒖S + (1 − 𝜒S(𝐶f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='Ω))𝒖D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' where 𝒖S and 𝒖D are the velocity obtained in the Stokes (𝐶f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='Ω = 0) and Darcy (𝐶f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='Ω = +∞) limits,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' and are given by 𝒖S(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑧) = 1 2 ������ sin(2𝜋𝑥) cos(2𝜋𝑦) cos(2𝜋𝑧) cos(2𝜋𝑥) sin(2𝜋𝑦) cos(2𝜋𝑧) −2 cos(2𝜋𝑥) cos(2𝜋𝑦) sin(2𝜋𝑧) ������ ∀(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑧) ∈ Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒖D = � −𝜈−1∇𝑝 if 𝜈 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 11 We notice that ∇·𝒖S = 0 and that 𝜈𝒖D + ∇𝑝 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' these are expected relations, respectively, for a solution of an incompressible Stokes equation, and for a solution of a Darcy equation in mixed form (when gravity is neglected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The meshes used for the test correspond to the families of Voronoi meshes “Voro-small-0”, of tetrahedral meshes “Tetgen-Cube-0” and of random hexahedral meshes “Random-Hexahedra” available on the HArDCore3D repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The errors as a function of ℎ are presented in Figures 1, 2 and 3, showing that the predicted convergence is observed in practice for all the considered mesh families and polynomial degrees, and that both orders of convergence and magnitudes of errors are robust in all regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑘 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑘 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑘 = 2 𝑘 = 3 10−1 100 10−3 10−2 10−1 100 1 1 1 2 1 3 1 4 (a) 𝜇 = 𝜈 = 1 10−1 100 10−3 10−2 10−1 100 1 1 1 2 1 3 1 4 (b) 𝜇 = 1, 𝜈 = 0 10−1 100 10−3 10−2 10−1 100 1 1 1 2 1 3 1 4 (c) 𝜇 = 0, 𝜈 = 1 Figure 1: Voronoi meshes: errors 𝐸𝑢,𝑝 with respect to ℎ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 Lid-driven cavity in porous medium The tests in this section are inspired by situations described in [1, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In these references, a V-crack is realised at the top of a homogeneous porous medium, and plays the role of a lid-driven cavity (with a Stokes-dominated model in this cavity, while the rest of the medium is modelled using pure Darcy flow), and low-order mixed finite elements on triangles/tetrahedra are used to simulate the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 12 𝑘 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑘 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑘 = 2 𝑘 = 3 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='6 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='4 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 100 10−3 10−2 10−1 100 1 1 1 2 1 3 1 4 (a) 𝜇 = 𝜈 = 1 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='6 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='4 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 100 10−3 10−2 10−1 100 101 1 1 1 2 1 3 1 4 (b) 𝜇 = 1, 𝜈 = 0 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='6 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='4 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 100 10−2 10−1 100 1 1 1 2 1 3 1 4 (c) 𝜇 = 0, 𝜈 = 1 Figure 2: Tetrahedral meshes: errors 𝐸𝑢,𝑝 with respect to ℎ We consider here a cavity, where a pure Stokes flow occurs, sitting in a porous medium, with pure Darcy flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' the porous medium is heterogeneous, with permeability equal to 10−7 in the surrounding “box” and 10−2 in a “wedge” at the outset of the cavity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' see Figure 4, left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The domain is Ω = (−1, 2) × (−1, 2) × (−2, 0), with the cavity being (0, 1)3 and the wedge {(𝑥, 𝑦, 𝑧) ∈ R3 : 1 < 𝑥 < 2 , 0 < 𝑦 < 1 , −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='75(𝑥 − 1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='25 < 𝑧 < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The domain has been meshed using gmsh (https://gmsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='info/), with cubic elements in the cavity, and mostly tetrahedral elements in the porous medium (together with a few pyramidal elements at the junctions cavity– porous medium);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' see Figure 4, right, for an example of mesh, and Table 1 for the characteristic of all meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The files describing the geometry are available in the HArDCore repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The forcing term 𝒇 = (0, 0, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='98) represents the gravity, while we fix 𝑔 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The boundary conditions on the velocity are 𝒖(𝑥, 𝑦, 𝑧) = (𝑥(1 − 𝑥), 0, 0) on top of the cavity, and 𝒖 = 0 elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Figure 5 presents the streamlines obtained on the third mesh in the family with 𝑘 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' These streamlines show the usual form of circulation inside the cavity for a pure Stokes lid-driven cavity, which drives some (slower) motion inside the wedge section of the porous medium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' given the very low permeability of the rest of the medium, little material is transferred 13 𝑘 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑘 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑘 = 2 𝑘 = 3 10−1 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='8 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='6 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='4 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 10−4 10−3 10−2 10−1 100 1 1 1 2 1 3 1 4 (a) 𝜇 = 𝜈 = 1 10−1 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='8 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='6 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='4 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 10−4 10−3 10−2 10−1 100 1 1 1 2 1 3 1 4 (b) 𝜇 = 1, 𝜈 = 0 10−1 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='8 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='6 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='4 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 10−3 10−2 10−1 100 1 1 1 2 1 3 1 4 (c) 𝜇 = 0, 𝜈 = 1 Figure 3: Random hexahedral meshes: errors 𝐸𝑢,𝑝 with respect to ℎ into this medium, in which the velocity remains almost zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' in the region 𝑧 < −1 below the cavity, for example, the maximum of the vertex values (obtained by averaging the potential reconstructions in each element surrounding the vertices) of the velocity is below 6 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' To qualitatively assess the impact of increasing the degree of approximation 𝑘 of the method, we evaluate for various meshes and degrees the flux across the interface Γ = {0} × (0, 1) × (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='75, 1) between the cavity and the wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' All the meshes Mℎ we consider are compatible with this interface, that is, setting Γℎ = {𝐹 ∈ Fℎ : 𝐹 ⊂ Γ} we have Γ = ∪𝐹∈Γℎ𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' We then consider the numerical convergence of the numerical flux defined by ∑︁ 𝐹∈Γℎ ∫ 𝐹 𝒖𝐹 · 𝒏Γ, where 𝒏Γ = (1, 0, 0) is the unit normal to Γ pointing inside the wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The values of this flux for different degrees of approximations 𝑘 are provided in Figure 6 (left: w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' the mesh size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' right: w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' the total wall time, including assembly and solution time – notice that the HArDCore library uses multi-threading processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' These results show that the lowest order of 14 Figure 4: Left: geometry of the cavity (green) inside the porous medium, comprising a wedge (green) and the surrounding box (shadow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Right: example of mesh used in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Mesh index 1 2 3 4 5 Mesh size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='17 Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' of elements 1,326 5,935 7,963 99,748 201,653 Table 1: Characteristics of the mesh family for the tests in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' approximation struggles to provide what seems to be a correct value of the flux, and that the mesh must be extremely fine to get close to this value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' on the contrary, for 𝑘 ≥ 1, all results, even on coarse meshes and with a low computational cost, seem to be very close to a given value, indicating that convergence has already occurred in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' These results corroborate a conclusion already highlighted in [3]: even on a problem where the solution is not expected to be very regular, slightly increasing the order of approximation of the scheme (here, going from 𝑘 = 0 to 𝑘 = 1) can lead to a vastly improved accuracy of the numerical outputs at a very low computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 5 Analysis 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1 Stability Proposition 9 (∥·∥𝜇,𝜈,ℎ-boundedness of the interpolator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' With 𝛽 as in Lemma 6, it holds, for all 𝒗 ∈ 𝑯1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑), 𝛽∥𝑰𝑘 ℎ𝒗∥𝜇,𝜈,ℎ ≲ ∥𝒗∥𝑯1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (36) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' It holds, by definition, ∥𝑰𝑘 ℎ𝒗∥2 𝜇,𝜈,ℎ = � 𝑇∈Tℎ [𝜇𝑇𝔗1(𝑇) + 𝜈𝑇𝔗2(𝑇) + 𝜈𝑇𝔗3(𝑇)] with 𝔗1(𝑇) ≔ ∥𝑮𝑘 𝑇 𝑰𝑘 𝑇𝒗∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑×𝑑) + min(1, 𝐶−1 f,𝑇) ℎ2 𝑇 ∥𝑰𝑘 𝑇 (𝒗 − 𝑷𝑘+1 S,𝑇 𝑰𝑘 𝑇𝒗)∥2 𝑼,𝑇, 𝔗2(𝑇) ≔ ∥�𝑷 𝑘 D,𝑇 𝑰𝑘 𝑇𝒗∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑), 𝔗3(𝑇) ≔ min(1, 𝐶f,𝑇)∥𝑰𝑘 𝑇 (𝒗 − 𝑷𝑘 D,𝑇 𝑰𝑘 𝑇𝒗)∥2 𝑼,𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' For the first term, combining (14) and [19, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='25)], we obtain 𝔗1(𝑇) ≲ |𝒗|2 𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' For 15 X Axis 2 1 0 0 1 0 Z Axis -1 + 2 Y Axis 0 Y Axis 1 2 01 1 X Axis 0 1Figure 5: Streamlines for the test case of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 (cavity and wedge displayed in shadow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='5e-01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 velocity Magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='0e+00𝑘 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑘 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝑘 = 2 𝑘 = 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='8 1 0 1 2 3 4 10−8 (a) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' mesh size 100 101 102 103 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='5 10−8 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='5 (b) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' wall time (seconds) Figure 6: Convergence of flux values from the cavity to the wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' the second term, if 𝐶f,𝑇 < 1, we can write 𝔗2(𝑇) = ∥𝝅𝑘 P,𝑇𝒗∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≤ ∥𝒗∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) using the boundedness of 𝝅𝑘 P,𝑇, while, if 𝐶f,𝑇 ≥ 1, (23) gives 𝔗2(𝑇) ≲ ∥𝒗∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + ℎ2 𝑇 |𝒗|2 𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≤ ∥𝒗∥2 𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑), where the conclusion follows observing that ℎ𝑇 ≤ 1 since Ω has unit diameter by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Finally, for the third term, the boundedness (8) of the interpolator in the ∥·∥𝑼,𝑇- norm followed by the approximation properties (22) of 𝑷𝑘 D,𝑇 ◦ 𝑰𝑘 𝑇 with (𝑟, 𝑚) = (0, 0) and (𝑟, 𝑚) = (0, 1) yield 𝔗3(𝑇) ≲ ∥𝒗 − 𝑷𝑘 D,𝑇 𝑰𝑘 𝑇𝒗∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + ℎ2 𝑇 |𝒗 − 𝑷𝑘 D,𝑇 𝑰𝑘 𝑇𝒗|2 𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ ℎ2 𝑇 |𝒗|2 𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Gathering the above estimates and recalling the bounds (1) on 𝜇 and 𝜈, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' □ Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Classical consequence of the continuous inf-sup condition for the divergence ∇· : 𝑯1 0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) → 𝐿2 0(Ω) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=', [8, 25, 27, 33]) along with the Fortin properties for the interpolator corresponding to (30) and (36);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=', [7, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='3] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 Convergence The purpose of this section is to prove Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The proof rests on consistency results for the Stokes, Darcy, and coupling bilinear forms as well as the forcing term linear form which make the object of the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1 Consistency of the Stokes bilinear form Lemma 10 (Consistency of the Stokes bilinear form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Given 𝒘 ∈ 𝑯1 0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) such that ∇·(𝜇∇𝒘) ∈ 𝑳2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑), let the Stokes consistency error linear form E𝑘 S,ℎ(𝒘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' ·) : 𝑼𝑘 ℎ,0 → R be such that, for all 𝒗ℎ ∈ 𝑼𝑘 ℎ,0, E𝑘 S,ℎ(𝒘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒗ℎ) ≔ − ∑︁ 𝑇∈Tℎ ∫ 𝑇 ∇·(𝜇𝑇∇𝒘) · 𝒗𝑇 − 𝑎𝜇,ℎ(𝑰𝑘 ℎ𝒘, 𝒗ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (37) 17 Then, further assuming 𝒘 ∈ 𝑯𝑟+2(Tℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) for some 𝑟 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' , 𝑘}, it holds ∥E𝑘 S,ℎ(𝒘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' ·)∥𝜇,𝜈,ℎ,∗ ≲ � ∑︁ 𝑇∈Tℎ 𝜇𝑇 min(1, 𝐶−1 f,𝑇)ℎ2(𝑟+1) 𝑇 |𝒘|2 𝑯𝑟+2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (38) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Let 𝒗ℎ ∈ 𝑼𝑘 ℎ,0 \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Proceeding as in [19, Point (ii) in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='18] using an integration by parts for the first term in the definition of E𝑘 S,ℎ along with the definitions (11) of 𝑎𝜇,ℎ and (9) of 𝑮𝑘 𝑇 for the second term, we get the following reformulation of the error: E𝑘 S,ℎ(𝒘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒗ℎ) = ∑︁ 𝑇∈Tℎ ∑︁ 𝐹∈F𝑇 𝜔𝑇𝐹 ∫ 𝐹 𝜇𝑇 (∇𝒘 − 𝑮𝑘 𝑇 𝑰𝑘 𝑇𝒘)𝒏𝐹 · (𝒗𝐹 − 𝒗𝑇) − ∑︁ 𝑇∈Tℎ 𝜇𝑇 min(1, 𝐶−1 f,𝑇) ℎ2 𝑇 (𝑰𝑘 𝑇 (𝒘 − 𝑷𝑘+1 S,𝑇 𝑰𝑘 𝑇𝒘), 𝒗𝑇 − 𝑰𝑘 𝑇 𝑷𝑘+1 S,𝑇 𝒗𝑇)𝑼,𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Using Cauchy–Schwarz and Hölder inequalities along with ∥𝒏𝐹∥𝑳∞(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≤ 1 for all 𝐹 ∈ Fℎ, we can write E𝑘 S,ℎ(𝒘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒗ℎ) ≲ ∑︁ 𝑇∈Tℎ [𝔗1(𝑇) + 𝔗2(𝑇)] (39) with 𝔗1(𝑇) ≔ 𝜇 1/2 𝑇 ℎ 1/2 𝑇 ∥∇𝒘 − 𝑮𝑘 𝑇 𝑰𝑘 𝑇𝒘∥𝑳2(𝜕𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑×𝑑) � 𝜇𝑇 ℎ𝑇 ∑︁ 𝐹∈F𝑇 ∥𝒗𝐹 − 𝒗𝑇 ∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) �1/2 , 𝔗2(𝑇) ≔ 𝜇𝑇 min(1, 𝐶−1 f,𝑇) ℎ2 𝑇 ∥𝑰𝑘 𝑇 (𝒘 − 𝑷𝑘+1 S,𝑇 𝑰𝑘 𝑇𝒘)∥𝑼,𝑇 ∥𝒗𝑇 − 𝑰𝑘 𝑇 𝑷𝑘+1 S,𝑇 𝒗𝑇 ∥𝑼,𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Let us estimate 𝔗1(𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Recalling that 𝑮𝑘 𝑇 ◦𝑰𝑘 𝑇 = 𝝅𝑘 P,𝑇 and using the approximation properties of this projector (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [15] and [19, Chapter 1] concerning the extension to non-star-shaped elements), it is readily inferred for the first factor 𝜇 1/2 𝑇 ℎ 1/2 𝑇 ∥∇𝒘 − 𝑮𝑘 𝑇 𝑰𝑘 𝑇𝒘∥𝑳2(𝜕𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑×𝑑) ≲ 𝜇 1/2 𝑇 ℎ𝑟+1 𝑇 |𝒘|𝑯𝑟+2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The estimate of the second factor depends on the regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' If 𝐶f,𝑇 < 1, using (15) we write 𝜇𝑇 ℎ𝑇 ∑︁ 𝐹∈F𝑇 ∥𝒗𝐹 − 𝒗𝑇 ∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ 𝜇𝑇 ∥𝒗𝑇 ∥2 S,𝑇 = 𝜇𝑇 min(1, 𝐶−1 f,𝑇)∥𝒗𝑇 ∥2 S,𝑇, (40) where the conclusion follows observing that 1 = min(1, 𝐶−1 f,𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' If, on the other hand, 𝐶f,𝑇 ≥ 1 (which implies, in particular, 𝜈𝑇 > 0), we insert ±𝑷𝑘 D,𝑇𝒗𝑇 into the norm and use triangle and discrete trace inequalities to write 𝜇𝑇 ℎ𝑇 ∑︁ 𝐹∈F𝑇 ∥𝒗𝐹 − 𝒗𝑇 ∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ 𝜈𝑇𝐶−1 f,𝑇 � ℎ𝑇 ∑︁ 𝐹∈F𝑇 ∥𝒗𝐹 − 𝑷𝑘 D,𝑇𝒗𝑇 ∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + ∥𝒗𝑇 − 𝑷𝑘 D,𝑇𝒗𝑇 ∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) � 18 ≲ 𝜈𝑇𝐶−1 f,𝑇 � ℎ𝑇 ∑︁ 𝐹∈F𝑇 ∩F b ℎ ∥𝒗𝐹 − 𝑷𝑘 D,𝑇𝒗𝑇 ∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + ∥𝒗𝑇 − 𝑰𝑘 𝑇 𝑷𝑘 D,𝑇𝒗𝑇 ∥2 𝑼,𝑇 � where we have additionally used the definition (4) of 𝐶f,𝑇 in the first inequality, and continued invoking the definition of ∥·∥𝑼,𝑇 (see (6)–(7)) to bound the element term and the non-boundary face terms in the second line by ∥𝒗𝑇 − 𝑰𝑘 𝑇 𝑷𝑘 D,𝑇𝒗𝑇 ∥2 𝑼,𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' For all 𝐹 ∈ F𝑇 ∩ F b ℎ , we have 𝒗𝐹 = 0 by definition of 𝑼𝑘 ℎ,0 and, using discrete trace inequalities and the mesh regularity to write card(F𝑇) ≲ 1, we infer that 𝜇𝑇 ℎ𝑇 ∑︁ 𝐹∈F𝑇 ∥𝒗𝐹 − 𝒗𝑇 ∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ 𝜈𝑇𝐶−1 f,𝑇 � ∥𝑷𝑘 D,𝑇𝒗𝑇 ∥2 𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + ∥𝒗𝑇 − 𝑰𝑘 𝑇 𝑷𝑘 D,𝑇𝒗𝑇 ∥2 𝑼,𝑇 � ≲ 𝜈𝑇𝐶−1 f,𝑇 ∥𝒗𝑇 ∥2 D,𝑇, where the last passage follows recalling the definitions (28) of the ∥·∥D,𝑇-norm, (25) of �𝑷 𝑘 D,𝑇 (which is equal to 𝑷𝑘 D,𝑇 since 𝐶f,𝑇 ≥ 1), and observing that 1 = min(1, 𝐶f,𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Hence, further observing that 𝐶−1 f,𝑇 = min(1, 𝐶−1 f,𝑇), we can go on writing 𝜇𝑇 ℎ𝑇 ∑︁ 𝐹∈F𝑇 ∥𝒗𝐹 − 𝒗𝑇 ∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ 𝜈𝑇 min(1, 𝐶−1 f,𝑇)∥𝒗𝑇 ∥2 D,𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (41) Gathering (40) and (41), we arrive at 𝔗1(𝑇) ≲ 𝜇 1/2 𝑇 min(1, 𝐶−1 f,𝑇) 1/2ℎ𝑟+1 𝑇 |𝒘|𝑯𝑟+2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) � 𝜇𝑇 ∥𝒗𝑇 ∥2 S,𝑇 + 𝜈𝑇 ∥𝒗𝑇 ∥2 D,𝑇 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (42) Moving to 𝔗2(𝑇), using the ∥·∥𝑼,𝑇-boundedness (8) of 𝑰𝑘 𝑇 followed by the approximation properties of 𝑷𝑘+1 S,𝑇 ◦ 𝑰𝑘 𝑇 (consequence, for each of its components, of [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='14) and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='48]), we have ∥𝑰𝑘 𝑇 (𝒘−𝑷𝑘+1 S,𝑇 𝑰𝑘 𝑇𝒘)∥𝑼,𝑇 ≲ ∥𝒘−𝑷𝑘+1 S,𝑇 𝑰𝑘 𝑇𝒘∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑)+ℎ𝑇 |𝒘−𝑷𝑘+1 S,𝑇 𝑰𝑘 𝑇𝒘|𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ ℎ𝑟+2 𝑇 |𝒘|𝑯𝑟+2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Plugging this estimate into the definition of 𝔗2(𝑇) and recalling the definition (13) of ∥·∥S,𝑇, we get 𝔗2(𝑇) ≲ 𝜇 1/2 𝑇 min(1, 𝐶−1 f,𝑇) 1/2ℎ𝑟+1 𝑇 |𝒘|𝑯𝑟+2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) 𝜇 1/2 𝑇 ∥𝒗𝑇 ∥S,𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (43) Using (42) and (43) to estimate the right-hand side of (39), we obtain E𝑘 S,ℎ(𝒘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒗ℎ) ≲ ∑︁ 𝑇∈Tℎ 𝜇 1/2 𝑇 min(1, 𝐶−1 f,𝑇) 1/2ℎ𝑟+1 𝑇 |𝒘|𝑯𝑟+2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) � 𝜇𝑇 ∥𝒗𝑇 ∥2 S,𝑇 + 𝜈𝑇 ∥𝒗𝑇 ∥2 D,𝑇 �1/2 ≤ � ∑︁ 𝑇∈Tℎ 𝜇𝑇 min(1, 𝐶−1 f,𝑇)ℎ2(𝑟+1) 𝑇 |𝒘|2 𝑯𝑟+2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) �1/2 ∥𝒗ℎ∥𝜇,𝜈,ℎ, where the conclusion follows using a discrete Cauchy–Schwarz inequality on the sum over 𝑇 ∈ Tℎ along with the definition (34) of ∥·∥𝜇,𝜈,ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Dividing by ∥𝒗ℎ∥𝜇,𝜈,ℎ and passing to the supremum concludes the proof of (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' □ 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2 Consistency of the Darcy bilinear form Lemma 11 (Consistency of the Darcy bilinear form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Given 𝒘 ∈ 𝑯1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑), let the Darcy consistency error linear form E𝑘 D,ℎ(𝒘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' ·) : 𝑼𝑘 ℎ → R be such that, for all 𝒗ℎ ∈ 𝑼𝑘 ℎ, E𝑘 D,ℎ(𝒘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒗ℎ) ≔ ∑︁ 𝑇∈Tℎ ∫ 𝑇 𝜈𝑇𝒘 · �𝑷 𝑘 D,𝑇𝒗𝑇 − 𝑎𝜈,ℎ(𝑰𝑘 ℎ𝒘, 𝒗ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (44) Then, further assuming 𝒘 ∈ 𝑯𝑟+1(Tℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) for some 𝑟 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' , 𝑘}, it holds ∥E𝑘 D,ℎ(𝒘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' ·)∥𝜇,𝜈,ℎ,∗ ≲ � ∑︁ 𝑇∈Tℎ 𝜈𝑇 min(1, 𝐶f,𝑇)ℎ2(𝑟+1) 𝑇 |𝒘|2 𝑯𝑟+1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (45) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Let 𝒗ℎ ∈ 𝑼𝑘 ℎ,0 \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Expanding 𝑎𝜈,ℎ according to its definition (26), we get E𝑘 D,ℎ(𝒘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒗ℎ) = ∑︁ 𝑇∈Tℎ [𝔗1(𝑇) + 𝔗2(𝑇)] , (46) with 𝔗1(𝑇) ≔ ∫ 𝑇 𝜈𝑇 (𝒘 − �𝑷 𝑘 D,𝑇 𝑰𝑘 𝑇𝒘) · �𝑷 𝑘 D,𝑇𝒗𝑇, 𝔗2(𝑇) ≔ −𝜈𝑇 min(1, 𝐶f,𝑇)(𝑰𝑘 𝑇 (𝒘 − 𝑷𝑘 D,𝑇 𝑰𝑘 𝑇𝒘), 𝒗𝑇 − 𝑰𝑘 𝑇 𝑷𝑘 D,𝑇𝒗𝑇)𝑼,𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The estimate of 𝔗1(𝑇) depends on the regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Let us start with the case 𝐶f,𝑇 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Recalling (25) to replace �𝑷 𝑘 D,𝑇 with 𝑷𝑘 D,𝑇 and applying a Cauchy–Schwarz inequality, we get |𝔗1(𝑇)| ≲ 𝜈𝑇 ∥𝒘 − 𝑷𝑘 D,𝑇 𝑰𝑘 𝑇𝒘∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑)∥𝑷𝑘 D,𝑇𝒗𝑇 ∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ 𝜈 1/2 𝑇 min(1, 𝐶f,𝑇) 1/2ℎ𝑟+1 𝑇 |𝒘|𝑯𝑟+1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) 𝜈 1/2 𝑇 ∥𝒗𝑇 ∥D,𝑇, where, to pass to the second line, we have used the approximation properties (22) of 𝑷𝑘 D,𝑇 ◦ 𝑰𝑘 𝑇 with 𝑚 = 0, the definition (28) of the ∥·∥D,𝑇-norm, and observed that 1 = min(1, 𝐶f,𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Let us now consider the case 𝐶f,𝑇 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Recalling that �𝑷 𝑘 D,𝑇 𝑰𝑘 𝑇𝒘 = 𝝅𝑘 P,𝑇𝒘 in this case, we can write 𝔗1(𝑇) = ∫ 𝑇 𝜈𝑇 (𝒘 − 𝝅𝑘 P,𝑇𝒘) · (𝒗𝑇 − 𝝅0 P,𝑇𝒗𝑇) and, using Cauchy–Schwarz inequalities, continue with |𝔗1(𝑇)| ≤ 𝜈𝑇 ∥𝒘 − 𝝅𝑘 P,𝑇𝒘∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑)∥𝒗𝑇 − 𝝅0 P,𝑇𝒗𝑇 ∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (47) Using the approximation properties of 𝝅𝑘 P,𝑇, it is readily inferred that the first factor is ≲ ℎ𝑟+1 𝑇 |𝒘|𝑯𝑟+1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' To estimate the last factor, we use a Poincaré–Wirtinger inequality to write ∥𝒗𝑇 − 𝝅0 P,𝑇𝒗𝑇 ∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ ℎ𝑇 ∥∇𝒗𝑇 ∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑×𝑑) ≲ ℎ𝑇 ∥𝒗𝑇 ∥S,𝑇, where the conclusion follows from (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Plugging the above estimates into (47), we can go on writing |𝔗1(𝑇)| ≲ 𝜈 1/2 𝑇 ℎ𝑟+1 𝑇 |𝒘|𝑯𝑟+1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) 𝜈 1/2 𝑇 ℎ𝑇 ∥𝒗𝑇 ∥S,𝑇 = 𝜈 1/2 𝑇 ℎ𝑟+1 𝑇 |𝒘|𝑯𝑟+1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) 𝜇 1/2 𝑇 𝐶 1/2 f,𝑇 ∥𝒗𝑇 ∥S,𝑇, = 𝜈 1/2 𝑇 min(1, 𝐶f,𝑇) 1/2ℎ𝑟+1 𝑇 |𝒘|𝑯𝑟+1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) 𝜇 1/2 𝑇 ∥𝒗𝑇 ∥S,𝑇, 20 where we have used the definition (4) of 𝐶f,𝑇 to pass to the second line and, after rearranging the factors, the fact that 𝐶f,𝑇 = min(1, 𝐶f,𝑇) to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Gathering the above estimates, we thus have |𝔗1(𝑇)| ≲ 𝜈 1/2 𝑇 min(1, 𝐶f,𝑇) 1/2ℎ𝑟+1 𝑇 |𝒘|𝑯𝑟+1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) � 𝜇𝑇 ∥𝒗𝑇 ∥2 S,𝑇 + 𝜈𝑇 ∥𝒗𝑇 ∥2 D,𝑇 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (48) To estimate 𝔗2(𝑇), we use a Cauchy–Schwarz inequality to write |𝔗2(𝑇)| ≤ 𝜈 1/2 𝑇 min(1, 𝐶f,𝑇) 1/2∥𝑰𝑘 𝑇 (𝒘 − 𝑷𝑘 D,𝑇 𝑰𝑘 𝑇𝒘)∥𝑼,𝑇 𝜈 1/2 𝑇 min(1, 𝐶f,𝑇) 1/2∥𝒗𝑇 − 𝑰𝑘 𝑇 𝑷𝑘 D,𝑇𝒗𝑇 ∥𝑼,𝑇 ≲ 𝜈 1/2 𝑇 min(1, 𝐶f,𝑇) 1/2 � ∥𝒘 − 𝑷𝑘 D,𝑇 𝑰𝑘 𝑇𝒘∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) + ℎ𝑇 |𝒘 − 𝑷𝑘 D,𝑇 𝑰𝑘 𝑇𝒘|𝑯1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) � 𝜈 1/2 𝑇 ∥𝒗𝑇 ∥D,𝑇 ≲ 𝜈 1/2 𝑇 min(1, 𝐶f,𝑇) 1/2ℎ𝑟+1 𝑇 |𝒘|𝑯𝑟+1(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) 𝜈 1/2 𝑇 ∥𝒗𝑇 ∥D,𝑇, (49) where we have used the ∥·∥𝑼,𝑇-boundedness (8) of 𝑰𝑘 𝑇 along with the definition (28) of the ∥·∥D,𝑇-norm in the second inequality and the approximation properties (22) of 𝑷𝑘 D,𝑇 ◦ 𝑰𝑘 𝑇 with 𝑚 = 0 and 𝑚 = 1 to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Plugging (48) and (49) into (46), using discrete Cauchy–Schwarz inequalities, dividing by ∥𝒗ℎ∥𝜇,𝜈,ℎ, and passing to the supremum, the conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='3 Consistency of the coupling bilinear form The quantity estimated in the following lemma can be interpreted as an adjoint consistency error for the discrete divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Lemma 12 (Consistency of the coupling bilinear form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Given 𝑞 ∈ 𝐻1(Ω), let the coupling consistency error linear form E𝑘 c,ℎ(𝑞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' ·) : 𝑼𝑘 ℎ,0 → R be such that, for all 𝒗ℎ ∈ 𝑼𝑘 ℎ,0, E𝑘 c,ℎ(𝑞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒗ℎ) ≔ ∑︁ 𝑇∈Tℎ ∫ 𝑇 ∇𝑞 · �𝑷 𝑘 D,𝑇𝒗𝑇 − 𝑏ℎ(𝒗ℎ, 𝜋𝑘 P,ℎ𝑞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (50) Then, further assuming, for some 𝑟 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' , 𝑘}, 𝑞 ∈ 𝐻𝑟+1+⟨𝐶f,𝑇 ≥1⟩(𝑇) for all 𝑇 ∈ Tℎ, it holds ∥E𝑘 c,ℎ(𝑞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' ·)∥𝜇,𝜈,ℎ,∗ ≲ � ∑︁ 𝑇∈Tℎ � 𝜇−1 𝑇 ⟨𝐶f,𝑇 < 1⟩ℎ2(𝑟+1) 𝑇 |𝑞|2 𝐻𝑟+1(𝑇) + 𝜈−1 𝑇 ⟨𝐶f,𝑇 ≥ 1⟩ℎ2(𝑟+1) 𝑇 |𝑞|2 𝐻𝑟+2(𝑇) �� 1/2 , (51) where 𝜈−1 𝑇 ⟨𝐶f,𝑇 ≥ 1⟩ ≔ 0 if 𝜈𝑇 = 0 as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Let 𝒗ℎ ∈ 𝑼𝑘 ℎ,0 \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' We start by noticing that, expanding the bilinear form 𝑏ℎ according to its definition (29), E𝑘 c,ℎ(𝑞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒗ℎ) = ∑︁ 𝑇∈Tℎ �∫ 𝑇 ∇𝑞 · �𝑷 𝑘 D,𝑇𝒗𝑇 + ∫ 𝑇 𝜋𝑘 P,𝑇𝑞 𝐷𝑘 𝑇𝒗𝑇 − ∑︁ 𝐹∈F𝑇 𝜔𝑇𝐹 ∫ 𝐹 𝑞 (𝒗𝐹 · 𝒏𝐹) � , (52) where the insertion of the last term in parenthesis is made possible by the single-valuedness of 𝑞 (𝒗𝐹 · 𝒏𝐹) at interfaces along with the fact that 𝒗𝐹 · 𝒏𝐹 = 0 for all 𝐹 ∈ F b ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Denote by 𝔗(𝑇) the 21 argument of the summation in (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' To estimate this quantity, we distinguish two cases based on the value of 𝐶f,𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' If 𝐶f,𝑇 < 1, �𝑷 𝑘 D,𝑇𝒗𝑇 = 𝒗𝑇 by (25), so that 𝔗(𝑇) = ∫ 𝑇 ∇𝑞 · 𝒗𝑇 + ∫ 𝑇 𝜋𝑘 P,𝑇𝑞 𝐷𝑘 𝑇𝒗𝑇 − ∑︁ 𝐹∈F𝑇 𝜔𝑇𝐹 ∫ 𝐹 𝑞 (𝒗𝐹 · 𝒏𝐹).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Thus, proceeding as in the derivation of [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='41)], we get 𝔗(𝑇) ≲ ℎ𝑟+1 𝑇 |𝑞|𝐻𝑟+1(𝑇) � 1 ℎ𝑇 ∑︁ 𝐹∈F𝑇 ∥𝒗𝑇 − 𝒗𝐹∥2 𝑳2(𝐹;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) �1/2 ≲ 𝜇−1/2 𝑇 ⟨𝐶f,𝑇 < 1⟩ 1/2ℎ𝑟+1 𝑇 |𝑞|𝐻𝑟+1(𝑇) 𝜇 1/2 𝑇 ∥𝒗𝑇 ∥S,𝑇, (53) where we have used (15) to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' If 𝐶f,𝑇 ≥ 1, on the other hand, we have �𝑷 𝑘 D,𝑇 = 𝑷𝑘 D,𝑇 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (25)), so that 𝔗(𝑇) = �∫ 𝑇 ∇𝑞 · 𝑷𝑘 D,𝑇𝒗𝑇 + ∫ 𝑇 𝜋𝑘 P,𝑇𝑞 𝐷𝑘 𝑇𝒗𝑇 − ∑︁ 𝐹∈F𝑇 𝜔𝑇𝐹 ∫ 𝐹 𝑞 (𝒗𝐹 · 𝒏𝐹) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Using the definition (19) of 𝑷𝑘 D,𝑇 to proceed as in [17, Theorem 11], we get 𝔗(𝑇) ≲ ℎ𝑟+1 𝑇 |𝑞|𝐻𝑟+2(𝑇) ∥𝒗𝑇 ∥D,𝑇 ≤ 𝜈−1/2 𝑇 ⟨𝐶f,𝑇 ≥ 1⟩ 1/2ℎ𝑟+1 𝑇 |𝑞|𝐻𝑟+2(𝑇) 𝜈 1/2 𝑇 ∥𝒗𝑇 ∥D,𝑇, (54) where we have additionally noticed that 𝐶f,𝑇 ≥ 1 implies 𝜈𝑇 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' To conclude, we plug (53) and (54) into (52), use a Cauchy–Schwarz inequality on the sum over 𝑇 ∈ Tℎ, recall the definition (34) of the ∥·∥𝜇,𝜈,ℎ-norm, and pass to the supremum after dividing by ∥𝒗ℎ∥𝜇,𝜈,ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' □ Remark 13 (Discretisation of the source term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The use of 𝑷𝑘 D,𝑇 in the discretisation of the source term when 𝐶f,𝑇 ≥ 1 (see (32) and (25)) is crucial to ensure that, in this case, the consistency error of the coupling bilinear form can be bounded above using the Darcy norm instead of the Stokes norm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' compare (54) and (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' This bound is key to establishing an error estimate in ℎ𝑟+1 that remains robust in the Darcy limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='4 Consistency of the forcing term linear form The following lemma estimates the difference between the standard HHO right-hand side linear form and the one obtained, as in (2a), using �𝑷 𝑘 D,𝑇𝒗𝑇 instead of 𝒗𝑇 as a test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Lemma 14 (Consistency of the forcing term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' For any 𝝋 ∈ 𝑳2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑), define the right-hand side consistency error linear form E𝑘 rhs,ℎ(𝝋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' ·) : 𝑼𝑘 ℎ → R such that, for all 𝒗ℎ ∈ 𝑼𝑘 ℎ, E𝑘 rhs,ℎ(𝝋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒗ℎ) ≔ ∑︁ 𝑇∈Tℎ ∫ 𝑇 𝝋 · (𝒗𝑇 − �𝑷 𝑘 D,𝑇𝒗𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (55) Further assuming 𝝋 ∈ 𝑯𝑟(Tℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' R𝑑) for some 𝑟 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' , 𝑘}, it holds ∥E𝑘 rhs,ℎ(𝝋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' ·)∥𝜇,𝜈,ℎ,∗ ≲ � ∑︁ 𝑇∈Tℎ 𝜇−1 𝑇 min(1, 𝐶−1 f,𝑇)ℎ2(𝑟+1) 𝑇 |𝝋|2 𝑯𝑟 (𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (56) 22 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Denote by 𝔗(𝑇) the argument of the summation in (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' If 𝐶f,𝑇 < 1, the definition (25) of �𝑷 𝑘 D,𝑇 yields 𝔗(𝑇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Consider now the case 𝐶f,𝑇 ≥ 1 (which implies, in particular, 𝜈𝑇 > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' We first notice that, letting 𝝅𝑘−1 P,𝑇 𝝋 ≔ 0 if 𝑘 = 0, ∥𝝋 − 𝝅𝑘−1 P,𝑇 𝝋∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ ℎ𝑟 𝑇 |𝝋|𝑯𝑟 (𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑), (57) where the result is trivial if 𝑘 = 0 (which imposes 𝑟 = 0) and otherwise follows from the approximation properties of 𝝅𝑘−1 P,𝑇, see [19, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Recalling that, for 𝐶f,𝑇 ≥ 1, we have �𝑷 𝑘 D,𝑇 = 𝑷𝑘 D,𝑇 by (25) and invoking (20) (which trivially holds also for 𝑘 = 0), we then write 𝔗(𝑇) = ∫ 𝑇 (𝝋 − 𝝅𝑘−1 P,𝑇 𝝋) · (𝒗𝑇 − 𝑷𝑘 D,𝑇𝒗𝑇) ≤ ∥𝝋 − 𝝅𝑘−1 P,𝑇 𝝋∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑)∥𝒗𝑇 − 𝑷𝑘 D,𝑇𝒗𝑇 ∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ 𝜇−1/2 𝑇 ℎ𝑟 𝑇 |𝝋|𝑯𝑟 (𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ℎ𝑇 𝜇 1/2 𝑇 ℎ−1 𝑇 ∥𝒗𝑇 ∥D,𝑇 = 𝜇−1/2 𝑇 ℎ𝑟+1 𝑇 |𝝋|𝑯𝑟 (𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) 𝜈 1/2 𝑇 min(1, 𝐶−1 f,𝑇) 1/2∥𝒗𝑇 ∥D,𝑇, where we have used Cauchy–Schwarz inequalities in the first passage, the approximation prop- erties (57) of the 𝐿2-orthogonal projector for the first factor together with the definitions (7) and (28) of ∥·∥𝑼,𝑇 and ∥·∥D,𝑇 to write ∥𝒗𝑇 − 𝑷𝑘 D,𝑇𝒗𝑇 ∥𝑳2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≤ ∥𝒗𝑇 − 𝑰𝑘 𝑇 𝑷𝑘 D,𝑇𝒗𝑇 ∥𝑼,𝑇 ≤ ∥𝒗𝑇 ∥D,𝑇 in the second passage, while the conclusion follows from the definition (4) of 𝐶f,𝑇 along with 𝐶−1 f,𝑇 = min(1, 𝐶−1 f,𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Using the above estimate in (55), applying a Cauchy–Schwarz inequality on the sum over 𝑇 ∈ Tℎ, and recalling the definition (34) of ∥·∥𝜇,𝜈,ℎ, (56) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='5 Proof of Theorem 7 Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Since 𝑎𝜇,ℎ + 𝑎𝜈,ℎ is 1-coercive and has norm 1 for the ∥·∥𝜇,𝜈,ℎ norm, Lemma 6 and [19, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='11] show that Aℎ is 𝛾-inf-sup stable for the norm in the left-hand side of (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Hence, in the spirit of the third Strang lemma [16], this error estimate follows if we bound the consistency error by the bracketed term in the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The consistency error for the scheme (32) is E𝑘 ℎ (𝒖, 𝑝;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒗ℎ) ≔ ∑︁ 𝑇∈Tℎ ∫ 𝑇 𝒇 · �𝑷 𝑘 D,𝑇𝒗𝑇 + ∫ Ω 𝑔𝑞ℎ − Aℎ((𝑰𝑘 ℎ𝒖, 𝜋ℎ P,𝑘 𝑝), (𝒗ℎ, 𝑞ℎ)) = ∑︁ 𝑇∈Tℎ ∫ 𝑇 ∇·(𝜇𝑇∇𝒖) · (𝒗𝑇 − �𝑷 𝑘 D,𝑇𝒗𝑇) − ∑︁ 𝑇∈Tℎ ∫ 𝑇 ∇·(𝜇𝑇∇𝒖) · 𝒗𝑇 − 𝑎𝜇,ℎ(𝑰𝑘 ℎ𝒖, 𝒗ℎ) + ∑︁ 𝑇∈Tℎ ∫ 𝑇 𝜈𝑇𝒖 · �𝑷 𝑘 D,𝑇𝒗𝑇 − 𝑎𝜈,ℎ(𝑰𝑘 ℎ𝒖, 𝒗ℎ) + ∑︁ 𝑇∈Tℎ ∫ 𝑇 ∇𝑝 · �𝑷 𝑘 D,𝑇𝒗𝑇 − 𝑏ℎ(𝒗ℎ, 𝜋𝑘 P,ℎ𝑝) + \x18\x18\x18\x18\x18\x18\x18\x18\x18\x18\x18 ∫ Ω 𝑔𝑞ℎ + 𝑏ℎ(𝑰𝑘 ℎ𝒖, 𝑞ℎ) = E𝑘 rhs,ℎ(∇·(𝜇∇𝒖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒗ℎ) + E𝑘 S,ℎ(𝒖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒗ℎ) + E𝑘 D,ℎ(𝒖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒗ℎ) + E𝑘 c,ℎ(𝑝;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 𝒗ℎ), (58) where we have we have replaced 𝒇 with the left-hand side of (2a), expanded Aℎ according to its definition (33), and used (30) along with (2b) to cancel the last term in the first passage, and used the definitions of the consistency errors (55) with 𝝋 = ∇·(𝜇∇𝒖), (37) and (44) with 𝒘 = 𝒖, and (50) with 𝑞 = 𝑝 to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 23 Using, respectively, (56) (further noticing that |∇·(𝜇𝑇∇𝒖)|𝑯𝑟 (𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) ≲ 𝜇𝑇 |𝒖|𝑯𝑟+2(𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='R𝑑) for all 𝑇 ∈ Tℎ), (38), (45), and (51) to estimate the terms in the right-hand side of (58), the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' □ Acknowledgements This research received support from the ANR “NEMESIS” (ANR-20-MRS2-0004) and the Australian Research Council’s Discovery Projects funding scheme (DP210103092).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The authors would also like to thank Ricardo Ruiz-Baier for sharing Gmsh geometry files at the source of the tests in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' References [1] M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1 (2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 61– 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1093/imanum/drn054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Boffi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Brezzi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Fortin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Mixed finite element methods and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Springer Series in Computational Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Heidelberg: Springer, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' xiv+685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1007/978-3-642-36519-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Bogovski˘ı.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' “Theory of cubature formulas and the application of functional analysis to problems of mathematical physics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In: vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 149(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Trudy Sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Soboleva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Novosibirsk, Russia: Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Nauk SSSR Sibirsk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Otdel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=', 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Solutions of some problems of vector analysis associated with the operators div and grad, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 5–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Botti and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Di Pietro.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='3 (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 783–822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1007/s42967-021-00142-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Botti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Di Pietro, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Droniou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' “A Hybrid High-Order discretisation of the Brinkman problem robust in the Darcy and Stokes limits”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In: Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Appl.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Guglielmana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' “A low-order nonconforming method for linear elasticity on general meshes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In: Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Mech.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 986–997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1002/num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='20076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [14] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Cáceres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Gatica, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Sequeira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' “A mixed virtual element method for the Brinkman problem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Models Methods Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='4 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 707–743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1142/S0218202517500142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [15] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Di Pietro and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Droniou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' “A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='307 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 2159–2191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1090/mcom/3180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Di Pietro and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Droniou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' “A third Strang lemma for schemes in fully discrete formulation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In: Calcolo 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='40 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1007/s10092-018-0282-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [17] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Di Pietro and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Droniou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' “An arbitrary-order discrete de Rham complex on poly- hedral meshes: Exactness, Poincaré inequalities, and consistency”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In: Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1007/s10208-021-09542-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Di Pietro and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Droniou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' “An arbitrary-order method for magnetostatics on poly- hedral meshes based on a discrete de Rham sequence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='109991 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='jcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='109991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [19] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Di Pietro and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Droniou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' The Hybrid High-Order method for polytopal meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Design, analysis, and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Modeling, Simulation and Application 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Springer International Publishing, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1007/978-3-030-37203-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [20] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Di Pietro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Droniou, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Rapetti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' “Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Models Methods Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='9 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 1809–1855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1142/S0218202520500372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Di Pietro and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Ern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' “A hybrid high-order locking-free method for linear elasticity on general meshes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In: Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Engrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 283 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 1–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='cma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' [22] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Di Pietro and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Ern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' “Arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In: IMA J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 37.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Ern, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Lemaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' “An arbitrary-order and compact-stencil dis- cretization of diffusion on general meshes based on local reconstruction operators”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' In: Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' Appl.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content='1142/S0218202518500057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} +page_content=' 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE1T4oBgHgl3EQfkQSg/content/2301.03272v1.pdf'} diff --git a/wtFRT4oBgHgl3EQfgzeb/content/tmp_files/2301.13581v1.pdf.txt b/wtFRT4oBgHgl3EQfgzeb/content/tmp_files/2301.13581v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2fa33e066d44008281a70ff1060031a4425c73cd --- /dev/null +++ b/wtFRT4oBgHgl3EQfgzeb/content/tmp_files/2301.13581v1.pdf.txt @@ -0,0 +1,985 @@ +arXiv:2301.13581v1 [cs.CR] 31 Jan 2023 +MACHINE LEARNING AND PORT SCANS: A SYSTEMATIC +REVIEW +A PREPRINT +Jason M. Pittman +ORCID: 0000-0002-5198-8157 +ABSTRACT +Port scanning is the process of attempting to connect to various network ports on a computing end- +point to determine which ports are open and which services are running on them. It is a common +method used by hackers to identify vulnerabilities in a network or system. By determining which +ports are open, an attacker can identify which services and applications are running on a device +and potentially exploit any known vulnerabilities in those services. Consequently, it is important +to detect port scanning because it is often the first step in a cyber attack. By identifying port scan- +ning attempts, cybersecurity professionals can take proactive measures to protect the systems and +networks before an attacker has a chance to exploit any vulnerabilities. Against this background, +researchers have worked for over a decade to develop robust methods to detect port scanning. While +there have been various surveys, none have focused solely on machine learning based detection +schemes specific to port scans. Accordingly, we provide a systematic review of 15 papers published +between February 2021 and January 2023. We extract critical information such as training dataset, +algorithm used, technique, and model accuracy. We also collect unresolved challenges and ideas for +future work. The outcomes are significant for researchers looking to step off from the latest work +and for practitioners interested in novel mechanisms to detect the early stages of cyber attack. +Keywords systematic review, machine learning, port scanning, cybersecurity, algorithms, training data +1 +Introduction +Cybersecurity incidents continue to plague digital life. +While a significant portion of incidents result from phish- +ing and malware, 45% are the result of network-based cy- +ber attacks [1]. These cyber attacks follow a pattern or +procedure. Existing models and methodologies vary in +the number of steps. However, the first step is universally +understood to be reconnaissance. In turn, reconnaissance +most often includes some type of port scanning. +Port scanning is a technique to enumerate target endpoints. +Confusingly, port scanning can be both a legitimate en- +gagement [2] or a malicious precursor to escalating intru- +sion [3, 4]. A general issue is differentiating between what +may be an authorized benign instance of host enumeration +and a malicious scanning of active hosts and their avail- +able ports. Furthermore, if we accept port scanning as a +necessary prelude to cyber attack, then we want to develop +a means to detect port scanning with high certainty. To +this end, there is a small but growing literature on detect- +ing port scanning. The literature ranges from early intru- +sion detection mechanisms [5] to sophisticated machine +learning techniques [6]. There have been several compar- +ative surveys during this time, most recently Aamir et al. +[7] and [8]. However, there has not been a systematic re- +view of the literature. +Literature reviews are invaluable to a field of study. Re- +views provide an understanding of the existing research +by establishing a foundation of knowledge. Reviews also +clarify existing knowledge related to a given problem. +Both functions guide new investigations and reduce over- +lap or unnecessary duplication of work. Yet, new reviews +are necessary as the field grows, new techniques are dis- +covered, and new technologies are released which impact +forms of inquiry. Accordingly, the purpose of this study is +to provide a systematic review of existing literature using +machine learning algorithms to detect port scanning. +The remainder of this work is organized in a way which +(a) situates the systematic review in existing knowledge +and (b) maximizes understanding of the cutting edge. The +first is achieved by discussing port scanning and detection +of such port scanning literature. Thereafter, we present +the research method and techniques used to find, organize, +and analyze research published since 2021. Finally, we + +Machine Learning and Port Scans: A Systematic Review +A PREPRINT +demonstrate the findings of the analysis in terms of quan- +titative results from the existing research. +2 +Related Work +The work most proximal to this study exists in two cat- +egories: scanning TCP/IP ports and detection of those +scans. The following discussion is not intended to be ex- +haustive. Rather, we offer background research that we +view as seminal and salient. +2.1 +Port Scanning +Port scanning uses features of TCP/IP to enumerate com- +puting systems on a network. As different network proto- +cols use different ports, it’s essential to scan a wide range +of ports to gather complete information. This is because +vulnerabilities can exist in all protocols. The total number +of ports that can be scanned is 65535, with ports 0 to 1023 +being well-known, ports 1024 to 49151 being registered, +and ports 49152 to 65535 being dynamic or private. +The origin of the phrase port scanning in the academic lit- +erature can be traced back to the early days of computer +networking. In the late 1980s and early 1990s, as the In- +ternet was growing and becoming more widely used, there +was an increasing need for tools to help network adminis- +trators and security professionals understand the state of +their networks. One of the key tasks for these profession- +als was identifying which network services were running +on which hosts, and which ports on those hosts were open +or closed. This process became known as port scanning. +One of the earliest references to port scanning in the liter- +ature is found in Fyodor [9], which described a method +for determining the operating system of a remote host +by sending probes to specific ports and analyzing the re- +sponses. The work explained the operating system of a +host can be determined by analyzing the TCP/IP stack’s +behavior and its responses to different types of probes, +such as the initial sequence numbers (ISNs) and the op- +tions in the TCP headers of the responses. Further, the pa- +per also described how to use this technique to fingerprint +the operating system of a remote host as well as the limi- +tations and challenges of the technique. Additionally, the +paper introduced the first version of an open-source tool +named nmap (Network Mapper) that implements this tech- +nique for Remote OS detection. While nmap is not the +only port scanner available, it is featured heavily through- +out the literature. +De Vivo et al. [10] generalizes from the port scanning +foundation provided in Fyodor [9] and several [11, 12] +others. The significance of De Vivo et al. [10] emerges +from the rigorous classification applied to port scanning +techniques and procedures. The paper described the dif- +ferent types of port scans, such as TCP connect scans and +SYN scans as classical. This is in relation to indirect and +stealth scanning. The latter is also referred to as a FIN, +XMAS, or NULL scan. The former is realized by bounc- +ing scans off of a zombie endpoint. The work goes on to +describe scanning techniques. These includes decoy scan- +ning, fragmented scanning, and coordinated or distributed +scanning, UDP scanning, and ICMP sweeping. +Barnett et al. [13] presented a classification system for net- +work scanning techniques. The significance of the work is +in establishing a clear and organized classification of the +different types of network scanning techniques that exist +and their use cases. To that end, the authors propose a tax- +onomy categorizing network scanning techniques based +on the level of interaction with the target system, the type +of information gathered, and the purpose of the scan. This +extends De Vivo et al. [10] in both types and techniques. +Barnett et al. +[13] add two additional network scan- +ning techniques to the three presented by De Vivo et +al. +These are vertical, horizontal. +Further, Barnett et +al. differentiate between OSI Model layer 2 scans and +layer 3 scans with overlaying attributes according to speed +(slow, medium, rapid) and distribution (one-to-one, one- +to-many, many-to-one, many-to-many). Barnett et al. also +describe how scan types from prior work (e.g., De Vivo et +al.) map to their categories. The mapping is more pro- +nounced when attributes encompassing speed and distri- +bution were considered. +Bou et al. [14] demonstrated a comprehensive overview +of the different types of cyber scanning techniques that are +used to identify various features of networks. The authors +divide port scanning techniques into two main categories: +passive and active. Passive scanning techniques involve +listening to network traffic to gather information about the +target network without sending any packets. Active scan- +ning techniques involve sending packets to a target host +to elicit a response, which can be used to determine the +host’s characteristics and identify vulnerabilities. +The work extends the categorization by describing differ- +ent techniques of passive and active scanning techniques. +This calls back to the organizational structure provided by +De Vivo et al. [10] and Barnett et al. [13] but differs in +semantics. For instance, the De Vivo et al. classical, indi- +rect, and stealth scans map under the nature of active and +passive scanning. Further, the semantic developed by [13] +around relations between scanner and target (e.g., one-to- +many) falls under approarch in Bou et al. [14]. Bou et +al. also offered strategy as a way to categorize directional +relationship between scanner and target. +Roy et al. [15] claimed a gap exists in the literature on +classifying and categorizing adversarial reconnaissance +processes. The claim stands to reason given the authors +first delineate between technical and non-technical recon- +naissance techniques. Technical reconnaissance included +network scanning, or cyber scanning as Roy et al. refer +to it, as a remote technique. The authors then differentiate +between host detection and port enumeration which stand +as a combined label for all of the scanning techniques out- +lined by De Vivo et al [10] (i.e., ICMP, SYN, Full Connect, +etc.). +2 + +Machine Learning and Port Scans: A Systematic Review +A PREPRINT +While Roy et al. [15] did not add anything new to the port +scanning taxonomy, the authors did connect the prior re- +search by De Vivo et al. [10] and Barnett et al. [13] to a +burgeoning literature around detection of port scanning. +2.2 +Detecting Port Scanning +Port scanning, as a reconnaissance technique, is de- +tectable. ML is a compelling solution to detecting oth- +erwise undetectable port scans because of its ability to +correlate seemingly unrelated features across enormous +datasets. Yet, not all ML algorithms work in the same +way or have the capability to address the same problem. +Furthermore, there are a variety of ML algorithms types- +classification, regression, deep learning, and so forth- with +a diversity of implementation variations. +The majority of works investigating ML for detecting port +scanning over the past decade and a half include at a re- +view of prior ML algorithm performance. Such work ex- +ists as a quasi review with an add-on quantitative analysis +of an algorithm not featured in the prior research. We use +the term quasi because these works review algorithms by +running each against a common training and evaluation +dataset. These studies do not rely on results from the prior +research responsible for introducing the algorithm to the +field. This is close to the notion of a meta-analysis but +not precisely so. Still, the quasi reviews are particularly +significant for researchers and practitioners looking to get +up to speed on the state of the field in short order. We dis- +covered two such works and include those as foundational +literature which we directly extend with our systematic re- +view. +Aamir et al. [7] investigated the detection of characteris- +tics of port scanning and analyzed the performance of 22 +ML algorithms. The algorithms included decision trees, +discriminant analysis, support vector machines (SVM), k- +nearest neighbors (KNN), and ensemble classifiers. The +authors used the CICIDS2017 dataset with a 70%training +and 30% evaluation split. Of the 22 ML algorithms ex- +amined, nine demonstrated more than 85% classification +(testing) accuracy. Specifically, Aamir et al. identified +Fine Gaussian SVM as best performing algorithm with +99% testing and 99% training accuracy. False negative +rates are provided for all 22 algorithm experiments. Fur- +ther, for fast training with high accuracy scores, discrimi- +nant analysis was more accurate and efficient in classify- +ing port scans. Aamir et al. did not discuss the type of port +scans detected nor what scanning techniques were present +in the dataset. +Krishna et al. [8] also investigated port scan detection +using a variety of ML algorithms. The authors analyzed +fewer algorithms but used the same training and evalua- +tion dataset as Aamir et al. [7]. Krishna et al. examined +two of the algorithms as Aamir et al., SVM and decision +trees. Krishna et al. also evaluated random forest and lo- +gistic regression. Unlike any other study we found in the +literature, Krishna et al. do not present results. Instead, +the authors include snapshots of their Juypter notebook as +figures. On one hand, including code makes the work re- +peatable and reproducible. On the other hand, one would +need to repeat and reproduce the study to obtain algorithm +performance values. +Both Aamir et al. [7] and Krishna et al. [8] represent +the predominant type of research in the literature. That +is, existing port scan detection research frequently exam- +ines ML algorithm performance by direct experimenta- +tion rather than reference to prior experimentation. Con- +sequently, the findings from these studies are scattered +throughout the literature. Anyone interested in extending +the field is left to trace through the forest to find relevant +trees. With that in mind, the goal of this work was to sys- +tematically review results published since 2021 to catalog +ML algorithm performance in detecting port scans. +3 +Method +This work employed a systematic literature review +methodology. +Systematic literature review is a well- +defined method that is used to identify, evaluate, and in- +terpret all of the available research on a particular topic +[16, 17]. The process is designed to be comprehensive, un- +biased, and transparent, and it involves a number of steps, +including formulating a research question, searching for +relevant literature, selecting studies, extracting data, and +synthesizing the results [16, 18]. Systematic literature re- +views are increasingly used in the field of software engi- +neering and other technical fields, but also in other scien- +tific fields, as a way to provide an in-depth understanding +of existing knowledge on a topic. +A systematic literature review differs from other literature +review methods in several ways. A traditional literature re- +view, also called a narrative review [16], is typically less +structured and less systematic. It is often used to provide +an overview of the current state of knowledge on a topic, +but it is not as rigorous as an SLR in terms of the search +and selection process. +With the design of systematic reviews in mind, we pose +four questions. +RQ1: What machine learning algorithms have been used +to detect port scanning? +RQ2: What were the detection rates and false positive +rates for those algorithms? +RQ3: What datasets were used for training and evaluation +of those algorithms? +RQ4: What port scanning types and techniques were used +for evaluation of those algorithms? +We constrained the literature search to 2021 and newer. +We did so based on the last relevant reviews being pub- +lished in 2021 [7, 8]. While we recognize the majority +of work in detecting port scans exists between 2010 and +2021, research is still progressing in this research area +3 + +Machine Learning and Port Scans: A Systematic Review +A PREPRINT +and a systematic review of published research since 2021 +holds significance for researchers and practitioners alike. +No literature search will produce perfect results. How- +ever, careful attention to search strings can yield sufficient +results so as to be thorough. We used "detecting port +scan" AND "machine learning" as a starting search string. +The search returned 61 papers. For comparison, searching +with "port scan" AND "machine learning" produced 952 +results. Meanwhile, "detect port scan" AND "machine +learning" produced 21 results. Often researchers use the +commonly accepted short form of machine learning, ML. +Thus, we used "ML" AND "detecting port scan" to cross- +check the search. This produced 43 results. The variant +of "detect port scan" AND "ML" found 12 papers. +We manually reviewed each work to ensure each study +included detection of port scanning. Manual review was +necessary because some work folds port scan detection +into an overarching intrusion detection framework. We +were left with 15 studies as our dataset after this step. +Data extraction from the selected papers is an important +step to properly answer the research questions. In this +study, we used the following data form to extract the +needed information: (a) year of publication; (b) authors; +(c) source of publication; (d) citation count; results (ac- +curacy as F1); (e) dataset source; and (f) algorithms as +task, technique, and procedure (TTP). We also included +the port scanning types and techniques when such were +available in the research. +4 +Results +We separate the results of the systematic review into two +sections. The first section provides an overview of our +dataset. We describe the literature features for ease of +future reference. Then, we present a breakdown of that +literature by algorithm. As with Aamir et al. [7] and Kr- +ishna et al. [8], some work experimented with more than +one ML algorithm. Such research appears in multiple cat- +egories below. +4.1 +The Literature +We analyzed 15 studies published since 2021 (Table 1). +Seven studies were from 2021, six were from 2022, and a +single study appeared in early 2023. The remaining study +was from 2020 which we included as a specific exception. +This is discussed in the Neural Network (NN) algorithm +section below. +The sample encompassed six total ML algorithms. The +majority (10) of studies examined a single ML algorithm +while five studies examined more than one. The litera- +ture published in 2021 spanned all six algorithms whereas +literature from 2022 focused on a single algorithm (with +one exception). Random Forest (RF) and SVM were the +most investigated algorithms in the 2021 subset. A vari- +ety of NN implementations appeared throughout the 2022 +subset. Nature-inspired (NI) appeared once while Regres- +sion (R) and Naive Bayes (NB) were studied three and +five times respectively. +Six studies have not been cited. Six studies have been +cited more than once with 25 being the highest citation +count. Only one study [19] included a paper [7] from the +literature population in its related work. The other 14 pa- +pers exist independent of one another with only indirect +relations from support research in general ML or cyberse- +curity. +Table 1: Literature using ML to detect port scans +Authors +Year +Cited +TTP +Hartpence et al. +2020 +6 +NN +Algaolahi et al. +2021 +1 +RF,SVM +Baah et al. +2021 +0 +RF,SVM,NB +Sirisha et al. +2021 +4 +RF,R,NB +Liu et al. +2021 +21 +NI +Bertoli et al. +2021 +25 +RF,SVM,R,NB,NN +Mohseni et al. +2021 +1 +RF +Al-Haija et al. +2021 +9 +NB +Bakaletz +2022 +0 +NB +Tojeiro et al. +2022 +0 +R +Singh et al. +2022 +2 +NN +Lv et al. +2022 +0 +NN +Kirtas et al. +2022 +1 +NN +SaiKiran et al. +2022 +0 +RF,SVM,NN +Henry et al. +2023 +0 +NN +4.2 +The Algorithms +We found six machine learning algorithms in the litera- +ture sample. The following sections present a summary +for each algorithm and the relative meaning of using it +to detect port scanning. We summarize each algorithms’s +performance in terms of accuracy and false positives. We +also present the dataset used to train and evaluate the mod- +els when such are revealed in the source literature. +4.2.1 +Random Forest +Random Forest is good for classification problems, partic- +ularly in cases where there are many features and interac- +tions among features. The algorithms is also useful for +feature selection and handles missing data well. +Six studies out of the 15 study sample experimented with +the RF algorithm [20, 21, 22, 23, 24, 25]. +Algorithm +performance ranged from 78.09% to 100% across those +studies. One paper [21] included source code or a link +to a source code repository (e.g., GitHub). Four different +datasets were used, three of which do not appear in other +algorithm categories. +Two studies discussed the types of port scans present in +training and evaluation data. SaiKiran et al. [25] men- +tioned port sweep but did not specify further. Bertoli et +al. [21] conducted training and evaluating against the full +4 + +Machine Learning and Port Scans: A Systematic Review +A PREPRINT +spectrum of port scan types. Further, the authors included +port scan data from five different port scan tools. +Table 2: Random Forest Algorithm Performance +Authors +Accuracy (F1) +Dataset +Algaolahi et al. +99.75 +CICIDS2017 +Baah et al. +99.98 +CICIDS2017 +Sirisha et al. +78.09 +NSLKDD +Sirisha et al. +84.14 +CICIDS2017 +SaiKiran et al. +99.93 +CICIDS2017 +Mohseni et al. +99.94 +CICIDS2017 +Bertoli et al. +96.00 +MAWILab +Bertoli et al. +100.00 +Bonafide +4.2.2 +Support Vector Machine (SVM) +Support Vector Machine (SVM) is good for classification +and regression problems, especially in cases where the +data has clear boundaries and is not noisy. It works well +for datasets with a limited number of features. +SVM is different from Random Forest in that it uses a +boundary (a hyperplane) to separate the data into classes, +whereas Random Forest creates multiple decision trees +and aggregates their predictions to make a final decision. +SVM is best suited for cases where the boundary between +classes is well defined and clear, whereas Random Forest +is better suited for complex, non-linear decision bound- +aries. +Four studies experimented with SVM [20, 21, 24, 25]. All +four also had explored RF performance. Results for the +SVM experiments ranged from 89.61% to 99.87% both +coming from the same dataset (of two total). Source code +availability and port scan details remained the same as in- +dicated in the RF algorithm category. +Table 3: Support Vector Machine Algorithm Performance +Authors +Accuracy (F1) +Dataset +Algaolahi et al. +89.61 +CICIDS2017 +Baah et al +99.87 +CICIDS2017 +SaiKiran et al. +93.29 +CICIDS2017 +Bertoli et al. +92.00 +Bonafide +4.2.3 +Regression +Regression algorithms are used for predicting a continu- +ous target variable based on one or more input features. +They are commonly used for tasks such as predictions and +forecasting. +Regression algorithms, including linear regression, are +different from Random Forest and SVM algorithms in that +they focus on establishing a linear or non-linear relation- +ship between the input features and the target variable. On +the other hand, Random Forest and SVM algorithms are +mainly used for classification problems. +In a regression problem, the aim is to predict a numerical +output, whereas in classification the output is categorical. +SVM can also be used for regression problems by using +a specific formulation called Support Vector Regression +(SVR). However, the emphasis and method used in re- +gression algorithms are different compared to SVM and +Random Forest. +Four different datasets were used by three studies [22, 21, +26]. The results span 59.21% to 94% accuracy (F1). Only +one study [21] discussed port scanning in detail and in- +cluded source code for the algorithm. +Table 4: Regression Algorithm Performance +Authors +Accuracy (F1) +Dataset +Tojeiro et al. +94.00 +CICIDS2017 +Sirisha et al. +74.05 +NSLKDD +Sirisha et al. +59.21 +CICIDS2017 +Bertoli et al. +70.00 +MAWILab +Bertoli et al. +92.00 +Bonafide +4.2.4 +Naive Bayes +Naive Bayes is a probabilistic algorithm that is good for +classification problems, especially when the assumption +of independence between features holds. It is fast and sim- +ple to implement and can handle large datasets well. +Naive Bayes is different from regression, SVM, and Ran- +dom Forest algorithms in that it makes a probabilistic pre- +diction based on Bayes’ theorem and the assumption of +independence between features, whereas the other algo- +rithms make predictions based on a boundary or a com- +bination of trees. In comparison, regression, SVM, and +Random Forest algorithms work well for more complex +problems where the relationship between features is not +necessarily independent and the decision boundary is not +clear. +Bakaletz [27] used a custom generated dataset for train- +ing and evaluation. +The author used two nmap scan +techniques- aggressive (NMAP-A) and stealth (NMAP- +S). There were five additional datasets used in three +[19, 22, 21] out of the remaining four studies [24] in +this category. Training and evaluation of NB algorithms +demonstrated performances in the range of 55% to 99.7%. +Table 5: Naive Bayes Algorithm Performance +Authors +Accuracy (F1) +Dataset +Al-Haija et al. +99.70 +PSA-2017 +Baah et al +93.02 +CICIDS2017 +Bakaletz +98.00 +NMAP-A +Bakaletz +82.00 +NMAP-S +Sirisha et al. +74.47 +NSLKDD +Sirisha et al. +37.92 +CICIDS2017 +Bertoli et al. +55.00 +MAWILab +Bertoli et al. +78.00 +Bonafide +4.2.5 +Neural networks +Neural networks (NNs) are good for a wide range of tasks +including classification, speech recognition, and natural +5 + +Machine Learning and Port Scans: A Systematic Review +A PREPRINT +language processing. They are particularly good for com- +plex and non-linear problems, such as recognizing pat- +terns where traditional algorithms struggle. +Neural networks are different from other machine learn- +ing algorithms in that they are based on a modeled struc- +ture inspired by the human brain. They consist of multiple +interconnected nodes, or artificial neurons, that process +information. These neurons are organized into layers and +the connections between them are associated with weights +that are learned during the training process. +In comparison, algorithms such as RF, SVM, NB, and +regression algorithms make predictions based on a clear +boundary or a combination of trees or linear relationships, +whereas NNs are capable of learning complex relation- +ships between the inputs and outputs through their inter- +nal structure. NN algorithms have the ability to learn and +make predictions based on the examples they are trained +on, which makes them highly flexible. However, they can +also be more difficult to interpret and train, and require a +large amount of data to achieve good performance. +Furthermore, we differentiate between two types of NN: +Artificial Neural Networks (ANNs) and Convolutional +Neural Networks (CNNs). Both types of deep learning +algorithms used for various tasks. The main difference be- +tween ANNs and CNNs is the structure and the way they +process information. ANNs are fully connected networks, +where each neuron in one layer is connected to all neurons +in the next layer. This structure makes ANNs computa- +tionally expensive and require a large amount of data to +achieve meaningful results. +On the other hand, CNNs are specifically designed for +classification tasks. They have a unique structure, con- +sisting of convolutional layers, activation layers, pooling +layers, and fully connected layers. Convolutional layers +are used to extract features, activation layers apply a non- +linear activation function to the output of the convolu- +tional layer, pooling layers reduce the spatial dimensions +of the output, and fully connected layers make the final +prediction. This unique structure makes CNNs efficient +and effective for classification tasks, as it allows them to +learn hierarchical representations of the data and identify +different objects and features. In comparison, ANNs are +not specifically designed for classification and may not +achieve the same level of performance as CNNs for these +types of tasks. +We did break our own time bounding constraint in this +category because Aamir et al. [7] specifically noted neu- +ral networks were not being investigated. Hartpence et +al. [28] published their work in 2020, therefore we felt +obliged to include the work here. +On that note, three of the four studies in this algorith- +mic category used non-standard datasets. Hartpence et +al. [28] provided immense detail in what port scan types +exist in the datasets the authors generated as part of their +experiments. The general (GEN) dataset contained typ- +ical network traffic with port scans intermingled. +The +second dataset contained port scans only (TCP). Lv et al. +[29] generated a custom dataset by capturing network traf- +fic while executing nmap full connect scans (NMAP-F). +Kirtas et al. [30] executed nmap SYN scans (NMAP-Y) +while capturing network traffic for the dataset. No study +included code snippets or links to code repositories. +Table 6: ANN Algorithm Performance +Authors +Accuracy (F1) +Dataset +Hartpence et al. +99.99 +GEN +Hartpence et al. +99.31 +TCP +SaiKiran et al. +99.11 +CICIDS2017 +Kirtas et al. +87.88 +NMAP-Y +Bertoli et al. +100.00 +Bonafide +Table 7: CNN Algorithm Performance +Authors +Accuracy (F1) +Dataset +Lv et al. +99.00 +NMAP-F +SaiKiran et al. +63.52 +CICIDS2017 +Singh et al. +99.94 +CICIDS2017 +Henry et al. +98.73 +CICIDS2017 +5 +Conclusion +We posed four research questions to guide this systematic +review of port scan detection literature. We discovered +five algorithms present in the literature: Random Forest, +Support Vector Machine, Regression, Naive Bayes, and +Neural Network. The literature revealed multiple ways +to accurately detect port scanning. Bertoli et al. [21] con- +firmed RF and ANN algorithms are capable of 100% accu- +racy against a plethora of port scan types and techniques. +Sirisha et al. [22] showed the poorest accuracy, 37.92%, +with the authors evaluation of the NB algorithm. ANN +seemed to be the strongest type of algorithm across all +of the sample papers, followed by RF. There were 11 dif- +ferent datasets used in 34 algorithm experiments with the +CICIDS2017 dataset being used in 47% of those exper- +iments. These results originated from 14 research stud- +ies published since 2021 and a single study coming from +2020. Unfortunately, we were unable to adequately ad- +dress the fourth research question (i.e., What port scan- +ning types and techniques were used for evaluation of +those algorithms?) as only a few studies discussed port +scanning in any detail. +Overall, we found a variety of existing work included port +scanning as a minor point compared to focuses areas such +as general intrusion detection, DDoS, malware, botnets, +and so forth. We assume any work demonstrating a ca- +pability to detect port scanning mentioned such explicitly +and thus would be discoverable in our search. Another as- +sumption underlying this work is research indexing. More +specifically, we assume existing work has been indexed +and thus was discoverable given our search strings. A +final assumption present throughout the literature seems +to be the stability of port scanning types and techniques. +The dominance of the CICIDS2017 dataset in training and +6 + +Machine Learning and Port Scans: A Systematic Review +A PREPRINT +evaluation supports this point. This assumption will con- +tinue to be reasonable as long as significant innovations +do not occur in the port scanning research. +It is interesting to note many existing papers experiment +with more than one machine learning algorithm. The di- +versity of results within an algorithm category, across the +sample papers in the category, is curious. Perhaps this +would not be so unexpected if every paper used a different +dataset for training and evaluation. Yet, much of the exist- +ing work leverages the CICIDS2017 data. As well, The +distinct shift to NN algorithms in 2022 is notable. The +prominence of NN over the past year suggests NN and +its variations represent a viable research pathway going +forward. As a final note, we found the vast majority of +studies do not include code snippets or links to GitHub +repositories. +Accordingly, replication and reproduction to include spe- +cific port scanning techniques with packet captures and al- +gorithm source code would be beneficial. This would be +significant because doing so would fill in an existing gap +and also enable adjacent research in cybersecurity such +as offensive cyber, network security, and so forth. +At +the same time, such future work might look to incorpo- +rate green compute measurements given the increased fo- +cus on sustainability in algorithm research. Foundational +compute resource metrics can be taken from Bertoli et al. +[21] who detailed the compute resource profiles associ- +ated with training and evaluating the ML algorithms in +their work. +A final area for future exploration is nature inspired or +artificial life (Alife) algorithms. Greensmith et al. [31] +showed how a dendritic cell algorithm can be used to de- +tect port scanning. The authors show a theoretical model +with pseudocode examples. More recently, Liu et al. [32] +extended the dendritic cell concept albeit without evalua- +tion against port scan datasets. Such algorithms should be +evaluated using the datasets utilized in the review sample +papers and additional Alife algorithms investigated. +References +[1] Statista Research Department. Most common action +varieties in data breaches worldwide in 2019, 2023. +[2] Weijie Wang, Baijian Yang, and Yingjie Victor Chen. +Detecting subtle port scans through characteristics +based on interactive visualization. In Proceedings of +the 3rd annual conference on Research in informa- +tion technology, pages 33–38, 2014. +[3] Habib Ullah Baig and Farrukh Kamran. Detection +of port and network scan using time independent fea- +ture set. In 2007 IEEE Intelligence and Security In- +formatics, pages 180–184. 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IEEE Transactions on Cybernet- +ics, 2021. +8 + diff --git a/wtFRT4oBgHgl3EQfgzeb/content/tmp_files/load_file.txt b/wtFRT4oBgHgl3EQfgzeb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cf80f8c3969a00c0307d4a27f3c8ce1894b36a84 --- /dev/null +++ b/wtFRT4oBgHgl3EQfgzeb/content/tmp_files/load_file.txt @@ -0,0 +1,537 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf,len=536 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='13581v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='CR] 31 Jan 2023 MACHINE LEARNING AND PORT SCANS: A SYSTEMATIC REVIEW A PREPRINT Jason M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Pittman ORCID: 0000-0002-5198-8157 ABSTRACT Port scanning is the process of attempting to connect to various network ports on a computing end- point to determine which ports are open and which services are running on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' It is a common method used by hackers to identify vulnerabilities in a network or system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' By determining which ports are open, an attacker can identify which services and applications are running on a device and potentially exploit any known vulnerabilities in those services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Consequently, it is important to detect port scanning because it is often the first step in a cyber attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' By identifying port scan- ning attempts, cybersecurity professionals can take proactive measures to protect the systems and networks before an attacker has a chance to exploit any vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Against this background, researchers have worked for over a decade to develop robust methods to detect port scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' While there have been various surveys, none have focused solely on machine learning based detection schemes specific to port scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Accordingly, we provide a systematic review of 15 papers published between February 2021 and January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We extract critical information such as training dataset, algorithm used, technique, and model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We also collect unresolved challenges and ideas for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The outcomes are significant for researchers looking to step off from the latest work and for practitioners interested in novel mechanisms to detect the early stages of cyber attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Keywords systematic review, machine learning, port scanning, cybersecurity, algorithms, training data 1 Introduction Cybersecurity incidents continue to plague digital life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' While a significant portion of incidents result from phish- ing and malware, 45% are the result of network-based cy- ber attacks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' These cyber attacks follow a pattern or procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Existing models and methodologies vary in the number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' However, the first step is universally understood to be reconnaissance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' In turn, reconnaissance most often includes some type of port scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Port scanning is a technique to enumerate target endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Confusingly, port scanning can be both a legitimate en- gagement [2] or a malicious precursor to escalating intru- sion [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' A general issue is differentiating between what may be an authorized benign instance of host enumeration and a malicious scanning of active hosts and their avail- able ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Furthermore, if we accept port scanning as a necessary prelude to cyber attack, then we want to develop a means to detect port scanning with high certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' To this end, there is a small but growing literature on detect- ing port scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The literature ranges from early intru- sion detection mechanisms [5] to sophisticated machine learning techniques [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' There have been several compar- ative surveys during this time, most recently Aamir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [7] and [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' However, there has not been a systematic re- view of the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Literature reviews are invaluable to a field of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Re- views provide an understanding of the existing research by establishing a foundation of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Reviews also clarify existing knowledge related to a given problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Both functions guide new investigations and reduce over- lap or unnecessary duplication of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Yet, new reviews are necessary as the field grows, new techniques are dis- covered, and new technologies are released which impact forms of inquiry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Accordingly, the purpose of this study is to provide a systematic review of existing literature using machine learning algorithms to detect port scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The remainder of this work is organized in a way which (a) situates the systematic review in existing knowledge and (b) maximizes understanding of the cutting edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The first is achieved by discussing port scanning and detection of such port scanning literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Thereafter, we present the research method and techniques used to find, organize, and analyze research published since 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Finally, we Machine Learning and Port Scans: A Systematic Review A PREPRINT demonstrate the findings of the analysis in terms of quan- titative results from the existing research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2 Related Work The work most proximal to this study exists in two cat- egories: scanning TCP/IP ports and detection of those scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The following discussion is not intended to be ex- haustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Rather, we offer background research that we view as seminal and salient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='1 Port Scanning Port scanning uses features of TCP/IP to enumerate com- puting systems on a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' As different network proto- cols use different ports, it’s essential to scan a wide range of ports to gather complete information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' This is because vulnerabilities can exist in all protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The total number of ports that can be scanned is 65535, with ports 0 to 1023 being well-known, ports 1024 to 49151 being registered, and ports 49152 to 65535 being dynamic or private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The origin of the phrase port scanning in the academic lit- erature can be traced back to the early days of computer networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' In the late 1980s and early 1990s, as the In- ternet was growing and becoming more widely used, there was an increasing need for tools to help network adminis- trators and security professionals understand the state of their networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' One of the key tasks for these profession- als was identifying which network services were running on which hosts, and which ports on those hosts were open or closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' This process became known as port scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' One of the earliest references to port scanning in the liter- ature is found in Fyodor [9], which described a method for determining the operating system of a remote host by sending probes to specific ports and analyzing the re- sponses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The work explained the operating system of a host can be determined by analyzing the TCP/IP stack’s behavior and its responses to different types of probes, such as the initial sequence numbers (ISNs) and the op- tions in the TCP headers of the responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Further, the pa- per also described how to use this technique to fingerprint the operating system of a remote host as well as the limi- tations and challenges of the technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Additionally, the paper introduced the first version of an open-source tool named nmap (Network Mapper) that implements this tech- nique for Remote OS detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' While nmap is not the only port scanner available, it is featured heavily through- out the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' De Vivo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [10] generalizes from the port scanning foundation provided in Fyodor [9] and several [11, 12] others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The significance of De Vivo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [10] emerges from the rigorous classification applied to port scanning techniques and procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The paper described the dif- ferent types of port scans, such as TCP connect scans and SYN scans as classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' This is in relation to indirect and stealth scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The latter is also referred to as a FIN, XMAS, or NULL scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The former is realized by bounc- ing scans off of a zombie endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The work goes on to describe scanning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' These includes decoy scan- ning, fragmented scanning, and coordinated or distributed scanning, UDP scanning, and ICMP sweeping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Barnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [13] presented a classification system for net- work scanning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The significance of the work is in establishing a clear and organized classification of the different types of network scanning techniques that exist and their use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' To that end, the authors propose a tax- onomy categorizing network scanning techniques based on the level of interaction with the target system, the type of information gathered, and the purpose of the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' This extends De Vivo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [10] in both types and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Barnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [13] add two additional network scan- ning techniques to the three presented by De Vivo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' These are vertical, horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Further, Barnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' differentiate between OSI Model layer 2 scans and layer 3 scans with overlaying attributes according to speed (slow, medium, rapid) and distribution (one-to-one, one- to-many, many-to-one, many-to-many).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Barnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' also describe how scan types from prior work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=', De Vivo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=') map to their categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The mapping is more pro- nounced when attributes encompassing speed and distri- bution were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Bou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [14] demonstrated a comprehensive overview of the different types of cyber scanning techniques that are used to identify various features of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The authors divide port scanning techniques into two main categories: passive and active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Passive scanning techniques involve listening to network traffic to gather information about the target network without sending any packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Active scan- ning techniques involve sending packets to a target host to elicit a response, which can be used to determine the host’s characteristics and identify vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The work extends the categorization by describing differ- ent techniques of passive and active scanning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' This calls back to the organizational structure provided by De Vivo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [10] and Barnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [13] but differs in semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' For instance, the De Vivo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' classical, indi- rect, and stealth scans map under the nature of active and passive scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Further, the semantic developed by [13] around relations between scanner and target (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=', one-to- many) falls under approarch in Bou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Bou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' also offered strategy as a way to categorize directional relationship between scanner and target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [15] claimed a gap exists in the literature on classifying and categorizing adversarial reconnaissance processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The claim stands to reason given the authors first delineate between technical and non-technical recon- naissance techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Technical reconnaissance included network scanning, or cyber scanning as Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' refer to it, as a remote technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The authors then differentiate between host detection and port enumeration which stand as a combined label for all of the scanning techniques out- lined by De Vivo et al [10] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=', ICMP, SYN, Full Connect, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2 Machine Learning and Port Scans: A Systematic Review A PREPRINT While Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [15] did not add anything new to the port scanning taxonomy, the authors did connect the prior re- search by De Vivo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [10] and Barnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [13] to a burgeoning literature around detection of port scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='2 Detecting Port Scanning Port scanning, as a reconnaissance technique, is de- tectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' ML is a compelling solution to detecting oth- erwise undetectable port scans because of its ability to correlate seemingly unrelated features across enormous datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Yet, not all ML algorithms work in the same way or have the capability to address the same problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Furthermore, there are a variety of ML algorithms types- classification, regression, deep learning, and so forth- with a diversity of implementation variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The majority of works investigating ML for detecting port scanning over the past decade and a half include at a re- view of prior ML algorithm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Such work ex- ists as a quasi review with an add-on quantitative analysis of an algorithm not featured in the prior research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We use the term quasi because these works review algorithms by running each against a common training and evaluation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' These studies do not rely on results from the prior research responsible for introducing the algorithm to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' This is close to the notion of a meta-analysis but not precisely so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Still, the quasi reviews are particularly significant for researchers and practitioners looking to get up to speed on the state of the field in short order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We dis- covered two such works and include those as foundational literature which we directly extend with our systematic re- view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Aamir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [7] investigated the detection of characteris- tics of port scanning and analyzed the performance of 22 ML algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The algorithms included decision trees, discriminant analysis, support vector machines (SVM), k- nearest neighbors (KNN), and ensemble classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The authors used the CICIDS2017 dataset with a 70%training and 30% evaluation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Of the 22 ML algorithms ex- amined, nine demonstrated more than 85% classification (testing) accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Specifically, Aamir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' identified Fine Gaussian SVM as best performing algorithm with 99% testing and 99% training accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' False negative rates are provided for all 22 algorithm experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Fur- ther, for fast training with high accuracy scores, discrimi- nant analysis was more accurate and efficient in classify- ing port scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Aamir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' did not discuss the type of port scans detected nor what scanning techniques were present in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Krishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [8] also investigated port scan detection using a variety of ML algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The authors analyzed fewer algorithms but used the same training and evalua- tion dataset as Aamir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Krishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' examined two of the algorithms as Aamir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=', SVM and decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Krishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' also evaluated random forest and lo- gistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Unlike any other study we found in the literature, Krishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' do not present results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Instead, the authors include snapshots of their Juypter notebook as figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' On one hand, including code makes the work re- peatable and reproducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' On the other hand, one would need to repeat and reproduce the study to obtain algorithm performance values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Both Aamir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [7] and Krishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [8] represent the predominant type of research in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' That is, existing port scan detection research frequently exam- ines ML algorithm performance by direct experimenta- tion rather than reference to prior experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Con- sequently, the findings from these studies are scattered throughout the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Anyone interested in extending the field is left to trace through the forest to find relevant trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' With that in mind, the goal of this work was to sys- tematically review results published since 2021 to catalog ML algorithm performance in detecting port scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 3 Method This work employed a systematic literature review methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Systematic literature review is a well- defined method that is used to identify, evaluate, and in- terpret all of the available research on a particular topic [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The process is designed to be comprehensive, un- biased, and transparent, and it involves a number of steps, including formulating a research question, searching for relevant literature, selecting studies, extracting data, and synthesizing the results [16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Systematic literature re- views are increasingly used in the field of software engi- neering and other technical fields, but also in other scien- tific fields, as a way to provide an in-depth understanding of existing knowledge on a topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' A systematic literature review differs from other literature review methods in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' A traditional literature re- view, also called a narrative review [16], is typically less structured and less systematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' It is often used to provide an overview of the current state of knowledge on a topic, but it is not as rigorous as an SLR in terms of the search and selection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' With the design of systematic reviews in mind, we pose four questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' RQ1: What machine learning algorithms have been used to detect port scanning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' RQ2: What were the detection rates and false positive rates for those algorithms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' RQ3: What datasets were used for training and evaluation of those algorithms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' RQ4: What port scanning types and techniques were used for evaluation of those algorithms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We constrained the literature search to 2021 and newer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We did so based on the last relevant reviews being pub- lished in 2021 [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' While we recognize the majority of work in detecting port scans exists between 2010 and 2021, research is still progressing in this research area 3 Machine Learning and Port Scans: A Systematic Review A PREPRINT and a systematic review of published research since 2021 holds significance for researchers and practitioners alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' No literature search will produce perfect results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' How- ever, careful attention to search strings can yield sufficient results so as to be thorough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We used "detecting port scan" AND "machine learning" as a starting search string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The search returned 61 papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' For comparison, searching with "port scan" AND "machine learning" produced 952 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Meanwhile, "detect port scan" AND "machine learning" produced 21 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Often researchers use the commonly accepted short form of machine learning, ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Thus, we used "ML" AND "detecting port scan" to cross- check the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' This produced 43 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The variant of "detect port scan" AND "ML" found 12 papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We manually reviewed each work to ensure each study included detection of port scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Manual review was necessary because some work folds port scan detection into an overarching intrusion detection framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We were left with 15 studies as our dataset after this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Data extraction from the selected papers is an important step to properly answer the research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' In this study, we used the following data form to extract the needed information: (a) year of publication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' (b) authors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' (c) source of publication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' (d) citation count;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' results (ac- curacy as F1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' (e) dataset source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' and (f) algorithms as task, technique, and procedure (TTP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We also included the port scanning types and techniques when such were available in the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 4 Results We separate the results of the systematic review into two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The first section provides an overview of our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We describe the literature features for ease of future reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Then, we present a breakdown of that literature by algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' As with Aamir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [7] and Kr- ishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [8], some work experimented with more than one ML algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Such research appears in multiple cat- egories below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='1 The Literature We analyzed 15 studies published since 2021 (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Seven studies were from 2021, six were from 2022, and a single study appeared in early 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The remaining study was from 2020 which we included as a specific exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' This is discussed in the Neural Network (NN) algorithm section below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The sample encompassed six total ML algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The majority (10) of studies examined a single ML algorithm while five studies examined more than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The litera- ture published in 2021 spanned all six algorithms whereas literature from 2022 focused on a single algorithm (with one exception).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Random Forest (RF) and SVM were the most investigated algorithms in the 2021 subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' A vari- ety of NN implementations appeared throughout the 2022 subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Nature-inspired (NI) appeared once while Regres- sion (R) and Naive Bayes (NB) were studied three and five times respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Six studies have not been cited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Six studies have been cited more than once with 25 being the highest citation count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Only one study [19] included a paper [7] from the literature population in its related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The other 14 pa- pers exist independent of one another with only indirect relations from support research in general ML or cyberse- curity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Table 1: Literature using ML to detect port scans Authors Year Cited TTP Hartpence et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2020 6 NN Algaolahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2021 1 RF,SVM Baah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2021 0 RF,SVM,NB Sirisha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2021 4 RF,R,NB Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2021 21 NI Bertoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2021 25 RF,SVM,R,NB,NN Mohseni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2021 1 RF Al-Haija et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2021 9 NB Bakaletz 2022 0 NB Tojeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2022 0 R Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2022 2 NN Lv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2022 0 NN Kirtas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2022 1 NN SaiKiran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2022 0 RF,SVM,NN Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 2023 0 NN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='2 The Algorithms We found six machine learning algorithms in the litera- ture sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The following sections present a summary for each algorithm and the relative meaning of using it to detect port scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We summarize each algorithms’s performance in terms of accuracy and false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We also present the dataset used to train and evaluate the mod- els when such are revealed in the source literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='1 Random Forest Random Forest is good for classification problems, partic- ularly in cases where there are many features and interac- tions among features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The algorithms is also useful for feature selection and handles missing data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Six studies out of the 15 study sample experimented with the RF algorithm [20, 21, 22, 23, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Algorithm performance ranged from 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='09% to 100% across those studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' One paper [21] included source code or a link to a source code repository (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=', GitHub).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Four different datasets were used, three of which do not appear in other algorithm categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Two studies discussed the types of port scans present in training and evaluation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' SaiKiran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [25] men- tioned port sweep but did not specify further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Bertoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [21] conducted training and evaluating against the full 4 Machine Learning and Port Scans: A Systematic Review A PREPRINT spectrum of port scan types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Further, the authors included port scan data from five different port scan tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Table 2: Random Forest Algorithm Performance Authors Accuracy (F1) Dataset Algaolahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='75 CICIDS2017 Baah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='98 CICIDS2017 Sirisha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='09 NSLKDD Sirisha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='14 CICIDS2017 SaiKiran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='93 CICIDS2017 Mohseni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='94 CICIDS2017 Bertoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='00 MAWILab Bertoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='00 Bonafide 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='2 Support Vector Machine (SVM) Support Vector Machine (SVM) is good for classification and regression problems, especially in cases where the data has clear boundaries and is not noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' It works well for datasets with a limited number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' SVM is different from Random Forest in that it uses a boundary (a hyperplane) to separate the data into classes, whereas Random Forest creates multiple decision trees and aggregates their predictions to make a final decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' SVM is best suited for cases where the boundary between classes is well defined and clear, whereas Random Forest is better suited for complex, non-linear decision bound- aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Four studies experimented with SVM [20, 21, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' All four also had explored RF performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Results for the SVM experiments ranged from 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='61% to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='87% both coming from the same dataset (of two total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Source code availability and port scan details remained the same as in- dicated in the RF algorithm category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Table 3: Support Vector Machine Algorithm Performance Authors Accuracy (F1) Dataset Algaolahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='61 CICIDS2017 Baah et al 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='87 CICIDS2017 SaiKiran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='29 CICIDS2017 Bertoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='00 Bonafide 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='3 Regression Regression algorithms are used for predicting a continu- ous target variable based on one or more input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' They are commonly used for tasks such as predictions and forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Regression algorithms, including linear regression, are different from Random Forest and SVM algorithms in that they focus on establishing a linear or non-linear relation- ship between the input features and the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' On the other hand, Random Forest and SVM algorithms are mainly used for classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' In a regression problem, the aim is to predict a numerical output, whereas in classification the output is categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' SVM can also be used for regression problems by using a specific formulation called Support Vector Regression (SVR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' However, the emphasis and method used in re- gression algorithms are different compared to SVM and Random Forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Four different datasets were used by three studies [22, 21, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The results span 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='21% to 94% accuracy (F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Only one study [21] discussed port scanning in detail and in- cluded source code for the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Table 4: Regression Algorithm Performance Authors Accuracy (F1) Dataset Tojeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='00 CICIDS2017 Sirisha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='05 NSLKDD Sirisha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='21 CICIDS2017 Bertoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='00 MAWILab Bertoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='00 Bonafide 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='4 Naive Bayes Naive Bayes is a probabilistic algorithm that is good for classification problems, especially when the assumption of independence between features holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' It is fast and sim- ple to implement and can handle large datasets well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Naive Bayes is different from regression, SVM, and Ran- dom Forest algorithms in that it makes a probabilistic pre- diction based on Bayes’ theorem and the assumption of independence between features, whereas the other algo- rithms make predictions based on a boundary or a com- bination of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' In comparison, regression, SVM, and Random Forest algorithms work well for more complex problems where the relationship between features is not necessarily independent and the decision boundary is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Bakaletz [27] used a custom generated dataset for train- ing and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The author used two nmap scan techniques- aggressive (NMAP-A) and stealth (NMAP- S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' There were five additional datasets used in three [19, 22, 21] out of the remaining four studies [24] in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Training and evaluation of NB algorithms demonstrated performances in the range of 55% to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Table 5: Naive Bayes Algorithm Performance Authors Accuracy (F1) Dataset Al-Haija et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='70 PSA-2017 Baah et al 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='02 CICIDS2017 Bakaletz 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='00 NMAP-A Bakaletz 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='00 NMAP-S Sirisha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='47 NSLKDD Sirisha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='92 CICIDS2017 Bertoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='00 MAWILab Bertoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='00 Bonafide 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='5 Neural networks Neural networks (NNs) are good for a wide range of tasks including classification, speech recognition, and natural 5 Machine Learning and Port Scans: A Systematic Review A PREPRINT language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' They are particularly good for com- plex and non-linear problems, such as recognizing pat- terns where traditional algorithms struggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Neural networks are different from other machine learn- ing algorithms in that they are based on a modeled struc- ture inspired by the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' They consist of multiple interconnected nodes, or artificial neurons, that process information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' These neurons are organized into layers and the connections between them are associated with weights that are learned during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' In comparison, algorithms such as RF, SVM, NB, and regression algorithms make predictions based on a clear boundary or a combination of trees or linear relationships, whereas NNs are capable of learning complex relation- ships between the inputs and outputs through their inter- nal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' NN algorithms have the ability to learn and make predictions based on the examples they are trained on, which makes them highly flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' However, they can also be more difficult to interpret and train, and require a large amount of data to achieve good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Furthermore, we differentiate between two types of NN: Artificial Neural Networks (ANNs) and Convolutional Neural Networks (CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Both types of deep learning algorithms used for various tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The main difference be- tween ANNs and CNNs is the structure and the way they process information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' ANNs are fully connected networks, where each neuron in one layer is connected to all neurons in the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' This structure makes ANNs computa- tionally expensive and require a large amount of data to achieve meaningful results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' On the other hand, CNNs are specifically designed for classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' They have a unique structure, con- sisting of convolutional layers, activation layers, pooling layers, and fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Convolutional layers are used to extract features, activation layers apply a non- linear activation function to the output of the convolu- tional layer, pooling layers reduce the spatial dimensions of the output, and fully connected layers make the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' This unique structure makes CNNs efficient and effective for classification tasks, as it allows them to learn hierarchical representations of the data and identify different objects and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' In comparison, ANNs are not specifically designed for classification and may not achieve the same level of performance as CNNs for these types of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We did break our own time bounding constraint in this category because Aamir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [7] specifically noted neu- ral networks were not being investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Hartpence et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [28] published their work in 2020, therefore we felt obliged to include the work here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' On that note, three of the four studies in this algorith- mic category used non-standard datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Hartpence et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [28] provided immense detail in what port scan types exist in the datasets the authors generated as part of their experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The general (GEN) dataset contained typ- ical network traffic with port scans intermingled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The second dataset contained port scans only (TCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Lv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [29] generated a custom dataset by capturing network traf- fic while executing nmap full connect scans (NMAP-F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Kirtas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [30] executed nmap SYN scans (NMAP-Y) while capturing network traffic for the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' No study included code snippets or links to code repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Table 6: ANN Algorithm Performance Authors Accuracy (F1) Dataset Hartpence et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='99 GEN Hartpence et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='31 TCP SaiKiran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='11 CICIDS2017 Kirtas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='88 NMAP-Y Bertoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='00 Bonafide Table 7: CNN Algorithm Performance Authors Accuracy (F1) Dataset Lv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='00 NMAP-F SaiKiran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='52 CICIDS2017 Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='94 CICIDS2017 Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='73 CICIDS2017 5 Conclusion We posed four research questions to guide this systematic review of port scan detection literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We discovered five algorithms present in the literature: Random Forest, Support Vector Machine, Regression, Naive Bayes, and Neural Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The literature revealed multiple ways to accurately detect port scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Bertoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [21] con- firmed RF and ANN algorithms are capable of 100% accu- racy against a plethora of port scan types and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Sirisha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [22] showed the poorest accuracy, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='92%, with the authors evaluation of the NB algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' ANN seemed to be the strongest type of algorithm across all of the sample papers, followed by RF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' There were 11 dif- ferent datasets used in 34 algorithm experiments with the CICIDS2017 dataset being used in 47% of those exper- iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' These results originated from 14 research stud- ies published since 2021 and a single study coming from 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Unfortunately, we were unable to adequately ad- dress the fourth research question (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=', What port scan- ning types and techniques were used for evaluation of those algorithms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=') as only a few studies discussed port scanning in any detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Overall, we found a variety of existing work included port scanning as a minor point compared to focuses areas such as general intrusion detection, DDoS, malware, botnets, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' We assume any work demonstrating a ca- pability to detect port scanning mentioned such explicitly and thus would be discoverable in our search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Another as- sumption underlying this work is research indexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' More specifically, we assume existing work has been indexed and thus was discoverable given our search strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' A final assumption present throughout the literature seems to be the stability of port scanning types and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The dominance of the CICIDS2017 dataset in training and 6 Machine Learning and Port Scans: A Systematic Review A PREPRINT evaluation supports this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' This assumption will con- tinue to be reasonable as long as significant innovations do not occur in the port scanning research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' It is interesting to note many existing papers experiment with more than one machine learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The di- versity of results within an algorithm category, across the sample papers in the category, is curious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Perhaps this would not be so unexpected if every paper used a different dataset for training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Yet, much of the exist- ing work leverages the CICIDS2017 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' As well, The distinct shift to NN algorithms in 2022 is notable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The prominence of NN over the past year suggests NN and its variations represent a viable research pathway going forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' As a final note, we found the vast majority of studies do not include code snippets or links to GitHub repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Accordingly, replication and reproduction to include spe- cific port scanning techniques with packet captures and al- gorithm source code would be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' This would be significant because doing so would fill in an existing gap and also enable adjacent research in cybersecurity such as offensive cyber, network security, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' At the same time, such future work might look to incorpo- rate green compute measurements given the increased fo- cus on sustainability in algorithm research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Foundational compute resource metrics can be taken from Bertoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [21] who detailed the compute resource profiles associ- ated with training and evaluating the ML algorithms in their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' A final area for future exploration is nature inspired or artificial life (Alife) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' Greensmith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' [31] showed how a dendritic cell algorithm can be used to de- tect port scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFRT4oBgHgl3EQfgzeb/content/2301.13581v1.pdf'} +page_content=' The authors show a theoretical model with pseudocode examples.' metadata={'source': 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